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Gravity
Terms in this set (577)
Harmonic motion (also called vibrational motion)
An object is in harmonic motion if it follows a repeated path at regular time intervals (ex:spring and pendulum)
As an object vibrates it has both
Period and frequency
Period (T)
Time it takes to complete 1 revolution
Frequency (f)
Number of revolutions the ball makes per unit of time (hertz). Cycles per unit of time
The period and frequency of a pendulum depend on only
The length of the pendulum and acceleration due to gravity
Equilibrium position
Lowest point in the swing of a pendulum
Amplitude
Maximum displacement from equilibrium. Can be measured by the maximum angle or by linear horizontal distance from equilibrium.
Restoring force
Force that is trying to restore the pendulum back to the center if the swing. Greatest at the amplitude and 0 as the pendulum passes through the equilibrium position
Where are acceleration and restoring force at their max in a pendulum swing?
Acceleration and restoring force are at their maximum at the amplitude
Where is velocity at its max in a pendulum swing?
The velocity is 0 at the amplitude and at its maximum as it passes through the equilibrium position
Energy in pendulum
At max amplitude the pendulum mass has max potential energy and 0 kinetic. As it passes through the equilibrium position, all of the potential energy has been converted to kinetic energy
Period of Oscillation of a Pendulum formula
2π√L/g (g is gravity and L is length of string)
The period and frequency of a mass vibrating on a spring depend on
The stiffness of the spring. The stiffer the spring, it takes more force to stretch the spring to a particular length
spring constant (k)
Amount of force needed per unit length. The stiffer the spring the greater value of k. Measured in newtons per meter.
Hooke's law
The force is proportional to the distance the spring is stretched
Hooke's spring formula
F_spring= -kx; k is the constant of the linear spring, x is the stretched length of the spring. The negative sign in Hooke's law tells us that F and x always point in opposite directions
How to find the constant in spring formula
Take the ratio of the force to the stretch for a particular interval (slope of F vs x graph)
As an object vibrates in harmonic motion, energy is transferred between
potential and kinetic energy
Consider a mass sitting on a surface of negligible friction and attached to a linear spring. If we stretch a spring from its equilibrium (unstretched) position to a certain displacement we do work on the mass against the spring force. By the work energy theorem the work done is equal to the store potential energy in the spring
If we release the mass and allow it to begin moving back toward equilibrium the PE changes into KE. As the mass passes through equilibrium all of the PE has been converted to KE and the speed of the mass is at its max. KE begins changing back to PE until all of the KE is converted into PE at max compression. The compressed spring then accelerates the mass back through equilibrium and total energy remains constant (see 141)
The period of oscillation of a simple harmonic oscillator described by Hooke's Law is T= and what it tells us
T=2π√m/k; This equation tells us that as the mass of the block, m, increases and the spring constant, k, decreases, the period increases. In other words a heavy mass attached to an easily stretched spring will oscillate back and forth very slowly, while a light mass attached to a resistant spring will oscillate back and forth very quickly.
Frequency is inversely proportional to period:
f=1/T
Whenever potential energy is lost in a spring
kinetic energy is gained and vice versa
When is velocity the greatest in a spring
When passing through equilibrium
When is PE greatest in a spring?
At the most compressed and stretched points (when KE is 0)
The potential energy of a spring (U_s) is sometimes called elastic energy, because it results from the spring being stretched or compressed. Mathematically, is defined by:
U_s= (1/2)kx^2 (unlikely that this will be on SAT)
If the amplitude is increased and the pendulum is released from a greater angle what will happen to the period and frequency?
They will not change. The period (and thus the frequency is not affected by a change in amplitude only a change in length or gravity
Where is acceleration greatest in a spring and where is it 0?
At the same point where the restoring force is the greatest at the amplitude (compressed area). It is 0 at equilibrium since it's going from positive to negative
Where is PE equal to 0 in spring?
At equilibrium
Gravitational force is proportional to the product of the masses and inversely proportional to
the square of the distance between their centers
Gravitational force equation
F_g= (Gm1m2)/r^2; F_g is the gravitational force, m1 and m2 are the masses in kg and r is the distance between their centers.
G constant
6.67 x 10^-11 Nm^2/ kg^2
Because gravitational force is inversely proportional to the square of the distance from its sources it is called an inverse square law
...
Gravitational acceleration is proportional to gravitational force and
inversely proportional to the square of the distance between the centers of the 2 masses
satellites travel in an elliptical orbit and are faster when
The object is closer to earth than farther. This fits in with the angular momentum is conserved theorem (see pg 149 for pic). Also, as the satellite orbits, PE and KE change but total energy is conserved. Angular momentum is also conserved
The acceleration due to gravity does not depend on the mass of orbiting (falling) object, but only on the mass of the Earth and the distance between the masses
...
Charge and symbol and SI unit
The fundamental quantity that underlies all electrical phenomena. The symbol for charge is q and the SI unit for charge is the Coulomb (C)
Carrier of negative charge and its charge
The electron with a charge of -1.6 x 10^-19
Carrier of positive charge and its charge
The proton with a charge of 1.6 x 10^-19
If an object has a surplus of electrons it is negatively charged and if an object has a deficiency of electrons it is positively charged
...
Charge is conserved during any process and so
any charge lost by one object must be gained by another object
Law of Charges
Like charges repel each other and unlike charges attract each other. This law is fundamental to understanding all electrical phenomena
Two charged spheres of equal size carry a charge of +6 C and -4 C. The spheres are brought in contact w/one another until reaching an equilibrium charge. What is the final charge of each sphere?
One of them has 4 extra electrons. If it gives it away to the +6 one, it will have a charge of 0 and the +6 one will have a charge of +2. When they reach equilibrium, the 0 charged one will give away an electron making both of the spheres have a charge of +1
Coulomb's Law
electric force is proportional to the magnitude of the charges and inversely proportional to the square of the distance between the charges
Coulomb's Law equation
F_electric= (Kq1q2)/r^2; F_e is the electric force, q1 and q2 are the charges, r is the distance between their centers and K is a constant. This is kind of like the gravitational force equation (F_e= Eq is another equation)
K constant number
9 x 10^9 Nm^2/C^2
electroscope
device that consists of a metal ball or plate connected to a metal rod with 2 thin metal leaves attached at the bottom. The rod and leaves are insulated so as not to pick up any extra charges from the air. Neutral objects still have charges; they just have an equal number of positive and negative charges. We can use other charged objects to redistribute the charges in a neutral object without actually changing the amount of the charge
use electroscope to study
How charges in the ball, rod and leaves separate from one another when we bring another charged object near the electroscope or touch the ball of the electroscope with the charged object. Used to study how charges distribute themselves
REVIEW PG 158 FOR ELECTROSCOPE PROCESS
...
Conduction
Transfer of charge by actual contact. Transferring charge from 1 object to another
difference between conductors and insulators
Conductors, like metals have electrons that are loosely bound to the outskirts of their atoms and can easily move from 1 atom to another. An insulator, like wood or glass, does not have many loosely bound electrons and therefore cannot pass charge easily
electric field(SEE pg 159 for pics)
The force per unit charge in a region of space. Condition of space around a charge (or distribution of charges) in which another charge will feel a force. Electric field lines always point in the direction in which a positive charge would feel a force
The magnitude of the electric field is given by the equation:
E= F/q; E is the electric field measured in Newtons per Coulomb, F is the force acting on the charge q, which is feeling the force in the electric field. The test charge q would feel a force radially outward anywhere around the source charge Q so we would draw the electric field lines around the positive Q charge
Electric field lines are always drawn in the direction
in which a positive charge will feel a force
Electrons (negative charges) are moved when charge is transferred, but electric field lines are drawn in the direction a positive charge would move (see pics on 159 and 160)
...
electric potential (V)
defined in terms of the work we would have to do on a charge to move it against an electric field
difference between electric field and electric potential
Electric field is the force per unit charge and electric potential is the work per unit charge
We say that here is a potential difference ∆V between points A and B, and the equation for potential difference between 2 points is
∆V= Work/ q and is measured in joules/coulomb or volts. When we apply potential difference to circuits we call it voltage
current
We say that positive charges naturally want to move from a point of high potential to low potential and we refer to the movement of the positive charges as current
Creating a Uniform Electric Field
We can create a UEF in a region of space by taking 2 metal plates, setting them parallel to each other and separating them by a distance d and placing a voltage V (as from a battery) across the plates so that 1 of the plates will be positive and the other negative. The + charges on the top will line up uniformly with - charge directly across from it which would cause an electric field lines and are uniformly spaced to produce a UEF everywhere between the plates
capacitors
Conducting plates that are connected in this way. Used to store charge and electric field in a circuit that can be used at a later time
The electric field, voltage and distance between the plates are related by the equation
E=V/d (volts per meter is equivalent to newtons/coulomb)
conventional current (current)
One end of a battery is positive and the other end is negative. When we connect the wires and light bulb to the battery in a complete circuit, charge begins to flow from one end of the battery through the wires and the bulb to the other end of the battery causing the bulb to light. We say the movement of + charge from the + end of the battery through the circuit and the - end of the battery is called a current. Current is the amount of charge moving through a conductor per second
current unit of measurement
coulomb/second or ampere. We use the symbol / for current
Technically what is moving, positive or negative charge through a circuit?
Negative (although it's conventional to speak of + charge flowing rather than - charge flowing). When the battery is connected to the circuit, an electric field is set up such that the - charges experience a force that push them in 1 direction through the wire. At the same time, positive charges experience a force in the opposite direction, but since the electrons that comprise the negative charge in matter are much more mobile than protons, electrons move through the circuit
resistance
As charge moves through the circuit, it encounters resistance, or opposition to the flow of current and is measured in ohms (Ω). Resistance is the electrical equivalent of friction. (ex:light bulb, wire)
Ohm's law (REVIEW PG 168 TABLE 12.1)
R= V/I; R is resistance, V is volts and I is Current. When a larger voltage is used, more charge moves through the wire and thus more current flows through the circuit. Current is directly proportional to the voltage. This implies that the ratio of voltage to current is a constant
ammeter
Measures current. Have low resistance and are always placed in series with a resistor. Have low resistance so as not to add to the total resistance of the circuit (and thus decrease the current in the circuit)
voltmeter
Used to measure voltage. Have high resistance so that the current will not want to flow through them and bypass the resistor
If we wanted to measure the current through the resistor and the voltage across the resistor we would
Connect an ammeter in series with the resistor and voltmeter in parallel with the resistor. We place the ammeter in series with the resistor so that the same current will pass through the ammeter and resistor and we place the voltmeter in parallel with the resistor so that the voltage will be the same across the voltmeter and resistor.
Power in a circuit
Energy used per unit time, measured in watts
Joule heating
When current flows through a resistor, heat is produced and the amount of heat produced in joules per second is equal to the power in the resistor. The heating in the resistor is called joule heating, and the power dissipated in the resistor follows Joule's law of heating
The equation that relates power to the current, voltage and resistance in a circuit. P=
P=IV, P=I^2R, P= V^2/R
The unit for power
Joule per second or Watt
When 2 or more resistors are placed in circuit, there are basically 3 ways to connect them
in series, in parallel, a combination of series and parallel
Series circuits
2 or more resistors of any value placed in a circuit in such a way that the same current passes through each of them
If there are 3 resistors, add each of their resistances up to get the total resistance (in series circuit)
R_total= R1+ R2+ R3+...Rn
Total current (only in series circuit)
I_total= I1= I2= I3; Current in all 3 places is the same
The voltage divides proportionally among the resistances according to Ohm's law (REVIEW PG 171) (only series circuit)
V1= IR1; V2= IR2; V3= IR3
I_total= (only series circuit)
I_total= (V_total)/ (R_total)
Parallel Circuits
2 or more resistors of any value placed in a circuit in such a way that each resistor has the same potential difference across
difference between series circuit and parallel circuit
In a series circuit, the resistors get the same current. In a parallel circuit, the resistors get the same voltage
emf (looks like backwards 3) is just an older term for voltage
...
The total resistance in a parallel circuit is given by the following equation:
1/R_total= (1/R1)+ (1/R2)+ (1/R3). Once you have the answer inverse it to get the total resistance
In a parallel circuit the voltage across each resistance is the same
V_total= V1=V2=V3
In a parallel circuit to find the current the equation is
I1=V1/R1; I2= V2/R2 ect where R's are equal to each other
Review page 173 and 174 for combination circuit
You first follow parallel rules then series rules
capacitance
charge per unit voltage and depends only on the geometry of the plates
capacitance of conducting parallel plates equation and unit
C= q/V; q is the charge on 1 of the plates, and V is the voltage across the plates. The unit for capacitance is the coulomb/volt or farad
capacitance of a capacitor is
proportional to the area of each plate and inversely proportional to the distance between the plates. In symbols it is C=A/d
capacitor purpose
The purpose of a capacitor is to store charge and electric field in a circuit that can be used at a later time
Resistance Capacitance Circuit
Circuit containing a battery, a resistor and capacitor in series with one another. Can store charge and release it later
In RC circuit the current is initially
V/R by Ohm's law but then decreases as time goes on until the capacitor is full of charge and will not allow any more charge to flow out of the battery
2 rules we can follow when dealing with capacitors in an RC circuit
1. An empty capacitor doesn't resist the flow of current and thus acts like a wire
2. A capacitor that is full of charge will not allow current to flow and thus acts like a broken wire
If we move the switch to position b (see pg 176) (with the battery taken out of the circuit)...
The capacitor begins to drain its charge through the resistor, creating a current in the opposite direction to the current flowing when the battery was connected. Eventually, the current will die out because of the heat energy lost through the resistor
In the beginning of a RC circuit the current is___ and at the end it's _____
V/R and at the end it's 0. The 0 current flow is only in the C area. It may be different in the resistor areas
lodestones
Pieces of iron ore found 2000 years ago in Greece that would attract or repel each other. When one of these magnetic stones was suspended from a string, one side of the stone would align itself to point in a notherly direction (labeled north)
magnetic field and unit
The space around a magnet in which another magnet will feel a force; measured in teslas (T). Have no beginning and no end but generally are drawn from the north pole of a magnet to its south pole
domains
clusters of atoms with similar magnetic orientations
A material become magnetic when
It is placed in a strong external magnetic field and the clusters of atoms with similar magnetic orientations called domains become aligned with the external magnetic field
Magnetic fields are produced by moving charges. That's why atoms
are all tiny magnets.
Magnetic Field Produced by a Current Carrying Wire
A current carrying wire is magnetic or creates a magnetic field around itself. The magnetic field produced by a current carrying wire circulates around the wire in a direction given by what we will refer to as the first magnetic right hand rule (the mechanics right hand rule is for torque for wheels)
First magnetic right hand rule
The magnetic field produced by a current carrying wire circulates around the wire in a direction given by this rule. Place right thumb in the direction of the current (I), and your fingers will curl around the direction of the magnetic field produced by that current. Currents are positive so this shows the proton flow. To show electron flow just make the arrows opposite
Second magnetic right hand rule
The direction of the force acting on the wire is given by this rule. The current carrying wire produces a magnetic field and if we place a current carrying wire in an external magnetic field it will experience a force. Place your finger in the direction of the magnetic field (north or south) with your thumb in the direction of the current in the wire and the magnetic force on the wire will come out of your palm
Use right hand for conventional current flow or moving positive charges and left hand for electron flow or moving negative charges
...
Equation for finding the force on a current carrying wire in a magnetic field
F= ILB sinΘ; I is the current in the wire, L is the length of the wire in the magnetic field, B is the magnetic field and Θ is the angle between the length of wire and magnetic field
The length must have a component that is perpendicular to the magnetic field or there will be no magnetic force on the wire. In other words
if the wire is placed parallel to the magnetic field (sin 0=0) and the force will also be 0
Force on a charged particle in a magnetic field direction
Since a moving charge creates a magnetic field around itself, it will also feel a force when it moves through a magnetic field. The direction of the force acting on such a charge is given by the second magnetic right hand rule, with the thumb pointing in the direction of the velocity of the charge
A charge must be moving across magnetic field lines to feel a force
...
The equation for finding the force on a charge moving through a magnetic field is
F=qvBsinΘ; q is the charge in Coulombs, v is the velocity in m/s, B is the magnetic field in teslas, and Θ is the angle between the velocity and the magnetic field
electromagnetic induction
generating a current by moving a magnet through a coil of wire or by moving a wire through a magnetic field
The amount of voltage and current produced in a coil of wire depends on
How quickly the magnetic field lines are crossed by wire. For example, if the magnet is moved through the coil slowly, hardly any current is produced in the coil. Also, a greater number of coils will produce a greater induced voltage and current
The direction of the current induced is dependent on
the direction in which the magnet or wire is moving
generator
Converts mechanical energy into electrical energy
How to create generator
Place a loop of wire on an axle in a magnetic field. As the loop is rotated the wire crosses magnetic field lines and generates a current in the loop. That current can be used to light a light bulb or power a city
What can increase the amount of voltage induced in a coil of wire? (5)
Move the magnet faster through the coil. Move a stronger magnet through the coils of wire. Move the magnet through more coils of wire. Move more coils of wire around a magnet. Move more magnets simultaneously through a coil of wire
When an electron or particle is moving towards a magnetic field the path of it is
It will orbit a magnetic field line but will also continue in the direction of the velocity
in a generator pulling a coil out in the opposite direction as it went into the coil, there will be a current produced in the coil in the opposite direction as before
...
center of mass formula
x_cm= (m1x1 + m2x2 +...+ mn xn) / (m1 + m2...mn) where the particles are located somewhere on x axis
center of mass when person's on a boat
If you've ever tried to walk from one end of a small boat to the other, you may have noticed that the boat moves backward as you move forward. That's because there are no external forces acting on the system, so the system as a whole experiences no net force. If we recall the equation , the center of mass of the system cannot move if there is no net force acting on the system. The fisherman can move, the boat can move, but the system as a whole must maintain the same center of mass. Thus, as the fisherman moves forward, the boat must move backward to compensate for his movement.
average angular velocity ω =
ω= ∆Θ/∆t
average angular acceleration α=
α= ∆ω/∆t
Angular frequency
f, is defined as the number of circular revolutions in a given time interval. It is commonly measured in units of Hertz (Hz), where 1 Hz = 1 s-1. For example, the second hand on a clock completes one revolution every 60 seconds and therefore has an angular frequency of 1 /60 Hz.
The relationship between frequency and angular velocity is given by the formula: f=
f= ω/2π
Angular period
T, is defined as the time required to complete one revolution
Angular period is related to frequency by the equation: T=
T=1/f
Period and angular velocity are related by the equation: T=
T=2π/ω; it's the inverse of the frequency-angular velocity formula
We can relate the angular position of P to the length of the arc of the circle between P and the x-axis by means of an easy equation: φ=
φ=l/r (l is arc length and r is the radius of the circle)
Similarly, for any angular displacement, Θ, we can say that the length, L, of the arc made by a particle undergoing that displacement is L=
L=Θr (L is arc length and r is radius); Note that the length of the arc gives us a particle's distance traveled rather than its displacement, since displacement is a vector quantity measuring only the straight-line distance between two points
We can express the instantaneous linear velocity of a rotating particle as v=
v=Lt (L is distance traveled around arc)
v= what in terms of angular velocity?
v=αr
Big 5 equations for rotational kinematics are
the same as mechanical except in terms of α, Θ, ω, and t
The direction of the vector for angular velocity (ω)
Actually perpendicular to the plane in which the object is rotating. This is determined using the mechanics right hand rule. See which way the object is spinning, wrap your fingers around it and the way your thumb points is the direction of ω
The direction of the vector for angular acceleration (α)
To find the direction of a rigid body's angular acceleration, you must first find the direction of the body's angular velocity. Then, if the magnitude of the angular velocity is increasing, the angular acceleration is in the same direction as the angular velocity vector. On the other hand, if the magnitude of the angular velocity is decreasing, then the angular acceleration points in the direction opposite the angular velocity vector
Just as the familiar version of Newton's Second Law tells us that the acceleration of a body is proportional to the force applied to it, the rotational version of Newton's Second Law tells us that the angular acceleration of a body is proportional to the torque applied to it
...
Torque =
T=FrsinΘ (where Θ is between F and r)
Of course, force is also proportional to mass, and there is also a rotational equivalent for mass: the moment of inertia, I, which represents
an object's resistance to being rotated.
Torque formula that is similar to newton's second law: T=
T=Iα (where I is the moment of inertia)
What might make a body more difficult to rotate? (moment of inertia)
First of all, it will be difficult to set in a spin if it has a great mass: spinning a coin is a lot easier than spinning a lead block. Second, experience shows that the distribution of a body's mass has a great effect on its potential for rotation. In general, a body will rotate more easily if its mass is concentrated near the axis of rotation
I= (moment of inertia) generally is
mr^2
kinetic energy of a rotating body formula KE=
KE= (1/2)Iω^2 (it's like the normal one except with I replacing m and ω replacing v)
An object, such as a pool ball, that is spinning as it travels through space, will have both rotational and translational kinetic energy: KE_total=
KE_total= (1/2)mv^2+ (1/2)Iω^2
ω in relation to V_tangential
V_t= ωr
a_centripetal= (in relation to ω and r)
a_c= ω^2r
angular momentum
L=Iω
angular momentum in relation to V_tangential
L=mvr
ΣW=∆KE which equals
ΣW=(1/2mVt^2) - (1/2mVo^2)
rolling without slipping conservation of energy equation includes:
(1/2)Iω^2, 1/2 mv^2 and mgh (Rot KE, KE and PE)
the moment of inertia is greater the farther the mass of a body is from its axis of rotation, we can maximize angular velocity by concentrating all the mass near the axis of rotation.
...
acceleration caused by gravity on a planet formula (a=)
a= (Gm_of planet)/r^2
Effect of mass on satellite's speed
A satellite's speed doesn't depend on its mass
Tangential velocity of object in orbit formula
V_T= √(G(M/r)) or V_T= 2πr/T (where T is the time it takes to complete 1 orbit)
The gravitational potential energy of two masses, M and m, separated by a distance r is:
PE_of gravity= (-GMm)/r
The work done getting the satellite from one place to another is equal to the change in the satellite's potential energy
...
KE of satellite formula
KE= (1/2)F_centripetal R (where R is the radius of the orbit) or KE=(GMm)/2R
the total energy of the satellite is the sum
E= (-GMm)/2R
Kepler's First Law
The path of each planet around the sun is an ellipse with the sun at one focus
Kepler's Second Law
Relates a planet's speed to its distance from the sun. The Second Law states that if a line is drawn from the sun to the orbiting planet, then the area swept out by this line in a given time interval is constant. This means that when the planet is farthest from the sun it moves much more slowly than when it is closest to the sun.
Kepler's Third Law
states that given the period, T, and semimajor axis, a, of a planet's elliptical orbit, the ratio T 2/a3 is the same for every planet. The semimajor axis is the longer one, along which the two foci are located.
During the oscillation the force on the block at equilibrium is
0 when the block is at equilibrium (x=0). Therefore the acceleration is also 0 (F=ma) (see pg 156 in PR)
PE of spring relative to its equilibrium position PE_spring=
PE_spring=(1/2)kx^2; notice that the farther you stretch or compress a spring, the more work you have to do
Energy in a spring is
Conserved. KE_spr+PE_spr (MAKE SURE YOU UNDERSTAND PG 156 AND 157 IN PR BOOK PICS BEFORE TAKING TEST)
Because F_spring varies as object moves the work done or against the spring
cannot be written as FdcosΘ. Just like gravity, we must calculate work in terms of potential energy
Work of spring if block moves from x1 to x2 formula
W_spr= (1/2)k (x2^2 - x1^2)
cycle in oscillation
round trip
T in oscillation
Amount of time it takes to complete cycle of oscillation. If T is short, the block is oscillating rapidly and if T is long the block is oscillating slowly
One of the defining properties of the spring-block oscillator is the frequency and period can be determined from the mass of the block and force constant of the spring. The equation are as follows: for f=
f= (1/2π) √(k/m)
One of the defining properties of the spring-block oscillator is the frequency and period can be determined from the mass of the block and force constant of the spring. The equation are as follows: for T=
T= 2π √(m/k); the T and f formulas are just inverses of each other
simple harmonic motion T and F relation to amplitude
In simple harmonic motion, both the frequency and period are independent of the amplitude
in simple harmonic motion, the position x can be expressed in equation:
x= Acos(2πft + Ø) or x= Asin (2πft+ Ø); Where A is the amplitude. Ø is called the phase and is determined by initial conditions
The max speed of the block in oscillation is given by the equation V=
V= A√(k/m)
If a spring is attached to the ceiling, it is being pulled down by gravity. The distance that it is being pulled when the mass is on is: (equation)
d= (mg)/k (where k is spring constant and d is distance from equilibrium without mass attached) (derived from equation kd=mg)
When the block is at a distance y below its equilibrium position, the spring is stretched a total distance of d+y, so the upward spring force is equal to (equation)
k(d+y); the downward force stays the same (mg)
The net force on the ceiling block (equation)
F=ky (derived from k(d+y) - mg, but since kd=mg, the mg's cancel out)
Restoring force formula in pendulum
F_restoring= mgsinΘ; restoring force in a pendulum is provided by gravity
Difference between simple harmonic motion in pendulum and spring-block oscillator
SHM results from restoring force that has a strength proportional to displacement in spring-block system. The magnitude of restoring force on a pendulum is mgsinΘ which is not proportional to the displacement Θ. Strictly speaking the motion of a simple pendulum is not really simple harmonic. However, it Θ is small, then sinΘ about=Θ so in this case the magnitude of the restoring force is approximately mgΘ which is proportional to Θ. This means if Θ_max is small the motion can be treated as simple harmonic
If the restoring force is given by mgΘ rather than mgsinΘ the frequency and period of the oscillations depend only on
th legth of the pendulum and gravitational acceleration
frequency of pendulum formula
f=(1/2π) √(g/L)
Period of pendulum formula
T= (2π) √(L/g); again the period and frequency formulas are inverses of each other
frequency and period (in pendulum) do not depend on
amplitude (max angular displacement) or mass of weight
Net charge can't be
created or destroyed
elementary charge (e)
Magnitude of charge on an electron (and therefore a proton) is denoted as e. It is the basic unit of electric charge. The charge of an ionized atom must be a whole number times e because charge can be added or subtracted only in lumps of size e. To remind us of quantized nature of electric charge, the charge of a particle is denoted by q
coulombs (C)
Charge is expressed as this.
1q= how many Coulombs
1.6 x 10^-19 C
Coulomb's law: The electric force between 2 particles which charges of q1 and q2 separated by distance r is given by the equation:
abs(F_e) = k abs (q1q2)/r^2 (similar to F_grav)
k in F_electric force and it's value
The value of the proportionality constant, k, depends on the material between the charge particles. In a vacuum of air the Coulomb's constant has the value k= 9x10^9 N x m^2/C^2
superposition
The fact that electric forces can be added (ex: F_on 2= F_1 on 2 + F_3 on 2
electric field
The presence of a charge creates an electric field in the space that surrounds it. Another charge placed in the electric field created by the first charge will experience a force due to the field
Consider a point charge Q in a fixed position and assume that it's positive. Now imagine moving a tiny positive test charge q around to various locations near Q. At each location, measure the force that the test charge experiences and call it F_on q. Divide this force by the test charge q; the resulting vector is the electric field vector E, at that location
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electric field vector formula
E= (F_e) / q
With electric fields the arrows are the directions a positive charge would move
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Since the force decreases as we get farther away from the charge (as 1/r^2) so does the electric field. This is why the electric field vectors farther from the source are
shorter than those that are closer
electric field formula for the strength of the electric field created by a point charge of magnitude Q. E=
E= k Q/r^2
electric dipole
2 equal but opposite charges form a pair called an electric dipole
conductors
Materials that permit the flow of excess charge and therefore conduct electricity
What are good conductors and why?
Metals conduct electricity because the structure of a typical metal consists of a lattice of nuclei and electrons with about 1 electron per atom not bound to it's nucleus. Electrons are free to move about the lattice creating a sort of sea of mobile electrons. This freedom allows excess charge to flow freely
Insulators and examples
Closely guard their electrons and even extra ones that might be added. Electrons are not free to roam throughout the atomic lattice. Examples of insulators are glass, wood, rubber, and plastic. If excess charge is placed on an insulator, it stays put
semiconductors and examples
Midway between conductors and insulators. They're less conducting than most metals but more conducting than most insulators. Examples of semiconducting materials are silicon and germanium
superconductor
Extreme example of conductor. Material that offers absolutely not resistance to the flow of charge; it is a perfect conductor of electric charge. Many metals and ceramics become superconducting when they are brought to extremely low temperatures
If a solid sphere is given a negative charge what is the electric field inside and outside the sphere?
The excess electrons that are deposited on the sphere move quickly to the outer surface. Any excess charge resides entirely on the outer surface. **There can be NO electric field within the body of a conductor
There can be no electric field within the body of a conductor. This is why you can
Shield yourself from electric fields simply by surrounding yourself with metal.
Electric field is always ______ to the surface, no matter what shape the surface may be
perpendicular
Electric field vectors point toward a negative source charge and the resulting electric force on an electron would point
in the opposite direction from the electric field vector
Electric potential energy
When a charge moves in an electric field unless its displacement is always perpendicular to the field, the electric force does work on the charge. If W_e is the work done by the electric force then the change in the charge's electrical potential energy is defined by ∆PE_e= -W_e
Gravitational potential energy
∆PE_G= -W_G
Other electric potential energy equation (when F_e and E are in same direction and F_e is constant)
∆PE_e= -qEr (W= Fd so Work done by electric field is equal to W_e= F_e r= qEr)
When there is positive work, electric potential energy is
negative (just like gravity: Gravity does positive work and the rock loses gravitational potential energy)
electric force naturally pushes negative charges
against field lines
electroscope (sparknotes)
When a negatively charged object is brought close to the metal bulb, the electrons in the bulb are repelled by the charge in the object and move down the rod to the two thin leaves. As a result, the bulb at the top takes on a positive charge and the two leaves take on a negative charge. The two metal leaves then push apart, as they are both negatively charged, and repel one another. http://www.sparknotes.com/testprep/books/sat2/physics/chapter13section1.rhtml
When electric field is linear or planar instead of a point charge the equations for electric field are
E= kq/r (no r^2) (for linear) and E=kq (for planar)
work done on a charge in electric field
W= qEd; d is displacement in the direction the force is exerted. This is equation is used when the particle doesn't go in a straight line. You just find the displacement (since ∆PE_e is -qEr, W= qEr)
When thinking about work and electric fields, keep these three rules in mind:
1. When the charge moves a distance r parallel to the electric field lines, the work done is qEr.
2.When the charge moves a distance r perpendicular to the electric field lines, no work is done.
3.When the charge moves a distance r at an angle Θ to the electric field lines, the work done is qEr cos Θ.
(q is positive)
Voltage (V)
the electric potential, or potential difference, V, between two points
Potential difference (Voltage) is the measure ____ measured in _______
of work per unit charge, and is measured in units of joules per coulomb, or volts (V).
One volt is equal
to one joule per coulomb
Voltage equation (voltage is electric potential on charge not field)
∆V= -W_e/q (work/charge)
Why does your hair stand on end when you touch an electric field, a charged metal ball for example?
Charge (either positive or negative) is brought to the woman by the metal ball. This charge then migrates to the ends of her hair. The repulsive force between like charges makes the hair separate and stand on end
PE_e of point particle 2nd equation
PE_e= (kq1q2)/r (keep in mind this is not ∆PE_e)
The change in electric potential ∆V=
∆V= (∆PE_e)/q
Units for electric potential energy
Joules per Coulomb or Volt
Consider the electric field that's created by a point source charge Q. The electric potential distance r from Q is V=
V= kQ/r
equipotential surfaces
area around electric field that create surfaces of constant potential. always perpendicular to electric field lines
Capacitor and its purpose
2 conductors separated by some distance that carry equal but opposite charges. Work must be done to create this separation of charge and as a result potential energy is stored. Storage devices for electrical potential energy. Can be used to store charge and electric field in a circuit that can be used at a later time
capacitance (C) equation and definition
C= Q/∆V; measure of the capacity for holding charge. The greater the capacitance the more charge can be stored on the plates at a given potential difference
The capacitor depends only on
the size, shape and separation of conductors
units for capacitance
1C/V= 1 farad
When a capacitor charges up, work must be done by an external force (ex: battery). This increases the potential energy stored by the capacitor. The potential energy stored is given by the formula:
PE= (1/2)Q∆V= (1/2) CV^2= (1/2) Q^2/C
parallel capacitors
A collection of capacitors are parallel if they all share the same potential difference (voltage) (do C1+C2... to find equivalent capacitance)
How to find equivalent capacitance of a collection of capacitors in parallel?
equivalent capacitance of a collection of capacitors in parallel is found by adding the individual capacitances
Series capacitors
A collection of capacitors are in series if they all share the same charge magnitude (current) (do 1/C1+ 1/C2.... to find equivalent capacitance)
dielectric
insulator between plates that maintains charge separation and stores potential energy. Always increases the capacitance of a capacitor.
How dielectric works
See pg 204 for clear explanation. The dielectric insulator switches the charges on the capacitor
The electric field decreases by κ when there is dielectric
E_with dielectric= E_without dielctric- Ei= E/ κ
Since ∆V= Ed for a parallel plate capacitor we see that ∆V must have decreased by a factor of κ. But C= Q/∆V, so if ∆V decreases by a factor of κ then C increases by a factor of κ so
C_ with dielectric = κC_without dielectric
κ is
the dielectric constant which varies from material to material but is always greater than 1
drift speed (V_d)
In a current electrons experience an electric force and would start to drift through the wire. Although the electric field would travel through the wire at nearly the speed of light the electrons themselves would still have to make their way through a crowd of atoms and other free electrons so their drift speed would be relatively slow
To measure the current we have to measure how much charge crosses a plane per unit time. If an amount of charge of magnitude ∆Q crosses an imaginary plane in a time interval ∆t, then the current is (equation)
I= ∆Q/∆t
unit for current
Because current is charge per unit time, it's expressed in coulombs per second. 1 coulomb per second is an ampere
current points in the direction
that positive charge carriers would move. So if the conduction electrons drift to the right we'd say the current points toward the left
Ω
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Resistance formula
R= ∆V/I; If the current is large, the resistance is low, and if the current is small, then resistance is high
Resistance unit
1 V/A (1 volt per amp) or ohm (Ω)
Resistivity depends on 2 things
Material and shape
circuit and what happens
An electric current is maintained when the terminals of a voltage source (a battery for ex) are connected by a conducting pathway in what's called a circuit. Remember currents are shown in positive directions so we say that a charge carriers drifts through the wire starting from the positive terminal of battery, enters wire, where it's pushed by electric field. It encounters other atoms (resistance) setting them into greater motion(heat). By the time the charge reaches the neg terminal, all of its original electrical potential energy is lost. To keep current going voltage source must do positive work on charge moving it from neg to pos terminal and cycle continues
direct circuit
If the current always travels in the same direction through the pathway it's called a direct current
electromotive force (emf)
The job of the voltage source is to provide a potential difference called emf which drives the flow of charge. The emf isn't really a force; it's the work done per unit charge and is measured in volts
Power of q in circuit basic equation
When a carrier of pos charge q drops by an amount V in potential, it loses potential energy in the amount qV. If this happens in time t, then the rate at which this energy is transformed is equal to P=qV/t
Power done on q in circuit 4 equations
P= qV/t; P=IV (since q/t is same as I); P=I^2R (sinceV=IR); P=V^2/r
Joule heat
When charge moves it bumps into stationary atoms that make up metal lattice and setting them into greater motion creating heat. Resistors become when current passes through them and the thermal energy generated is called joule heat
In a circuit resistor system, ε (emf) indicating battery power is the same as V so the current equation is thus
I= ε/R
ε is
emf
resistors
Since resistance of ordinary metal wire is negligible; resistance is provided by devices that control the current: resistors
What symbol on a circuit shows that there is a resistor?
zig zags
Batteries are denoted by the symbol of
A long line and short line. The long line represents the positive (higher potential) terminal and the shorter line is the negative (lower potential) terminal
Difference between placing resistors in series or parallel
In series is placing the resistors one after another. In parallel you place the resistors side by side
To simplify the circuit, our goal is to find the equivalent resistance of the resistors (This is similar to the capacitors)
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Resistors in series share the same current (I) but not the same
Voltage. The total voltage drop across them is equal to the sum of the individual voltage drops. This is the same concept in capacitors
Series resistors formula for equivalent resistance
R_eq= V/I= (V1+V2)/I= (V1/I)+ (V2/I)= R1+ R2. This idea can be applied to any number of resistors in series: R_eqseries= R1+R2 (As you can see this is the reciprocal of the capacitor formula since the capacitor formula is Q/V. The voltage is the denominator and is also divided so capacitors have their formulas as the reciprocals)
Resistors are said to be in parallel if they all share the same
Voltage drop and the total current (I) entering the combination is split among the resistors. Imagine that a current I enters the combination. It splits; some of the current I1 would go through R1 and the remainder I@ would go through R2 (see pic pg 217)
Parallel resistors formula for equivalent resistance
Since I= I1+I2--> (V/R_equiv)= (V/R1) + (V/R2)--> (1/R_equiv)= 1/R1 + 1/R2. This idea can be applied to any number of resistors in parallel (Also like capacitance except the opposite)
REDO PROBLEM ON PG 218. IT GIVES A GOOD OVERVIEW FOR THE UNDERSTANDING OF THE CHAPTER
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Resistance capacitance circuits
Capacitors are typically charged by batteries. (see pg 228 PR) electrons are attracted to the positive terminal of the battery and leave the top plate of the capacitor. Electrons also accumulate on the bottom plate of the capacitor and this continues until the voltage across capacitor plates matches emf of battery. When this condition is reached the current stops and the capacitor is fully charged
For a wire of length L and cross sectional area A made of a material with resistivity p resistance is given by the formula
R= (pL)/A
MAGNETIC force on a moving charge formula
F_B= abs(q) vBsinΘ; If a particle with charge q moves with velocity v through a magnetic field B, it will experience a magnetic force, F_B, with magnitude F_B= abs(q) vBsinΘ where Θ is the angle between v and B.
F_B= abs(q) vBsinΘ info on equation
If the charge is at rest (v=0) that means F_B is 0. Magnetic charges act only on MOVING charges. If v is parallel or antiparallel to B then F_B=0 since in either of these cases, sinΘ=0. Only charges that cut across the magnetic field lines will experience a magnetic force.
The direction of F_B (given by the right hand rule of magnetism) is always
perpendicular to both v and B and depends on the sign of the charge q
Right hand rule of magnetism
With you right hand (palm up) point your thumb in the direction of v and your fingers in the direction of B. If q is positive, F_B points out of the palm. If q is negative F_B points into the palm
Differences between electric force and magnetic force on a charge (2)
(1) a magnetic force acts on a charge only if the charge is moving; the electric force acts on a charge whether it moves or not. (2) The direction of the magnetic force is always perpendicular to the magnetic field while the electric force is always parallel or antiparallel to the electric field
SI units for magnetic field
Tesla (T) which is 1 N/A·m is the SI unit. Gauss (G) is also used but is not an SI unit
1 T equals how many G
1T=10,000 G
F_B is always perpendicular to v so
Since F_B is always perpendicular to v, the particle will undergo uniform circular motion; F_B will provide the centripetal force. Because F_B is always perpendicular to v, the magnitude of v will not change. Just the direction
magnetic forces cannot change _____ only
Magnetic forces cannot change the speed of an object only its direction
How much work does a magnetic field do on a charge
The magnetic field does no work on any charge
The radius of the particle's circular path is found from the equation F_B= F_centrip is
R= (mv) / (qB)
Since magnetic fields affect moving charges they should also affect current carrying wires. After all, a wire that contains a current contains charges that move. Let a wire of length L be immersed in magnetic field B. If the wire carries a current I, then the magnitude of the magnetic force it feels is (equation)
F= ILBsinΘ where Θ is the angle between L and B. Here the direction of L is the direction of the current I. The direction of F_B is given by the right hand rule as before remembering that the direction of the current is the direction that positive charges would flow
For a straight wire that carries a current I the current generates a magnetic field in the surrounding space that's proportional to I and inversely proportional to r, the distance from the wire:
B∝ I/r
The magnetic field lines are actually circles whose centers are on the wire. The direction of these circles is determined by the second magnetic right hand rule:
Imagine grabbing the wire in your right hand with your thumb pointing in the direction of the current. Then the direction in which your fingers curl around the wire gives the direction of the magnetic field lines
A charged particle can move through a magnetic field without feeling a magnetic force if:
It's velocity is parallel or antiparallel to the magnetic field lines
With the first magnetic right hand rule, it is in reference to a positive charge. If the charge is said to be negative it's just the opposite
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radius of a particle's motion in magnetic field due to centripetal magnetic force equation
r = (mv)/ (qB) (mv^2= qvB)
Period T=
2πr/v
Period in magnetic field
T= (2πm)/ (qB)
Review pg 252: When a conducting wire of length L, moves with a constant velocity (v) in the plane of the page through a uniform magnetic field B that's perpendicular to the page, the magnetic field exerts a downward force on the ELECTRON (not proton). As a result, electrons will be pushed to the lower end of the wire which will
leave an excess positive charge at its upper end. This separation of charge creates a uniform electric field E within the wire pointing downward.
The charge q in the wire feels 2 forces, an electric force (F_e= qE) and a magnetic force
F_B= qvB; The electron force and magnetic force act in opposite directions.
Once the magnitude of F_E equals the magnitude of F_B, the charges in the wire are in electromagnetic equilibrium. This occurs when
qE= qvB ; that is when E=vB
The presence of the electric field creates a potential difference between the ends of the rod. Since negative charge accumulates at the lower end, the upper end is at a higher electric potential. The potential difference V is equal to
V_across rod= EL and since E=vB, the potential difference can be written as V_across rod= vBL
Now imagine that the rod is sliding along a pair of conducting rails connected at the left by a stationary bar. The sliding rod now completes a rectangular circuit and the potential difference V_across rod cause current to flow. The motion of the sliding rod through the magnetic field creates an electromotive force, called
motional emf
motional emf formula
ε= vBL; The existence of a current in the sliding rod causes the magnetic field to exert a force on it. An external agent must provide this same amount of force to the right to maintain the rod's constant velocity and keep the current flowing
The power the external agent must supply is P=____ and the electric power delivered to the circuit is P=____
P=Fv=ILBv and the electric power delivered to the circuit is P=IV_across rod = Iε= IvBL. Notice that these 2 expressions are identical
The energy provided by the external agent is transformed first into electrical energy and then
into thermal energy as the conductors making up the circuit dissipate heat
Faraday's Law of electromagnetic induction
Electromotive force can be created by the motion of a conductor through a magnetic field but there's another way to create an emf from a magnetic field. According to this law the emf induced in a circuit is equal to the rate of change of the magnetic flux through the circuit
Faraday's Law of electromagnetic induction equation
ε= -(∆Φ_B)/∆t
Magnetic flux (ΦB) through an area A is equal to the product of A and the magnetic field perpendicular so the formula is:
Φ_B= BAcosΘ
Magnetic flux measures
the density of magnetic field lines that cross through an area (Note that the direction of A is taken to be perpendicular to the plane of the loop)
REVIEW PG 258 #6
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Induced emf can produce a current which will then create its own magnetic field
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The direction of the induced current is determined by the polarity of the induced emf and is given by Lenz's Law:
The induced current will always flow in the direction that opposes the change in the magnetic flux that produced it. If this were not so then the magnetic flux created by the induced current would magnify the change that produced it, and energy would not be conserved.
Note that it is common practice to refer to the direction of flux. Keep in mind that flux is
scalar and has no direction. Talking about flux having a direction makes applying Lenz's law easier but what we are really referring to is the direction of the field producing the flux
In a magnet, which side is the positive side?
The north pole
The magnetic field created by a long straight wire carrying a current, I
B= (M I)/ (2πr); M is called the permeability of free space; You don't need to memorize this equation. Just know that B is inversely prop to distance away from wire and I is proportional to B
When there's both a magnetic and electric field and they ask to calculate the force acting on the atom do
the electric force plus the magnetic force
M in a vacuum
4π x 10^7 N/A^2
Traveling waves (for rope or sound)
wave with peaks and valleys
At any point (x) along the wave, it has a certain and varying vertical displacement. It's this variation in the vertical displacement that defines the shape of the wave(see pg 268) This can be
positive (above dotted line) or negative (below dotted line)
For a traveling wave the displacement y of each point depends not only on x
but also on t
Because y depends on 2 independent variables wave analysis can be difficult. Instead of looking at a wave in which both variables (x and t) are changing, we'll allow only 1 of the variables to change. We'll use 2 points of view:
POV #1: x varies, t does not (see pg 268)
POV#2: t varies, x does not (see pg 269)
crests
Points at which the wave has its maximum vertical displacement above horizontal (highest point in wavelength)
trough
points at which the wave has its maximum vertical displacement below horizontal (lowest point in wavelength)
wavelength (λ)
distance between 2 adjacent crests (or 2 adjacent troughs) is the length of wave
amplitude (A)
max displacement from horizontal equilibrium position of the wave. (distance from the MIDDLE of the crest not form trough to crest)
POV#2 how to think of it as
think of putting 2 screens with a gap in front of the wave. the point will just oscillate up and down as time changes
propagate
traveling horizontal
transverse
when direction the wave oscillates is perpendicular to the direction that the wave travels horizontally
period of wave (T)
Time it takes for one complete vertical oscillation of a point on the rope is called the period (T)
frequency (f) of wave
the amount of cycles it completes in 1 sec
5 important characteristics of wave
wavelength, amplitude, period, frequency, velocity
Velocity of wave formula ***really important for test
v=λf (like the same as v = rate x time since λ=vT, v=λ/T so v=λf)
frequency, period, wavelength and wave speed relation to amplitude
frequency, period, wavelength and wave speed have nothing to do with amplitude
Don't be fooled by the important wave velocity formula. The speed of a wave does not depend on the frequency or wavelength. It depends on properties of the
medium (what wave travels through such as air water or other object)
Wave rule #1:
All waves of the same type in the same medium have the same speed. (If you move wave up and down at a certain frequency and then suddenly increase, that frequency, the speed remains unchanged. The wavelength therefore decreases since you are creating more pulses per second, each pulse won't have as much time to move down the rope before the next pulse is created. The pulses are closer together, meaning the wavelength has decreased
We can also derive an equation for the speed of a transverse wave on a stretched string or rope. Let the mass of the string be m and its length L; then its linear mass density (μ) is m/L. If the tension of the string is F_T then the speed of a traveling transverse wave on this string is given by:
v= √(F_T)/ μ (where μ=m/L)
Notice that v in wave or rope depends on the physical characteristics of rope:
its tension and linear density. So because v=λf for a given stretched string, varying f will create different waves that have different wavelengths but v will not vary
Wave Rule #2
When a wave passes into a new medium, its frequency stays the same
Superposition
When 2 or more waves meet, the displacement at any point of the medium is equal to the sum of the displacements due to individual waves
interfere
when waves meet and overlap
When waves interfere, the displacement of the string is equal to the
sum of the individual displacements, but after they pass, the wave pulses continue unchanged
constructive interference
If the 2 waves have displacements of the same sign (sign is means either crest or trough, positive or negative)when they overlap, the combined wave will have a displacement of greater magnitude than either individual wave
destructive interference
If the waves have opposite displacements when they meet, the combined waveform will have a displacement of smaller magnitude than either individual wave
in phase
If the waves travel in the same direction the amplitude of the combined wave depends on the relative phase of the 2 waves. If the waves are exactly in phase- that is if crest meets crest and trough meets trough- then the waves will constructively interfere completely and the amplitude of the combined wave will be the sum of individual amplitudes
out of phase
If the waves are exactly out of phase- that is if crest meets trough and trough meets crest- then they will destructively interfere completely and the amplitude of the combined wave will be the difference between the individual amplitudes.
standing waves
When our prototype traveling waves on a string strikes the wall, the wave will reflect and travel back toward us. The string now supports 2 traveling waves which have the same frequency, amplitude and wavelength. If the length of the string is just right the resulting pattern will oscillate vertically and remain fixed. This is a standing wave
nodes and amplitude (pg 277)
The interference of 2 traveling waves in standing waves results in complete destructive interference at some points. It has a zero amplitude. No nodes, no displacement
antinodes
have the greatest amplitude in standing wave, opposite of node
difference between nodes and antinodes
In a regular wave the amplitudes are the same. In a standing wave, each point on a string supporting a wave has an individual amplitude
Nodes and antinodes are always
alternate, they're equally spaced
distance between 2 successive nodes (or antinodes) is equal to
(1/2)λ
A standing wave can only form when the length of the string is a multiple of
(1/2)λ so L=n(1/2)λ
finding wavelength of standing wave using length of string formula and what they're called
λ_n= (2L)/n; these are harmonic (or resonant) wavelengths and n is known as the harmonic number
To figure out the frequencies that generate a standing wave formula and what they're called
f_n= nv/2L (derived from f=v/λ); These are the harmonic (or resonant) frequencies
fundamental standing wave
The first standing wave, the one for which the harmonic number n is 1
From the equation for the harmonic frequencies we see that the nth harmonic frequency is simply n times the fundamental frequency
f_n= nf_1; likewise the nth harmonic wavelength is equal to λ divided by n
sound waves
produced by vibration of object which cause pressure variations in conducting medium.
The variations in the conducting medium can be either
compressions or rarefactions
compressions
positions at which molecules of medium are bunched together (where pressure is above normal)
rarefactions
positions where the pressure is below normal
frequency for vibration to be picked up by human ears
20 Hz and 20,000 Hz
difference between sound waves and rope waves
The molecules of the medium transmitting a sound wave move parallel to the direction of wave propagation rather than perpendicular to it. For this reason sound waves sound waves are said to be LONGITUDINAL
It's very difficult to draw a pic of longitudinal waves. Instead graph the pressure as a function of position
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The speed of a sound wave depends on
The medium though which it travels. In particular it depends on the density (ρ) and bulk modulus (B)
bulk modulus (B)
Measure of medium's response to compression. A medium that is easily compressible like a gas has a low bulk modulus; liquids and solids which are much less easily compress have significantly greater bulk modulus values. For this reason, sound generally travels faster through solids that through liquids and faster through liquids than through gases
This equation that gives a sound wave's speed in terms of ρ and B is
v= √B/ρ; The speed of sound through air can also be written in terms of air's mean pressure which depends on its temperature.
At room temp (20 degrees C) and normal atmospheric pressure, sound travels at
343m/s; This value increases as air warms or pressure increases
Wave speed depends on characteristics of medium not frequency
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intensity
How loud we perceive a sound depends on both frequency and amplitude. Given a fixed frequency we can say loudness is measured by intensity
intensity formula
I= P/A; Where P is the power produced by the source and A is the area over which the power is spread
Relation of intensity to distance
At a distance the A is (π r^2). Therefore, I∝1/r^2. If listener doubles distance to source the sound will be heard one-fourth as a loud
decibel level
Alternate way of measuring loudness (sometimes called relative intensity);
Decibel level formula (β)
β= 10log (I/ I_o); I_o is the threshold of hearing and is equal to 10^-12 watts. Note that while β is measured in decibels (dB), it is dimensionless
**If sound increases by 10 dB, the intensity
increases by a factor of 10
Beats
If 2 sound waves whose frequencies are close but not identical interfere, the resulting sound modulates in amplitude, becoming loud, then soft, then loud, then soft. This is due to the fact that as individual waves travel they are in phase then out of phase then in phase and so on. The waves interfere constructively, then destructively then constructively, with the amp increasing, decreasing and then increasing. Each time the waves interfere constructively producing an increase in sound level, we say that a beat has occurred
The number of beats per second, known as beat frequency, is equal to the difference between the frequencies of the 2 combining sound waves with the formula
f_beat= abs(f1- f2); If the frequencies f1 and f2 match, the combined waveform doesn't waver in amplitude and no beats are heard. For example pianos are tunes using this. A key is struck and the corresponding tuning fork is struck; If the piano is in tune there should be no beats as the 2 sounds interfere
standing waves for sound
pretty similar to rope standing waves, the waves reflect off the far end and the superposition of the forward and reflected waves can produce a standing wave pattern if the length of the tube and frequency of the waves are related in a certain way
Nodes and antinodes in standing sound wave
Since air molecules at the end of the tube can't oscillate horizontally because they're up against a wall, the nodes are located there. The farther end of the tube is where the antinode is
Difference in rope standing wavelength and sound standing wavelength
In a rope the length between the node and antinode is (1/2)λ but in a sound wave the length between the node and antinode is (1/4)λ
Since L=λ/4 ----> λ1= 4L --->
f1= (v/λ1)= v/(4L)
Standing sound waves can be established in a tube that's closed at 1 end if the tube's length is equal to an ODD multiple of (1/4)λ. The resonant wavelengths and frequencies for a tube with 1 closed end are given by the equations:
λ_n= (4L)/n and f_n= (nv)/(4L)
If the far end of a standing wave is not sealed, the harmonic number does not have to be odd:
It can be any integer: λ= (2L)/n and f_n= (nv)/(2L)
Notice that while an open ended tube can support any harmonic, a closed end tube can only support odd harmonics
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Doppler effect
The shift in frequency that occurs when the source an detector are in relative motion.
If detector moves toward the source then the detector intercepts the waves at
a higher rate than the one at which they were emitted; the detector hears a higher frequency than the source emitted.
If the source moves toward the detector the wavefronts pile up and this results in detector
receiving waves with shorter wavelengths and higher frequencies (see pg 289).
Note that when the detector moves and not the source there is no change in wavelength. Instead
Instead there is a change to the speed with which the detector receives wavefronts
In general relative motion ____ each other results in a frequency shift upward and relative motion ____ from each other results in a frequency shift downward
In general relative motion toward each other results in a frequency shift upward and relative motion away from each other results in a frequency shift downward
To determine the frequency the detector (f_D) and frequency of the source (f_s) use the equation
f_D= (v±v_D)/ (v∓v_s) · f_s ; where v is the speed of sound, v_D is the speed of the detector and v_s is the speed of the source. The signs in the numerator depend on the directions in which the source and detector are moving: Use the higher sign in the numerator or denominator for motion towards and the lower for motion away
Ex: if a car is moving towards a moving police car what would be the signs be?
+ in the numerator since the detector is moving towards the police. + in the denominator since the police is moving away from the car. If the detector and source were moving in the same direction, the numerator and denominator cancel out and there is no Doppler effect since f_D=f_s
Doppler effect for light
Same concepts as sound. Motion toward corresponds to frequency shift upward and motion away corresponds to frequency shift downward. The difference is that for light the wavelength will change regardless of who moves
Through a vacuum all electromagnetic waves travel at a fixed speed c=
3 x 10^8 m/s regardless of their frequency
Electromagnetic (EM) spectrum
EM waves can be categorized by their frequency (or wavelength); the full range of waves is called the EM spectrum
Types of EM waves include (7)
radiowaves, microwaves, infrared, visible light, ultraviolet, X-rays and gamma (γ) rays
visible light order of spectrum
red orange yellow green blue violet
equation showing relation between c, λ and f
λf=c
coherent (define in wave terms)
the wave's phase difference remains constant over time and does not vary
If waves that have the same wavelengths meet then the difference in the distances they've traveled determine whether they are in phase. Assuming the waves are coherent if the difference in their path lengths, ∆L, is a whole number of wavelengths (0, ±λ, ±2λ ect) they'll
arrive in phase at the meeting point. However, if this difference is a whole number plus 1 half a wavelength (± .5λ, ±(1+.5)λ, ect) then they'll arrive exactly out of phase
constructive interference: ∆L=
∆L=mλ (where m is an integer
destructive interference: ∆L=
∆L= (m+ .5)λ
diffraction (303)
When a wave encounters a slit with a width that's comparable with its wavelength, the wave will fan out after it passes through. This is called diffraction. In this step, the waves will diffract through the slits and spread out and interfere as they travel toward the screen
fringes
When the light goes through the 2 slits, it will come up on the screen as bright bands called fringes. They are centered at those points at which the waves interfere constructively, alternating with dark fringes where the waves interfere destructively
To locate the positions of the bright fringes on the screen use the equation:
y_m= (mλL)/d ; Where y measures the vertical displacement along the screen from the center of the screen (y=0, the point directly across from the midpoint of the slits)
central maximum
The bright fringe directly opposite the midpoint of the slits (central max) will have the greatest intensity, the bright fringes with m= ±1 will have a lower intensity, those with m= ±2 will be fainter still and so on
If more than 2 slits are cut in the barrier, the interference pattern becomes sharper and the distinction between dark and bright fringes become more pronounced
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diffraction gratings
Barriers containing thousands of tiny slits per centimeter (diffraction gratings) are used to show distinction between dark and light fringes
Single aperture diffraction
A diffraction pattern will also form on the screen if the barrier contains only 1 slit. The central maximum will be very pronounced, but lower intensity maxima will also be seen because of interference from waves arriving from different locations within the slit itself. The central max will become wider as the width of slit is decreased
Reflection and Refraction of light: If a beam of light is directed toward a smooth transparent surface, some of its energy will be reflected off the surface and some will be transmitted into the new medium. We can figure out the directions of the reflected and transmitted beams by
calculating the angles that the beams make with the normal to the interface. The normal is a line perpendicular to the interface (pg306)
incident beam and angle of incidence
original beam of light and the angle of incidence is the angle the beam hits away from the normal and the angle looks like θ1
reflected beam
beam that creates the angle of reflection (θ1') which is an angle away from the normal
transmitted beam
beam absorbed by medium that creates the angle of refraction (θ2).
Law of reflection
The relation between θ1 and θ1': θ1=θ1'
To describe how θ1 and θ2 are related we first need to talk about a medium's index of refraction equation:
When light travels through a vacuum (empty space), its speed is c=3 x 10^8 m/s; but when it travels through a medium it's constantly being absorbed and re-emitted by the atoms of the material and as a result its apparent speed v is some fraction of c. The reciprocal of this fraction is: n=(c/v) is called the medium's index of refraction
n has no units and it's also never less than 1
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The equation that relates θ1 and θ2 involves the index of refraction of the incident medium (n1) and the index of refraction of the refracting medium (n2); it's called the SNELLS LAW
n1sinθ1= n2 sinθ2; If n2>n1 (when the light slows down in n2) then Snell's law tells us that θ2<θ1; that is the beam will bend (refract) toward the normal as it enters the medium. On the contrary if n2<n1 (when the light speeds up n2) then θ2>θ1 and the beam will bend away from the normal
the index of refraction (n) for air is
1
Dispersion of light (not sound): In sound we studied that wave speed is independent of frequency (Wave Rule #1). When light travels through a material medium, it displays dispersion:
A variation in wave speed with frequency (or wavelength). So the definition of the index of refraction, n=c/v, should be accompanied by a statement of the frequency of the light used to measure v, since different frequencies have different speeds and different index of refractions
As wavelength decreases thus increasing frequency, the refractive index _____
increase; In general, higher frequency waves have higher indices of refraction.
When white light hits glass prism, the beam is split into component colors in this order:
red orange yellow green blue violet (since red has longest wavelength it has the smallest index of refraction). Each color emerges from the prism at a slightly different angle so the light disperses into its component colors
Critical angle θ_c
When a beam of light strikes the boundary to a medium that has a lower index of refraction, the beam bends away from the normal (remember, θ and n are inverse concepts, if θ is large, n is small). As the angle of incidence increases the angle of refraction becomes larger. At some point when the angle of incidence reaches a critical angle θ_c, the angle of refraction becomes 90°, which means the refracted beam is directed along the surface
For angles of incidence greater than θ_c, what's the angle of refraction? What is total internal reflection (TIR)?
There is NO angle of refraction; the entire beam is reflected back into the original medium. This phenomenon is called total internal reflection
Total internal reflection occurs when there are 2 requirements:
1) n1>n2 AND 2)θ1>θ_c where θ_c= sin^-1 (n2/n1); Notice that total internal reflection cannot occur if n1<n2 cause then θ1>θ2 (pg 309) If it does not meet these conditions the light will reflect the angle same as the angle of incidence but will refract a smaller angle
critical angle equation (θ_c)
sinθ_c= (n2/n1)
Plane Mirrors
Light is reflected off the object standing in front of mirror and is reflected back to our eyes. The directions of the rays reflected off the mirror determines where we perceive the image to be.
Where is the image in a plane mirror?
It seems like the image is behind the plane mirror (if we step back so does the mirror image). The law of reflection can be used to show that the image seems as far behind the mirror as the object is in front of the mirror
Is the image real or virtual in a plane mirror?
An image is said to be real if light rays actually focus at the image. A real image can be projected onto a screen. For a flat mirror, light rays bounce off the front of the mirror; so no light focuses behind it. Therefore the images produced by flat mirror are not real they are virtual
Is the image upright or inverted in a plane mirror?
When we look into a flat mirror our image isn't upside down; flat mirrors produce upright images
What is the height of the image (compared with that of the object)?
Finally the image formed by a flat mirror is neither magnified nor diminished (minified) relative to the size of the object
Spherical mirror
Mirror that's curved in such a way that its surface forms part of a sphere
center of curvature and radius of curvature in spherical mirror
The center of imaginary sphere is the mirror's center of curvature (C), and the radius of the sphere is called the mirror's radius of curvature, R.
focus (or focal point) of spherical mirror
Halfway between the mirror and the center of curvature, C, is the focus, F
axis and vertex of spherical mirrors
The intersection of the mirror's optic axis (axis of symmetry) with the mirror itself is called the vertex, V
focal length of spherical mirrors
The distance from V to F is called the focal length, f, which is equal to 1/2 of the radius of the curvature
paraxial rays
incident light rays near the axis
concave mirror
mirror whose reflective side is caved in toward the center of curvature
convex mirror
reflective side that curves away from center of curvature
Ray tracing
Representative rays of light are sketched in a diagram that depicts the object and the mirror; the point at which the reflected rays intersect (or appear to intersect) is the location of the image
Rules governing rays from concave mirrors (3)
1) An incident ray parallel to the axis is reflected through the focus
2) An incident ray that passes through the focus is reflected parallel to the axis 3) An incident ray that strikes the vertex is reflected at an equal angle to the axis
Rules governing rays from convex mirrors (3)
1) An incident ray parallel to the axis is reflected away from the virtual focus 2) An incident ray directed toward the virtual focus is reflected parallel to the axis 3) An incident ray that strikes the vertex is reflected at an equal angle to the axis
difference between real and virtual
It is determined by seeing on which side of the mirror the image is formed. If the image is formed on the same side of the mirror as the object, then the image is real, but if the image is formed on the opposite side of the mirror it's virtual. Most of the time, concave mirrors form real images and convex mirrors form virtual images
mirror equation (used to find focal point of mirror) (pg 315)
(1/s_o) + (1/s_i) = (1/f) where s_o is the object's distance from the mirror, s_i is the image's distance from the mirror, and f is the focal length of the mirror. The value of s_o is always positive for a real object but s_i can be positive or negative.
The sign of s_i tells us whether the image is real or virtual:
If s_i is positive, the image is real; and if s_i is negative, the image is virtual
magnification equation and height
m= - (si/so); This gives the magnification; the height of the image, h_i, = abs(m) · h_o (h_o is the height of the object); If m is positive then the image is upright relative to the object; if m is negative, it's inverted
You need the mirror and magnification equations to answer the 4 physics questions
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If m from magnification equation is negative then:
si is positive, If si is negative then m is positive
Real images are always ______ and virtual images are always______________
Real images are always inverted and virtual images are always upright
f (focal point) in mirror equation for concave mirrors is always ________ and _______ for convex mirrors (SEE PG 316 FOR GREAT REVIEW)
f (focal point) in mirror equation for concave mirrors is always positive and negative for convex mirrors
What is the radius of curvature and focal length for a plane mirror?
A plane mirror can be considered a spherical mirror with an infinite radius of curvature (and infinite focal length) making f =∞
How do the 2 mirror equations apply to plane mirrors?
Since f=∞ then 1/f= 0 and the mirror equation becomes: (1/s_o)+(1/s_I)=0---> s_i= s_o. The image is as far behind the mirror as the object is in front. Also since s_o is always positive, s_i is negative, so the image is virtual. The magnification is m= -(s_i/s_o)=1 and the image is upright and has the same height as the object.
How to lens work and what are the 2 major categories of lenses?
A lens is an optical device that forms an image by refracting light. The 2 major categories of lenses are converging and diverging
converging lens (SEE PG 322 for pic)
Converges parallel paraxial rays of light to a focal point on the far side. Converging lenses all have at least 1 convex face.
real focus of converging lens
Because light rays actually focus at F, this point is called a real focus. Its distance from the lens is the focal length, f
diverging lens (SEE PG 322 for pic)
Cause parallel paraxial rays of light to diverge away from a virtual focus, F on the same side as the incident rays. Diverging lenses all have at least 1 concave face
optical center (O) of lenses
the central point within the lens where the axis intersects the lens (see pg 324)
Converging lenses rules that govern light rays
An incident ray parallel to the axis is refracted through the real focus. Incident rays pass undeflected through the optical center, O
Diverging Lenses rules that govern light rays
An incident ray parallel to the axis is refracted away from the virtual focus. Incident rays pass undeflected through the optical center, O
You can use mirror rules for lens to answer the 4 except:
Everything is opposite. The rules for converging lens are the same as concave rules and the rules for diverging lens are the same as convex rules since s_i is positive in converging lens and negative in diverging lens. However in converging lens, the image is still on the opposite side of the object whereas in diverging lens the image is still on the same side as the object. Think of your eyes as being converging lens since our eyes see images inverted
Difference between mirrors and lenses
Mirrors reflect light whereas lens refract it
converting Celsius to Kelvin
add 273°
triple point temperature of water in Kelvin and define
273.16° K; point where water is a liquid, solid and vapor
The change in temperature that a substance experiences upon a transfer of heat depends on 2 things:
the identity and the amount of the substance present
The equation that connects the amount of heat, Q, and the resulting temperature change, ∆T, is
Q= mc∆T; where m is the mass of the sample and c is an intrinsic property of the substance called its specific heat. Notice that positive Q is interpreted as heat coming in, while negative Q corresponds to heat going out
difference between specific heat and heat capacity
specific heat is the amount of heat to change the temperature of unit mass (1kg or 1gm) of substance by 1°C. Heat capacity is the amount of heat required to change temperature of substance by 1°C
When an object absorbs or loses heat,
either its temperature will change or the phase of the object will start to change but NOT both
The equation that applies during phase transition is Q=
Q=mL (where L is the latent heat of transformation, such as heat of fusion or vaporization); This equation tells us how much heat must be transferred to cause a sample of mass m to completely undergo a phase change
conduction and temperature
An iron skillet is sitting on a hot stove, and you accidentally touch the handle. You notice that there's been a transfer of thermal energy to your hand. The process by which this happens is known as conduction. The highly agitated atoms in the handle of the hot skillet bump into the atoms of your hand making them vibrate more rapidly, thus heating up your hand
convection
As the air around a candle flame warms, it expands, becomes less dense than the surrounding cooler air and rises. As a result heat is transferred away from the flam by a large scale motion of air
Radiation
Radiant energy from the sun's fusion reactions is transferred across millions of kilometers via electromagnetic waves. Absorption of the energy carried by these light waves defines heat transfer by radiation.
sublimation
solid to vapor
deposition
vapor to solid
When the temperature of an object changes so does its size. For example, heating a balloon makes it bigger. The equation for the change in length of objects that experience a change in temperature is
∆L= αL_o ∆T; Where ∆L is length change, L_o is initial length, α is the coefficient of linear expansion
Change in volume when heat is added to object
∆V= βV_o ∆T; where V_o is the sample's initial volume and β is the coefficient of volume expansion of the substance. For most solids β is about equal to 3α.
Kinetic theory of gases
The atoms that make up a gas do not move around in relatively fixed positions. Rather, the molecules of a gas move freely and rapidly in a chaotic swarm. A confined gas exerts a force on the walls of its container because the molecules are zipping around inside the container, striking the walls and rebounding causing pressure
Equation for pressure caused by movement of gases
P= F/A (The magnitude of force per unit are is called pressure) The faster molecules are moving the more pressure they exert
SI unit for pressure
N/m^2, the pascal (abbreviated Pa).
Avagadro's number (called N_a in physics) 1 mole of gas is equal to
1 mole= 6.022 x 10^23 molecules of gas
Ideal Gas Law
Ideal gases are when the volume of the gas molecules is negligible compared with that of the container that holds them, they experience no electrical forces and undergo elastic collisions
Ideal Gas Law Equation
PV= nRT; where n is the number of moles of gas and R is the universal gas constant; This equation tells us that for a fixed volume of gas, an increase in P gives a proportional increase in T. The pressure increases when the gas molecules strike the walls of their container with more force which occurs if they move more rapidly
Universal gas constant (R)
8.3 J/mol·K
Equation that relates translational kinetic energy and temperature
K_avg= (3/2) k_b T; Tells us that the average translational kinetic energy of the gas molecules is directly proportional to the absolute temperature of the sample
k_B is
k_b= R/N_a which is 1.38 x 10^-23
average speed of gas molecule (called rms speed)
v= √(3k_b T)/m or v=√(3RT)/M (M=mN_a which is the mass of 1 mole of the molecules)
make sure you use KELVINS when using ideal gas law, the rms speed formula and Kinetic energy of gas formula
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thermodynamics
The study of energy transfers involving work and heat and resulting changes in internal energy, temperature, volume and pressure
zeroth law of thermodynamics
When 2 objects are brought into contact heat will flow from the warmer object to the cooler one until they reach thermal equilibrium. If objects 1 and 2 are each in thermal equilibrium with object 3 then objects 1 and 2 are in thermal equilibrium with each other
First Law of Thermodynamics
Statement of conservation of energy that includes heat. Energy (in the form of heat) is neither created nor destroyed in any thermodynamic system
First Law of Thermodynamics equation
∆U= Q-W; where ∆U is the change in internal energy of the system, Q is the heat added to the system and W is the work done by the system. U depends on the state of the system
(see bottom of pg 345 and 346 for review) Work equation for gases on enclosed space
W= P∆V (work equals pressure times change in volume); Work is positive when the system does work against its surroundings and W is negative when the surroundings do work on the system. This equation assumes that P is constant. If it isn't constant, then work is equal to the work under the curve in P-V diagram. Moving left to right is a positive area and work and right to left is negative
isothermal
Process where the temperature remains constant. see graph on pg 349
adiabatic process
If there is no heat exchanged between the system and its surroundings. see graph on pg 349. It has a steeper slope than isothermal
entropy
disorder; A closed system that shows a high degree of order tends to evolve in such a way that its degree of order decreases . Time flows in such a way that ordered systems become disordered
second law of thermodynamics
The total amount of disorder- the total entropy- of a system plus its surroundings will never decrease
It is possible for entropy of a system to decrease but it'll always be at the expense of a greater increase in entropy in the surroundings. For example with water
When water freezes its entropy decreases. Ice is more structured than water. But when water freezes it releases heat into its environment which creates disorder in the surroundings
Work can easily be converted into heat but heat is harder to completely convert into work. Heat engines try to use heat to produce work.. Basic components of cyclic heat engine:
Energy in the form of heat comes into the engine from a high temperature source, some of this energy is converted into work but the remainder is ejected as exhaust heat into a low temperature sink and the system returns to its original state so ∆U is always 0. So Q_net=W, the heat absorbed by the system is equal to the work performed
Heat engine cycle Q_H and Q_C
The heat that's absorbed from the high temperature source is Q_H (hot) and the heat that is discharged into the low temperature reservoir is Q_C (cold). Because heat coming in is positive and heat going out is negative, Q_H is positive and Q_C is negative, and the heat absorbed is Q_H+Q_C. Normally write Q_H- abs(Q_C) to show that Q_net is less than Q_H.
Work of engine
W= Q_net= Q_H - abs(Q_C)
The thermal efficiency of the heat engine is (equation)
e= 1- abs(Q_C)/ Q_H; For any cyclic heat engine, some exhaust heat is always produced. Because Q_C cannot equal 0 no cyclic heat engine can operate at 100% efficiency; it is impossible to completely convert heat into useful work
Carnot Cycle
The most efficient heat engine follows what is known as the Carnot cycle; isothermal expansion, followed by adiabatic expansion, followed by isothermal compression, followed by adiabatic compression
Carnot cycle efficiency equation
e= 1- (T_C/T_H) where T_H and T_C are temperatures of the hot and cold reservoirs, respectively (don't really need)
raisin pudding model of atom
the pudding was the positively charged part of the atom and the raisins were the electrons
Rutherford's experiment to figure out atom structure
Rutherford fired alpha particles (carry electric charge of +2 since it consists of 2 protons and 2 neutrons) at an extremely thin sheet of gold foil. If the raisin pudding model is true than the alpha particle should sail right through the target atoms. However the experiment revealed that the alpha particles deflected at large angles (90° to 180°). This showed positive charge was in the middle of the atom (nucleus) with a swarm of electrons
Rutherford model
Proton nucleus in the middle with electrons swarming around it
quanta
electromagnetic radiation is emitted and absorbed by matter as though it existed in individual bundles called quanta
photon
quantum of electromagnetic energy
photoelectric effect
light behaves like a stream of photons and this is illustrated by the photoelectric effect
photoelectrons
When a piece of metal is illuminated by electromagnetic radiation, the energy absorbed by electrons near the surface of the metal can liberate them from their bound state and these electrons can fly off. The released electrons are known as photoelectrons
Light behaves as a particle and wave. In the 1900's there was a wave only theory of light that would predict the following 3 results (which all ended up being wrong)
1) There would be a significant time delay between the moment of illumination and the ejection of photoelectrons as the electrons absorbed incident energy until their kinetic energy was sufficient to release them from the atom's grip
2)Increasing the intensity of the light would cause the electrons to leave the metal surface with greater kinetic energy
3)Photoelectrons would be emitted regardless of the frequency of the incident energy as long as the intensity was high enough
Things that proved the 3 predictions of the wave only theory of light wrong
1) Photoelectrons were ejected within just a few billionths of a second after illumination
2) Increasing the intensity of light did not cause photoelectrons to leave the metal surface with greater KE. Although more electrons were ejected as the intensity increased
3) for the metal there was a certain threshold frequency (f_0), if light of frequency lower than f_0 were used to illuminate the metal surface no photoelectrons were ejected regardless of how intense the incident radiation
Einstein explained these observations by
postulating that the energy of the incident electromagnetic wave was absorbed in individual bundles (photons)
the energy of a photon is proportional to the frequency of the wave,
E= hf (where h is Planck's constant)
Planck's constant (in Joules not electronvolts)
6.63 x 10^-34 J · s
metal's work function
A certain amount of energy had to be imparted to an electron on the metal surface in order to liberate it; this was known as the metal's work function (ϕ)
If an electron absorbed a photon whose energy E was greater than ϕ
it would leave the metal with a max KE equal to E-ϕ. This process could occur very quickly which accounts for the rapidity with which photoelectrons are produced after illumination
max kinetic energy of electron that absorbs a photon whose energy E was greater than ϕ
If an electron absorbed a photon whose energy E was greater than ϕ it would leave the metal with a max KE equal to E-ϕ. or K_max= hf-ϕ
Increasing the intensity of the incident energy means bombardment with more photons and results in the ejection of more photoelectrons, but since the energy of each incident photon is fixed by the equation E=hf the value of
K_max will still be E-ϕ
If the incident energy had a frequency that was less than ϕ/h the incident photons would each have an energy that was less than ϕ; this would not be enough energy to liberate electrons regardless of
how many more photons you blast
electronvolt (eV)
unit for energy used when talking about light instead of Joules since Joules is too big to be convenient in light. The eV is equal to the energy gained or lost by an electron accelerated through a potential difference of 1 volt
Planck's constant in eV's
4.14 x10^-15 eV·s
***threshold frequency (f_0) equation
f_0= ϕ/h (where h is planck's constant)
wavelength of light equation
λ= c/f (where c is the speed of light)
speed of light (c)
3x10^8 m/s
the only way to increase K_max is to
increase the frequency- not the brightness- of the incident light
atomic spectra
Atoms in a gas discharge tube emitted and absorbed light only at specific wavelengths. The light from a glowing gas passed through a prism to disperse the beam into its component wavelengths produced patterns of sharp lines called atomic spectra
Bohrs model
Bohr postulated that the electron orbits the nucleus only at certain discrete radii. When the electron is in one of these special orbits it does not lose energy. However if the electron absorbs a certain amount of energy, it's excited to a higher orbit, one with a greater radius. When it returns to its lower orbit it emits a photon in the process. Since each allowed energy level has a specific radius (and energy) the photons emitted in each jump also have specific energies and wavelengths.
Electron's energy levels are
quantized
The energy levels within an atom are given by:
E_n= ((Z^2)/(n^2)) (-13.6eV); where Z is the number of protons in the atom's nucleus and n is the energy level.
When an excited electron drops from energy level n=j to a lower one, n=I, the transition causes a photon of energy to be emitted and the energy of a photon is:
the difference between the 2 energy levels
E_emitted photon=
E_emitted photon= abs(∆E)= E_ j- E_i
wave particle duality
Electromagnetic radiation propagates like a wave but exchanges energy like a particle
Since an electromagnetic wave can behave like a particle a particle can also behave like a wave
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de Broglie wavelength formula
λ= h/p; Particles in motion can display wave characteristics and behave as they had a wavelength λ= h/p where p=mv and h is plank's constant in J·s (6.63x10^-34)
Since the value of h is so small, ordinary macroscopic objects do not display wavelike behavior
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atomic number, neutron number, mass number and symbols
Atomic number is the amount of protons in atom with symbol of Z. The number of neutrons is denoted by N. Mass number (nucleon number) is Z+N and is denoted by A.
nuclide
The notation for a nuclide- The term for a nucleus with specific numbers of protons and neutrons- is to write Z and A one above the other before the chemical symbol of the element (367)
strong nuclear force
which binds together neutrons and protons to form nuclei
deuteron (d)
The nucleus of deuterium, an isotope of hydrogen that contains 1 proton and 1 neutron. The mass of a deuteron is 2.01356 amu which is a little less than the sum of the individual masses of the proton and neutron
mass defect (∆m)
(mass of proton + mass of neutron) - mass of deuteron (∆m= (m_p + m_n) - m_d)
What happens to the missing mass in mass defect?
It is converted to energy when the deutreron was formed
binding energy
the energy of mass defect. Mass defect represents the amount of energy needed to break the deuteron into separate proton and neutron. Since this tells us how strongly the nucleus is bound it is called the binding energy of the nucleus
Einstein's mass energy equivalence equation
The conversion between mass and energy is given by Einstein's mass energy equivalence equation, E=mc^2 (where c is the speed of light); the binding energy, E_B, is equal to: E_B= ∆mc^2
Using E=mc^2 the energy equivalent of 1 atomic mass unit is
about 931 MeV
Which element has the lowest binding energy per nucleon?
deuteron (hydrogen)
Which element has the highest binding energy per nucleon?
Nickel; When nuclei smaller than nickel are fused to form a single nucleus, the binding energy per nucleon increases, which tells us that energy is released in the process. On the other hand when nuclei larger than nickel are split binding energy per nucleon again increases releasing energy
Alpha Decay
Occurs when nucleus is too large or neutron to proton ratio is unfavorable or radioactive. When nucleus undergoes alpha decay it emits an alpha particle which consists of 2 protons and 2 neutrons and is the same as the nucleus of a helium-4 atom. Decreases the mass number by 4 and atomic number by 2. Turns into different element Represented as: α, α^4_2 or He^4_2 (see pg 370)
In any nuclear reaction what is conserved?
1) Mass number 2)Charge
The decaying nuclide is known as _____ and the resulting nuclide is known as ____
parent; daughter
Beta negative decay (more common than positive)
Occurs when neutron to proton ratio is too large (not enough protons). β- decay occurs when neutron transforms into a proton and an electron is ejected from the nucleus (beta particle). The atom turns into a different element. Another particle is also emitted called the electron antineutrino ( ̅ν_e). (atomic mass doesn't change)
Beta positive decay
Occurs when neutron to proton ratio is to small (too many protons). Proton is transformed into a neutron and a positron (e^0_+1) plus another particle the electron neutrino (ν_e) are ejected. Atom turns into different element (atomic mass doesn't change)
Electron capture
Another way in which a nucleus can increase its neutron to proton ratio is to capture an orbiting electron and cause the transformation of a proton into a neutron. Electron neutrino is still emitted. Atomic mass does not change. Turns into different element
Gamma decay (review pg 372)
Say that β- decay occurs changing K into an excited Ca (Ca*). For this excited nucleus to drop to its ground state it must emit a photon of energy, a gamma ray symbolized by γ. Atomic mass does not change. Gamma rays have no charge or mass
decay constant
As a radioactive sample disintegrates the number of decays per second decreases, but the fraction of nuclei that decay per second -the decay constant- does not change. The decay constant is determined by the identity of the radioisotope.
Activity (A) of radioactive sample equation
A= A_0 e^(-λt); The activity (A) of a radioactive sample is the number of disintegration it undergoes per second; it decreases with time. Where A_0 is the activity at time t=0 and λ is the decay constant (not wavelength in this case) (exponential decay)
activity is expressed as
Disintegrations per second: 1 disintegration per second is 1 Becquerel (Bq).
The greater the value of λ in decay
The faster the sample decays. This equation also describes the number (N) of radioactive nuclei in a given group
the number (N) of radioactive nuclei in a given group in radioactive decay sample equation
N= N_0 e^(-λt), or the mass (m) of the sample, m=m_0 e^(-λt)
The most common way to indicate the rapidity with which radioactive samples decays is to
give their half life which is the time required for half of a given sample to decay
Half life equation
Half life, T_(1/2), is inversely proportional to the decay constant, λ, and in terms of the half life, the exponential decay of a sample's mass (or activity can be written as:
m= m_0 (1/2)^(t/T_1/2)
nuclear fission
Bombardment of target nuclei with subatomic particles to artificially induce radioactivity
nuclear fusion
fusing small nuclei at extremely high temperatures
To balance nuclear reactions, we write
p^1_1 or H^1_1 for a proton and n^1_0 for a neutron
a deuteron has
an atomic mass of 2 and atomic number of 1 (1proton)
Nuclear reaction not only produce new nuclei and other subatomic but also involve the absorption or emission of energy. Nuclear reactions must
conserve total, so changes in mass are accompanied by changes in energy according to Einstein's equation, ΔE= Δmc^2
disintegration energy
A general nuclear reaction is written as: A+B ---> C+D+Q where Q is the disintegration energy
How do you know if a nuclear reaction is exothermic or endothermic?
If Q is positive the reaction is exothermic and the reaction can occur spontaneously; if Q is negative, the reaction is endothermic and the reaction cannot occur spontaneously
The energy Q is calculated as follows (equation)
Q= [(m_A + m_B) - (m_C + m_D)] c^2
2 postulates of special relativity
1) All the laws of physics are the same in all inertial reference frames 2)The speed of light in vacuum always has the same value (c= 3x 10^8 m/s), regardless of the motion of the source or observer
inertial reference frame
One in which Newton's first law holds. Given one inertial reference frame, any other reference that moves with a constant velocity relative to the first one will also be inertial
Relativity of Velocity formula (must reread pg 377)
v_you= (u+v)/ (1+ uv/c^2)
If a spaceship emitted a light pulse toward a planet, the speed of that light pulse is not u+v but
c. Postulate 2 says that the speed of light would be c regardless of the motion of the spaceship or the position of the person watching
Relativity of time formula (must reread pg 379 it's really cool)
In my reference frame let me call the time between my sneezes ΔT_1. If my velocity past you is v then the time that you'd measure between my sneezes would be: ΔT_2= γ · ΔT_1;
γ is the relativistic factor
relativistic factor (γ) formula
γ= 1/√(1-(v/c)^2); unless the spaceship is still relative to you, the time between 2 things like sneezes would be longer than the time I'd measure. For all ordinary speeds where v is very small compared to c the value of γ is pretty much equal to 1 so ΔT_2=ΔT_1 and we don't notice any difference in time intervals. As v gets closer to c, γ gets bigger and bigger along with the time difference. The faster the object goes the larger the γ
Relativity of length formula
Let's say I measure the length of my ship to be 100 meters. If you on earth were watching my ship fly by you would not measure its length to be 100 m. Use the formula: L_2= L_1/γ; Because γ is greater than 1 the length you'd measure would be shorter than what I'd measure.
Length contraction
lengths that are parallel to the velocity v are shortened by a factor of γ
rest energy
E=mc^2 tells us that mass and energy are equivalent and the formula tells us how much energy is equivalent to a given amount of mass. This energy is called rest energy because an object resting on your desk has energy-in fact is energy- simply by virtue of the fact the it exists and has mass.
kinetic energy of particles moving almost as fast as c
KE= (γ-1)mc^2
total energy (including new KE formula)
E_tot=γmc^2; E_tot= E_rest+KE= mc^2+ (γ-1)(mc^2)
see pg 383 it has a graph
and a paragraph. As v approaches c the object's kinetic energy approaches infinity. This is why it is impossible for a massive particle to move at the speed of light
Einstein's Equivalence Principle
it is impossible to differentiate between an accelerating reference frame and a reference frame in a gravitational field; such frames are equivalent
Einstein's theory general relativity
account for accelerated motion and for motion in the strong gravitational fields near large masses. Equivalence principle is the guiding principle to general relativity
Because of the equivalence principle when a person shines a flashlight in an elevator in space the light would appear to bend down since it is going straight as the person accelerates up.
The Equivalence Principle dictates that the same bending must occur in strong gravitational field as well. That is gravity makes light bend. Einstein went on to point out that this is not just a light property- light bends because space itself bends. 4 dimensional spacetime also bends with the result that time dilates in a gravitational field just as it would for any object moving at very high speed
Since planets have a lot of mass the bending of spacetime is significant in the vicinity of these objects. For example
the orbit of Mercury the closest planet to the Sun deviates measurably from the ellipse predicted by Newtonian gravitation
black holes
Very massive objects may have so much gravitational pull that the velocity required to escape from their attraction is greater than the speed of light. In general relativity because light is subject to the curvature of space that affects matter as well light cannot escape from these objects either. They don't emit or reflect light.
quasars
Extremely bright but extremely distant objects that are caused by black holes that are pulling matter in at extremely high speeds. The matter rubs against other falling in matter and the frictional effects generate light. When all the nearby matter is gone and the hole runs out of fuel, it stops generating light and this is the reason that the only ones we can see are very distant since they are from the past (time relativity)
Galaxies are always expanding
Edwin Hubble saw that galaxies outside our own send light to us that appears redshifted by the Doppler Effect for light and must be moving away from us showing that the universe is always expanding
Erwin Schrodinger
introduced the equation that described the way waves propagated in space and time
Werner Heisenberg and his principle
One cannot know both where a particle is and its momentum at the same time
Pauli's Exclusion principle
Certain types of particles cannot be in the same quantum states
superconductivity
The resistivity of material is temperature dependent- an object will conduct better at lower temperatures and at very low temperature the resistivity will drop to 0 making it a superconductor
Standard Model of Particle Physics and Quantum field theory
describes 3 of the 4 fundamental forces of nature: electromagnetic force, strong nuclear force and weak nuclear force
electromagnetic force, strong nuclear force and weak nuclear force
electromagnetic force is responsible for all of the forces wee see in daily life (friction normal force ect). Strong nuclear force is what holds nucleus of atom together since one would think that protons should repel one another. Weak nuclear force mediates radioactive decay
Cosmic Microwave Background (CMB)
In the early universe a great deal of matter and a great deal of energy (including light) were in a very small space and as a result photons that were emitted from one source just hit particles of matter and were absorbed. As the universe expanded the density of the universe decreased such that light could travel farther before hitting matter and being absorbed. Eventually at a certain point light because able to travel almost freely. Since this light was everywhere in the universe at the time it is still everywhere forming the background of the cosmos
dark matter
Based on Newton and Einstein's theory of gravity, stars should rotate in a certain way but do not. It is if there is a mass there that cannot be seen. This missing mass is called dark matter
dark energy
Based on general relativity the expansion of the universe should be decreasing. However observing the actual expansion shows that the universe is increasing. The source of the energy is unknown (dark energy)
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