# Physics Equations

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MCAT physics equations

### Displacement (d)

d = Δx = final position - initial positon = net distance (plus direction)

v = Δx/Δt = d/Δt

a = Δv/Δt

### What assumption do we make with the BIG 5 equations? What are the 3 most frequently used equations?

That acceleration is constant (uniformly accelerated motion)

2. Missing d
3. Missing vf (v final, v)
5. Missing t

d = ½(v₀ + v)t

v = v₀ + at

d = v₀²t + ½at²

d = vt - ½at²

### For projection motion, we assume that the downward direction is...

NEGATIVE direction.

DOWNWARD = NEGATIVE DIRECTION.

### PROJECTILE MOTION - Horizontal Motion (in the x-direction) Give the equation for: 1. Displacement 2. Velocity 3. Acceleration

1. d = v₀t (in the x-direction)

2. v₀x = vx (velocity is CONSTANT in the x-direction!!)

v₀x = v₀cosθ₀

3. ax = 0 (acceleration in the x-direction is ZERO!)

### PROJECTILE MOTION - Vertical Motion Give the equation for: 1. Displacement 2. Velocity 3. Acceleration

1. y = v₀yt + ½(-g)t² (in the y-direction)

2. vy = v₀y + (-g)t (in the y-direction)

v₀y = v₀sinθ₀

3. ay = -g (accerleration in the y-direction)

### Newton's 1st Law

Fnet = 0

velocity (v) = constant!
∴a = 0 --> F = 0

Fnet = m a

### Newton's 3rd Law

F₁₂ = F₂₁

Action-Reaction pair

### Weight

w = mg

Force due to gravity (Fg) = weight = mg

Fg = (GMm)/r²

since w = Fg,
mg = (GMm)r²
∴g = GM/r²

### Kinetic friction

Ff = μkFN

μk = coefficient of KINETIC friction
FN = normal force

### Static friction

Ff, max = μsFN

μs = coefficient of STATIC friction
FN = normal force

μs, max > μk

ALWAYS!

### What is the force due to gravity acting PARALLEL to the inclined plane? (sliding down the plane)

F = mgsinθ

θ = angle between incline and horizontal

### What is the force due to gravity acting PERPENDICULAR to the inclined plane?

F = mgcosθ

θ = angle between incline and horizontal

### Center of mass xCM

xCM = (m₁x₁ + m₂x₂ + m₃x₃...)/(m₁ + m₂ + m₃...)

### Center of gravity xCG

xCG = (w₁x₁ + w₂x₂ + w₃x₃...)/(w₁ + w₂ + w₃...)

xCM = xCG

ac = v²/r

Fc = mac = mv²/r

### Torque (τ)

τ = rFsinθ

r = distance from pivot point to F
F = force applied (Fapp)
θ = angle between r and F

When θ =90°
τ= r F

### What does it mean to be in ROTATIONAL EQUILIBRIUM?

τcw = τccw
∴τnet = 0 (net torque = 0)
clockwise torque = counterclockwise torque

### τnet (net torque)

τnet = I α

I = inertia
α = rotational acceleration
*compare to Fnet = ma

### Is the inertia greater for a solid sphere or a hollow sphere? If the same net torque was applied to each sphere, which would spin faster?

Hollow sphere has GREATER inertia than solid sphere.

Solid sphere would spin faster because it has less inertia!

### What does inertia depend on?

It depends on how a mass is distributed in an object.

### Work (W)

W = Fdcosθ

Unit = Joules (J)

F = force applied
d = displacement
θ = angle between F and d

Wmax occurs when θ = 0

### Power (P)

P = W/t

Unit: J/s = Watts (W)

W = Work
t = time

### Kinetic Energy (KE)

KE = ½mv²

Unit: Joules (J)

KE always has a + value; object has to be in motion!

### Work-energy theorem (Wtotal)

Wtotal = ΔKE = KEf - KE₀

### Gravitational Potential Energy (PE)

PE = mgh

m = mass
g = acceleration due to gravity
h = height

E = KE + PE

### Conservation of Energy (ideal frictionless world)

E₀ = E

KE₀ + PE₀ = KE + PE

W = ΔKE = -ΔPE

### Conservation of Energy (when friction is taken into account)

KE₀ + PE₀ - Wf = KE + PE

Wf = work done by FRICTION!

### Momentum (p)

p = mv

*p is always in the same direction as v

Unit = kg∗m/s

### Conservation of total momentum

p₀ = p
m₁v₁ = m₁v₁'

total p₀ = total pf

### Impulse-momentum theorem (J)

J = Δp = FΔt

change in momentum (Δp) = impulse (J) = FΔt

### During a collision (short duration of interaction between 2 objects)...

1. Total momentum (p) is ALWAYS conserved!
2. KE is NOT always conserved!

### 1. Elastic collision

(think of 2 pool balls)
1. p is conserved
2. KE is ALSO conserved!

∴p₀ = pf
∴KE₀ = KEf

### 2. Inelastic collision

(think of a car crash)
1. p is conserved
2. KE is NOT conserved: one object becomes deformed; loss in energy

m₁v₀₁ + m₂v₀₂ = m₁v₁' + m₂v₂'

### 3. Perfectly inelastic collision

(think of 2 objects becoming stuck together)
1. p is conserved
2. objects get stuck together and move with the same v' (final velocity!)

m₁v₀₁ + m₂v₀₂ = (m₁ + m₂)v'

### During the collision, there is a short duration of interaction between the 2 objects so we can apply...

Newton's 3rd Law: action-reaction pair!
F₁₂ = F₂₁

### Angular momentum (L)

L = Iω

I = moment of inetia
ω = angular velocity

*compare to p = mv

L₀ = L'

ρ = m/V

m = mass
V = volume

### Specific gravity (sp gr)

sp gr = ρ/ρH₂O

ρ = density of object or fluid
ρH₂O = density of water

### What is the ρ of water?

ρH₂O = 1000 kg/m³ = 1 g/cm³ = 1 kg/L

### Pressure (P)

F/A

F = perpendicularly applied force
A = area

Unit = N/m₂ = Pascal (Pa)
1atm = 100 kPa

### Gauge Pressure (Pg)

Pressure felt at a particular depth = hydrostatic pressure
Pg = ρ g D

ρ = density of fluid
g = accl due to gravity
D = depth

### Total hydrostatic pressure (Ptotal)

Ptotal = Patm + Pg

Patm = pressure at the surface
Pg = gauge pressure

### Buoyant Force (Fb) *Archimedes Principle

Fb = ρ V g

ρ = density of fluid
V = Vsub = volume of object submerged = volume of fluid displaced
g = accl due to gravity

### Archimedes Principle

There is an upward acting force on an object that is rising, sinking, floating, or completely submerged in a fluid. This force is called the buoyant force (Fb) and it equals the weight of the fluid displaced.

### Case 1: Object is sinking

ρ object > ρ fluid

### Case 2: Object is completely submerged and reaches the bottom of the tank

Fnet = Fn + Fb - Fg = 0
∴Fn = Fg - Fb

ρ object > ρ fluid

*Fn = normal force: weight of the object inside the fluid = apparent weight

### Case 3: Object is rising up in the fluid

Fnet = Fb - Fg = ma

ρ fluid > ρ object

### Case 4: Object is floating on the surface

Fnet = Fb - Fg = 0
∴Fb = Fg (= weight of object)
ρ V g = m g
ρ V = m = ρ₀V₀ (object)
ρ fluid > ρ object

V/V₀ = ρ₀/ρ = specific gravity

### Remember, ice is always

90% submerged! Think of the Titanic...

### What are the 4 conditions to have an IDEAL FLUID?

1. No or negligible viscosity.
2. Laminar flow (continuous, streamline flow).
3. Fluid must be incompressible.
4. Flow must be continuous.

### Flow rate (f)

f = A v
A: cross-sectional area
v: velocity of fluid
Unit: m³/s

A₁v₁ = A₂v₂

### Bernoulli's equation (ideal fluid)

describes ideal fluid flow through a pipe
*looks similar to conservation of energy equation

P₁ + ½ρv₁² + ρgh₁ = P₂ + ½ρv₂² + ρgh₂

*pgh is NOT gauge pressure, but the PE of the fluid at a certain height, h.

### Apply Bernoulli's equation to a HORIZONTAL PIPE.

Since h₁ = h₂, the equation looks like this:
P₁ + ½ρv₁² = P₂ + ½ρv₂²

### For a horizontal pipe, describe the relationships between pressure and velocity.

1. If v₁ > v₂, then P₁ < P₂.
2. If v₂ > v₁, then P₂ < P₁.

### Objects will move from _______ pressure to _______ pressure. Think of an example.

High P --> Low P

Think of air planes and the lift force!

v = √(2gD)

stress = F/A

### Strain

strain = ΔL/L₀

ΔL = change in length
L₀ = initial length

### Hooke's Law

stress = modulus ∗ strain

ΔL = FL₀/EA

*FLEA

X = FL₀/AG

*FLAG

1.6 ∗ 10⁻¹⁹ C

### Coulomb's law (Electric force: force on a charged particle)

F = kQq/r²

k = Coulomb's constant = 9 ∗10⁹ Nm²/C²
Q = charge that applies force
q = test charged being acted on by Q
r = distance between 2 charges

Positive F = repelling force
Negative F = attractive force

1μC = 10⁻⁶ C
1C = 10⁶μC

### State the principal of superposition.

The net electric force on a charge (q) due to a collection of other charges (Q's) is equal to the sum of the individual forces that each of the Q's alone exerts on q

Example: