PHY105 Chpt 5 Circular Motion & Gravity

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Physics MWCC Prof. E. Stevens Griffith, W. Thomas & Brosing, Juliet W. The Physics of Everyday Phenomena. New York: McGraw Hill, 2009.

Centripetal Force

Force(s) that make a body follow a curved path.
It is directed orthogonally to the velocity of the body, toward the instantaneous center of curvature of the path.

Centripetal Acceleration

Rate of change of tangential velocity
Text: rate of change in v of an object that is associated with the change in direction of the velocity. It is always perpendicular to the v vector and toward the center of the curve.
Example: Whirling a ball on a string
The string acts to pull the ball inward. Without it, the ball will fly off in the direction of travel at the point the string breaks. Gravity pulls it downward. The string changes the v vector continually.

Find v & centripetal

v₂-v₁=∆v
Each of the three will have a different direction
If the time span is short enough, ∆v will point toward center of curve
a(c)=v²/r
a sub c indicates that the acceleration vector is toward center of curve
Speed directly related to ∆v length
Speed directly related to direction of v vector change
Radius of curve inversely related to change in velocity

Centripetal Acceleration Cause

Ball on string has both horizontal and vertical components since string isn't entirely within plane of motion
Horizontal component pulls ball toward center of horizontal circle producing centripetal acceleration

Total tension on string determined by both H and V components
V component is equal to w of ball, therefore vertical Fnet=0; thus the ball stays in the horizontal plane of the circle and does not accelerate vertically

centripetal acceleration proportional to square of speed of ball

Centripetal equations

a_c=v²/r; v²=a_c*r; r=v²/a_c
Proportional to square of speed, inversely proportional to radius of curve

F_net=T_h=ma_c
net force=horizontal tension=mass*centripetal acceleration

F_net=mv²/r
net force = mass * velocity squared / radius

Centripetal Force

Any force or combination of forces that act on an object to produce centripetal acceleration

Can be caused by: pull from string, push from contacting another object, friction, gravity, etc.

Coefficient of friction

μ Greek mu
static or kinetic is denoted with subscript s or k
static: no motion in direction of force
kinetic: motion in direction of force
static usually > kinetic

Car on flat curve

Friction produces centripetal acceleration
Tendency of car to move in straight line causes tires to pull against pavement as car turns. Newton's 3d law: pavement pulls in opposite direction of tires. Frictive force points toward center of curve; thus, the car turns.
static: part of the tire in contact with road is at rest for a short time, it is not sliding
kinetic: the part of the tire in contact with the road is sliding against the surface

Fnet<ma_c means that car will begin to slide
The car sliding is related to it's mass, square of velocity, and turning radius
Smaller turns require lower speed

Car on banked curve

Horizontal component of normal force contributes to centripetal acceleration
With no friction, a car going too slowly will tend to slide inward while a car going to quickly will tend to slide outward.

Ferris Wheel

circular motion is vertical
Heavier at bottom due to greater upward pressure from seat, much like elevator passenger
Lighter at top because rider weight only produces centripetal acceleration

Motion on a slope (not in text)

The motion of an object on a slope must be broken into horizontal and vertical vectors.

X
Imagine the \ leg has an object at the center of the X. It is the slope on which the object lies. It represents the horizontal (x) component of the vectors. Kinetic force F_k lies uphill of this point.
The / leg represents the vertical (y) component of the vectors. F_n (normal force) lies uphill, w lies downhill.
Notice that if this is tilted such that the horizontal vector is level that it is the same as a force diagram for an object lying on a table.
Sketch a base level that the two legs lie upon. Through the center point, drop a line perpendicular to the base. Note that this forms two complementary right triangles. For the downslope triangle, theta represents the angle at the base. For the triangle under the uphill portion, theta represents the apex angle.
F_n (top of y vector)=w*cos(theta)
w (bottom of y vector)=cos(theta)
w (bottom of perpendicular line) = m*g
F_k (top of x vector)=Mu_kFn=Mu_kw*cos(theta)
F_? (bottom of x vector)=w*sin(theta)
Mu_s=tan(theta)
a_x=F_net/m
F_net=F_?-F_k=wsin(theta)-Mu_kw*cos(theta)

Kepler's Laws

1 Planets all move in elliptical orbits about a sun, which located at one focus of the ellipse
2 An imaginary line drawn from the sun to any planet moves through equal areas in equal intervals of time
3 If T is the amount of time of a period and r is the average radius of the orbit then T²/r³ is the same for all known planets
r³_earthT²_mars=r³_marsT²_earth
Period: time to complete a full orbit around the sun. On earth, a year.

Newton's Law of Universal Gravitation

F=Gm₁m₂/r²
Gravitation force between objects 1 and 2=Gmass of object 1mass of object 2 / square of the distance between the centers of the objects
G: constant=6.6710⁻¹¹ Nm²/kg²

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