1. Uniform Circular Motion
Chapter 4
AP Physics C: Mechanics
Mrs. Warren

2. Uniform Circular Motion
Uniform circular motion is the motion of an object traveling at a constant (uniform) speed in a circular path.
Does an acceleration exist if a particle is undergoing uniform circular motion? Why or why not?
Yes!! Because acceleration deals with velocity and while the speed is constant…the direction would be constantly changing in a circular path. Thus…since acceleration is a vector and velocity is a vector that depend on magnitude AND direction….if the velocity direction changes…there must be an acceleration.
Speed may be constant but direction is not
While moving in a circular path, an object is constantly being pulled “towards the center” of the circle away from its tangential path. Envision a stopper on the end of a string being twirled over your head in a horizontal circle. If the string were to break, the stopper would fly off at a tangent. The tension in the string is forcing the stopper to constantly be pulled back towards the center of the circle to follow the circular path instead of linear.

3. Changing Velocity in Uniform Circular Motion
The change in the velocity vector is due to the change in direction.
The direction of the change in velocity is toward the center of the circle.
The vector diagram shows:
The constant-magnitude velocity vector is always tangent to the path of the object.
As an object moves in a circle, it is constantly changing direction. At all instances, the object is moving tangent to the circle. Since the direction of the velocity vector is the same as the direction of the object’s motion, the velocity vector is directed tangent to the circle as well.
In a certain amount of time, an object moves from A to B…note that the magnitude of vf = vi because the speed is constant…only direction is changing.

4. Centripetal Acceleration
Magnitude ac of the centripetal acceleration depends on the speed v of the object and the radius r of the circular path. ac=v2/r
The acceleration always points toward the center of the circle of motion.
The acceleration is always perpendicular to the path of the motion.
The direction of the centripetal acceleration vector is always changing, to stay directed toward the center of the circle of motion.
When the acceleration points toward the center it’s because the velocity vector is changing direction. The acceleration vector is always centripetal (along the radial direction) and the velocity vector is always tangent.
If the acceleration vector were not perpendicular, there would be a component of acclereation parallel to that of the velocity vector…and that would lead to a change in speed of the particle along the path.

5. Derivation of Centripetal Acceleration

6. Centripetal Force
“Center-seeking” Force
Formula?

7. Period
Period, T, is defined as the time interval required for one complete revolution of the particle.
Formula?
V = 2pir/T
Thus…period must equal
T = 2pir/v

8. Angular Speed/Translational Speed
One full revolution around the circle corresponds to an angle of 2 pi radians, the product of 2 pi and the rotation rate gives the angular speed, w, measured in radians/s
W = 2pi/T

9. Quick Check
A particle moves in a circular path of radius, r, with speed, v. It then increases its speed to 2v while traveling along the same path. The centripetal acceleration of the particle has changed by what factor?
0.25
0.5 2 4
From the same choices above, by what factor has the period changed?

10. Example 1
What is the centripetal acceleration of the Earth as it moves in its orbit around the Sun?

11. Example 2
A skater moves with 15 m/s in a circle of radius 30m. The ice exerts a central force of 450 N. What is the mass of the skater?

12. Tangential and Radial Acceleration
This happens if velocity vector changes in direction and magnitude.
Tangential and Radial Acceleration
As a particle moves along a curved path, the direction of the total acceleration vector, a, changes from point to point. At any instant we can resolve it into components.
What are these components called?

13. Total Acceleration
The tangential acceleration causes the change in the speed of the particle.
The radial acceleration comes from a change in the direction of the velocity vector.
The tangential acceleration:
The radial acceleration:
The total acceleration:
Magnitude
Direction
Same as velocity vector if v is increasing, opposite if v is decreasing
The radial component, ar, is along the radius of the circle (arises from a change in direction of the velocity vector) ar = -ac
The tangential component, at, is perpendicular to the radius.
In uniform circular motion, where v is constant, at = 0 and the accleration is always completely radial
If the direction if v does not change, there is no radial acceleration and the motion is one dimensional.

14. Example 3
A car leaves a stop sign and exhibits a constant acceleration of m/s2 parallel to the roadway. The car passes over a rise in the roadway such that the top of the rise is shaped like an arc of a circle of radius 500 m. At the moment the car is at the top of the rise, its velocity vector is horizontal and has a magnitude of 6.00 m/s. What are the magnitude and direction of the total acceleration vector for the car at this instant?

15. Quick Check
A particle moves along a path, and its speed increases with time. In which of the following cases are its acceleration and velocity vectors parallel?
When the path is circular
When the path is straight
When the path is a parabola
Never
From the same choices above, in which cases are its acclereation and velocity vectors perpendicular everywhere along the path?

16. Homework Problem
An athlete rotates a 1.00 kg discus along a circular path of radius 1.06 m. The maximum speed of the discus is 20.0 m/s. Determine the magnitude of the maximum radial acceleration of the discus.