Uniform Circular Motion

Chapter 4

AP Physics C: Mechanics

Mrs. Warren

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?

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:

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.

Derivation of Centripetal Acceleration

Centripetal Force

“Center-seeking” Force

Formula?

Period

Period, T, is defined as the time interval required for one complete revolution of the particle.

Formula?

Angular Speed/Translational Speed

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?

Example 1

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

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?

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?

Tangential and Radial Acceleration

Total AccelerationThe 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

Example 3

A car leaves a stop sign and exhibits a constant acceleration of 0.300 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?

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?

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.