4.5.1 The Uniform Circular Motion Model

The uniform circular motion model is especially important because it applies not only to particles moving in circles but also to the uniform rotation of solid objects.

MODEL 4.2 Uniform circular motion

For motion with constant angular velocity $ω$ .

Applies to a particle moving along a circular trajectory at constant speed or to points on a solid object rotating at a steady rate.
Mathematically:
- The tangential velocity is $v_{t} = ω r$ .
- The centripetal acceleration is $v_{t}^{2} / r or ω^{2} r$ .
- $ω$ and $v_{t}$ are positive for ccw rotation, negative for cw rotation.
Limitations: Model fails if rotation isn’t steady.

The figure illustrates the motion diagram of an object to explain uniform circular motion model.

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Workbook Exercise 20

Example 4.12 The acceleration of a Ferris wheel

A typical carnival Ferris wheel has a radius of 9.0 m and rotates 2.0 times per minute. What speed and acceleration do the riders experience?

MODEL Model the rider as a particle in uniform circular motion.

SOLVE The period is $T = \frac{1}{2} min = 30 s$ . From Equation 4.21, a rider’s speed is

v_{t} = \frac{2 π r}{T} = \frac{2 π (9.0 m)}{30 s} = 1.88 m/s

Consequently, the centripetal acceleration has magnitude

a = \frac{v_{t}^{2}}{r} = \frac{(1.88 {m/s)}^{2}}{9.0 m} = 0.39 {m/s}^{2}

REVIEW This was not intended to be a profound problem, merely to illustrate how centripetal acceleration is computed. The acceleration is enough to be noticed and make the ride interesting, but not enough to be scary.