Nonuniform Circular Motion

Section 11.8 Nonuniform Circular Motion

An object moving in a circle with a non-constant speed is said to undergo nonuniform circular motion. Since the speed is changing, the object must have a non-zero tangential acceleration. The magnitudes of the tangential and angular acceleration are related by \(\alpha = \frac{a_t}{r}\text{.}\) An object undergoing nonuniform circular motion still has a centripetal acceleration, which will change as the speed changes. When using the Law of Motion to relate acceleration to the net force, the entire acceleration vector must be used, including both the tangential and centripetal components.

Assumption 11.8.1. Motion with Constant Angular Acceleration.

When you assume an object is moving with constant angular acceleration, you can derive the following equations of motion from the definitions of angular acceleration and angular velocity:

\begin{equation*} \theta(t) = \theta_{i} + \omega_{i}t + \frac{1}{2}\alpha t^2 \end{equation*}

\begin{equation*} \omega(t) = \omega_{i} + \alpha t \end{equation*}

Exercises Activity - The Go-kart

You are on a Go-kart at a circular track such that the Go-kart’s angular acceleration is a constant \(0.6 \mathrm{~rad/s^2}\text{.}\) Your initial angular speed is \(-1.2 \mathrm{~rad/s}\text{.}\)

1. Representations.

Sketch graphs of the angular acceleration, angular velocity, and angular position vs. time.

2. Calculations.

Find equations for the angular position, angular velocity, and angular acceleration as functions of time.

3. Understanding.

How does the centripetal acceleration of your Go-kart change as you go?

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