Physics

Problem

The angular position

\theta

(in radians) of a rotating rigid body is given by

\theta = 3t^{3} - 2t^{2} + t,

where

t

is in seconds. Use this information to answer the following. a. (i) Calculate the average angular velocity during the interval from

t = 1\,\text{s}

t = 3\,\text{s}

. b. (ii) Find the instantaneous angular velocity at

t = 2\,\text{s}

. c. (iii) Determine the instantaneous angular acceleration at

t = 2\,\text{s}

and state whether the body is speeding up or slowing down at that instant, giving a brief reason.

Intuitive explanation

The key to rotational kinematics is recognizing that angular position, velocity, and acceleration are related by derivatives—just like linear motion. Find the angular velocity by differentiating $\theta$ with respect to time, then differentiate again to get angular acceleration. These derivatives tell you how fast the body is spinning and how that spin rate is changing at any moment.

Intuitive explanation

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