Question

A wheel is rotating with an angular velocity of 3 rad/s. If the angular acceleration is 2 rad/s, then what is the angular velocity after 5 seconds?
${\text{A}}{\text{.}}$ 3 rad/s
${\text{B}}{\text{.}}$ 13 rad/s
${\text{C}}{\text{.}}$ 10 rad/s
${\text{D}}{\text{.}}$ 5 rad/s

Answer 1

Hint: Here, we will proceed by identifying whether the given angular acceleration is constant or not (for this particular problem, it is constant). If yes, then we will apply the kinematic equations for angular motion.

Complete step-by-step answer:

Formula Used- ${\omega _{\text{f}}} = {\omega _{\text{i}}} + \alpha t$.
Given, Initial angular velocity of the wheel ${\omega _{\text{i}}}$ = 3 rad/s
Angular acceleration of the wheel $\alpha $ = 2 rad/s

Clearly, we can see that the angular acceleration of the wheel is constant. This constant angular acceleration will increase the angular velocity of the wheel as the time passes.
According to kinematic equations for angular motion, we can say that the final angular velocity ${\omega _{\text{f}}}$ rad/s at any time t seconds will be equal to the sum of the initial angular velocity ${\omega _{\text{i}}}$ rad/s (at zero time) and the product of constant angular acceleration acting on the object and time t seconds.

i.e., ${\omega _{\text{f}}} = {\omega _{\text{i}}} + \alpha t{\text{ }} \to {\text{(1)}}$

The above kinematic equation for angular motion is valid only when the angular acceleration is constant (i.e., independent of the time t and position x)

Here, t = 5 seconds because we have to find the final angular velocity at that particular time

By substituting ${\omega _{\text{i}}}$ = 3 rad/s, $\alpha $ = 2 rad/s and t = 5 s in equation (1), we have
Final angular velocity after 5 seconds, $
   \Rightarrow {\omega _{\text{f}}} = 3 + \left( 2 \right)\left( 5 \right) \\
   \Rightarrow {\omega _{\text{f}}} = 3 + 10 \\
   \Rightarrow {\omega _{\text{f}}} = 13 \\
 $
Therefore, the angular velocity of the wheel after 5 seconds will be equal to 13 rad/s

Hence, option B is correct.

Note- In these types of problems, if we are given with non-constant angular acceleration (i.e., if angular acceleration is a function of time). Then, we will proceed by using the general concept which is that angular acceleration is simply rate of change of angular velocity with respect to time t i.e., $\alpha = \dfrac{{d\omega }}{{dt}}$.