Question

A rotating object has an angular acceleration of α = 0 rad/s2. Which one or more of the following three statements is consistent with a zero angular acceleration?
A. The angular velocity is ω = 0 rad/s at all times.
B. The angular velocity is ω = 10 rad/s at all times.
C. The angular displacement θ has the same value at all times.

Answer 1

Answer:

A,B and C

Explanation:

Statement A  

At all times, angular velocity is $\omega = 0\,{\rm{rad/s}$  

Angular acceleration is the rate of change in angular velocity with respect to time.  

Angular velocity and angular acceleration are related by  

${\omega _{\rm{f}}} = {\omega _{\rm{i}}} + \alpha t$

Which when re-arranged becomes  

$\alpha = \frac{{{\omega _{\rm{f}}} - {\omega _{\rm{i}}}}}{t}$

There’s no change in angular velocity anytime when the angular velocity is $\omega = 0\,{\rm{rad/s}}$

The equation can be modified as follows:  

$\begin{array}{c}\\\alpha = \frac{{0\,{\rm{rad/s}} - 0\,{\rm{rad/s}}}}{t}\\\\ = 0\\\end{array}$

Therefore, the angular acceleration becomes zero hence statement A is valid.  

Statement B  

Angular acceleration is the rate of change in angular velocity with respect to time.  

Angular velocity and angular acceleration are related by  

${\omega _{\rm{f}}} = {\omega _{\rm{i}}} + \alpha t$

Which when re-arranged becomes  

$\alpha = \frac{{{\omega _{\rm{f}}} - {\omega _{\rm{i}}}}}{t}$

There’s no change in angular velocity anytime when the angular velocity is $\omega = 10\,{\rm{rad/s}}$.The final and initial velocities remain the same.  

The equation can be modified as follows:  

$\begin{array}{c}\\\alpha = \frac{{10\,{\rm{rad/s}} - 10\,{\rm{rad/s}}}}{t}\\\\ = 0\\\end{array}$

Therefore, the angular acceleration becomes zero and statement B is valid  

Statement C  

Angular velocity is defined as the change in the angular position with respect to time.  

Angular velocity and angular displacement are related by  

$\theta = \omega t$

Which can also be modified as:  

${\theta _{\rm{f}}} - {\theta _{\rm{i}}}$

Note that the final position is ${\theta _{\rm{f}}}$and initial position is ${\theta _{\rm{i}}}$

Modifying the equation to find the angular velocity we obtain  

$\omega = \frac{{{\theta _{\rm{f}}} - {\theta _{\rm{i}}}}}{t}$

When the angular displacement has the same value at all times, the equation becomes  

$\begin{array}{c}\\\omega = \frac{{{\theta _{\rm{i}}} - {\theta _{\rm{i}}}}}{t}\\\\ = 0\\\end{array}$

The angular velocity becomes zero.  

Angular acceleration and angular velocity are related by  

${\omega _{\rm{f}}} = {\omega _{\rm{i}}} + \alpha t$

The expression above can be rearranged as follows:  

$\alpha = \frac{{{\omega _{\rm{f}}} - {\omega _{\rm{i}}}}}{t}$

At all times, the angular velocity is $\omega = 0\,{\rm{rad/s}}$ hence initial and final velocities remain the same  

We obtain  

$\begin{array}{c}\\\alpha = \frac{{0\,{\rm{rad/s}} - 0\,{\rm{rad/s}}}}{t}\\\\ = 0\\\end{array}$

Therefore, the angular acceleration becomes zero and statement C is valid.  

Therefore, statements A,B and C are consistent .