Question

For the angular velocity vs. time graph depicted above, at what time do the red and blue objects share the same angular displacement?

• A. $\frac{t}{4}$
• B. $\frac{t}{2}$
• C. $\frac{3t}{4}$
• D. $t$
• E. The angular displacements never match

Answer 1

The correct option is B $\frac{t}{2}$

The angular displacement in an angular velocity graph is given by the area between the velocity graph and the time axis.

at time $t/4$

area of red graph $=\frac{\omega }{2}\times \frac{t}{4}=\omega t/8$

area of blue graph $=\frac{1}{2}\times \frac{\omega }{2}\times \frac{t}{4}=\omega t/16$

at time $t/2$

area of red graph $=\frac{\omega }{2}\times \frac{t}{2}=\omega t/4$

area of blue graph $=\frac{1}{2}\times \omega \times \frac{t}{2}=\omega t/4$

at time $3t/2$

area of red graph $=\frac{\omega }{2}\times \frac{3t}{4}=3\omega t/8$

area of blue graph $=\frac{1}{2}\times \omega \times \frac{t}{2}+\omega \times \frac{t}{4}=\omega t/2$

at time $t$

area of red graph $=\frac{\omega }{2}\times t=\omega t/2$

area of blue graph $=\frac{1}{2}\times \omega \times \frac{t}{2}+\omega \times \frac{t}{2}=3\omega t/4$

it is clear that angular displacements (areas) are same at $t/2$ .