Question

The square of the angular velocity $\omega$ of a certain wheel increases linearly with the angular displacement during $100$ revolutions of the wheel’s motion as shown. Compute the time $t$ required for the increment.

• A. $\frac{40\pi }{7}\text{sec}$
• B. $\frac{20\pi }{7}\text{sec}$
• C. $\frac{25\pi }{7}\text{sec}$
• D. $\frac{40\pi }{3}\text{sec}$

Answer 1

The correct option is A $\frac{40\pi }{7}\text{sec}$

From graph, we get
${\omega }^2=1600{\text{rad/s}}^2,{\omega }_o^2=900{\text{rad/s}}^2$
$\theta =2\pi N=2\pi \times 100$
Given that ${\omega }^2\propto \theta$, so wheel's angular acceleration must be constant.
Now, by the equation of motion,${\omega }^2={\omega }_0^2+2\alpha \theta$
$\Rightarrow 1600=900+2\alpha (2\pi \times 100)$
$\Rightarrow \alpha =\frac{700}{4\pi \times 100}=\frac{7}{4\pi }$
Again, $\omega ={\omega }_0+\alpha t$
$\Rightarrow 40=30+\left(\frac{7}{4\pi }\right)t$
$\Rightarrow t=\frac{10\times 4\pi }{7}=\frac{40\pi }{7}\text{sec}$