Question

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

Answer 1

${\omega }_o^2=900$ $ra{d}^2/s^2$

${\omega }^2=1600$ $ra{d}^2/s^2$

$\theta =100$ revolution

$⟹\theta =100\times 2\pi$ $rad$

From rotational kinematics:

${\omega }^2-{\omega }_o^2=2\alpha \theta$

where, $\alpha$ is angular acceleration

$⟹1600-900=2\times 2\times 100\times 2\pi$

$⟹\frac{700}{2\times 2\pi \times 100}=\alpha$

$⟹\alpha =\frac{7}{4\pi }$ $rad/s^2$

Also, $\omega ={\omega }_o+\alpha t$

$⟹t=\frac{\omega -{\omega }_o}{\alpha }$

$⟹t=\frac{40-30}{7}\times 4\pi$

$⟹t=\frac{40\pi }{7}$

$⟹t\approx 18$ $s$