Question

Starting from rest, a fan takes four seconds to attain the maximum speed of 400 rpm (revolution per minute). Assuming uniform acceleration, calculate the time taken by the fan in attaining half the maximum speed

• A. $2$
• B. $4$
• C. $8$
• D. $16$

Answer 1

The correct option is A $2$

Initial angular velocity, ${\omega }_o=0$ $rad/s$

Final angular velocity, $\omega =400$ $rpm$

$⟹\omega =\frac{400}{60}=\frac{20}{3}$ $rps$

$⟹\omega =\frac{20}{3}\times 2\pi$

$⟹\omega =\frac{40\pi }{3}$ $rad/s$

Time, $t=4$ $s$

From kinematics:

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

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

$⟹\alpha =\frac{40\pi }{3}\times \frac{1}{4}$

$⟹\alpha =\frac{10\pi }{3}$ $rad/s^2$

Time taken to attain half the maximum speed:

Here, ${\omega }^{\prime }=\frac{20\pi }{3}$ $rad/s$

$t=\frac{{\omega }^{\prime }-{\omega }_o}{\alpha }$

$⟹t=\frac{20\pi }{3}\times \frac{3}{10\pi }$

$⟹t=2$ $s$