Question

When an old lp turntable was revolving at 33.3 rpm, it was shut off and uniformly slowed down and stopped in 5.5 seconds. through how many rotation did it turn while stopping?

Answer 1

First, let's convert the initial angular speed from rpm (rev/min) into rad/s, keeping in mind that $1 rev = 2 \pi rad$ and 1 min=60 s:
$\omega_i = 33 \frac{rev}{min}=33\frac{rev}{min} \cdot \frac{2 \pi rad}{60 s}=3.45 rad/s$

Now we can find the angular acceleration of the lp, keeping in mind that the final speed is zero:
$\alpha = \frac{\omega_f - \omega_i}{t}= \frac{0-3.45 rad/s}{5.5 s}=-0.63 rad/s^2$
and the acceleration is negative because the LP is decelerating.

Now we can find the angle covered by the LP from the beginning to the end of its motion:
$\theta (t)= \omega_i t + \frac{1}{2}\alpha t^2 = (3.45 rad/s)(5.5 s)+ \frac{1}{2}(-0.63 rad/s^2)(5.5 s)^2=$
$=9.45 rad$

And finally, we can convert it into number of revolutions:
$1 rev : 2 \pi rad = x: 9.55 rad$
$x= \frac{9.55}{2 \pi}= 1.52 rev$