Answer
(a) 1.50 revolutions per minute are needed.
(b) 0.919 revolutions per minute are needed.
Work Step by Step
(a) $g = \frac{v^2}{r}$
$v = \sqrt{gr}$
We can use the speed $v$ to find the time $T$ for one revolution.
$T = \frac{2\pi ~r}{v} = 2\pi ~\sqrt{\frac{r}{g}}$
$T = 2\pi~\sqrt{\frac{400~m}{9.80~m/s^2}}$
$T = 40.1~s$
We can use $T$ to find the number of revolutions per minute.
$\frac{60~s}{40.1~s} = 1.50~rpm$
1.50 revolutions per minute are needed.
(b) $T = 2\pi ~\sqrt{\frac{r}{g}}$
$T = 2\pi~\sqrt{\frac{400~m}{3.70~m/s^2}}$
$T = 65.3~s$
We can use $T$ to find the number of revolutions per minute.
$\frac{60~s}{65.3~s} = 0.919~rpm$
0.919 revolutions per minute are needed.
