Answer
Please see the work below.
Work Step by Step
(a) We know that the velocity depends on the ratio $\frac{I}{mr^2}$.
Now the ratio of $\frac{I}{mr^2}$ for the first hoop is given as
$\frac{I}{mr^2}=\frac{MR^2}{MR^2}=1$
Similarly, for the second hoop
$\frac{I}{mr^2}=\frac{(2M)R^2}{(2M)R^2}=1$
Thus, the two hoops finish the race simultaneously.
(b) We know that if the hoops have different radii, they will still finish the race at the same time because we can see from part (a) that the ratio $\frac{I}{mr^2}$ for both hoops remains the same.
