Physics GRE - #16

Two circular hoops, X and Y, are hanging on nails in a wall. The mass of X is four times that of Y, and the diameter of X is also four times that of Y. If the period of small oscillations of X is $T$, the period of small oscillations of Y is:

• $T$
• $\dfrac{T}{2}$
• $\dfrac{T}{4}$
• $\dfrac{T}{8}$
• $\dfrac{T}{16}$

Solution :

The answer is $\dfrac{T}{2}$.

The key thing to note here is that the period of small oscillations of a circular hoop is the same as that of a normal pendulum with the mass where the centre of the hoop is.
Thus, the period is given by the formula (we'll disregard the constant of $2\pi$ at the front, since it will not matter to our calculations):
$$T=\sqrt{\frac{r}{g}}.$$ For hoops X, we have $$T_x=\sqrt{\frac{r_x}{g}}.$$ For hoop Y, we have  $$T_y=\sqrt{\frac{r_y}{g}}.$$ Since $r_y = \frac{1}{4}r_x$, we have: $$T_y=\sqrt{\frac{1}{4}\frac{r_x}{g}}=\frac{1}{2}\sqrt{\frac{r_x}{g}}=\frac{T}{2}.$$