Question

A ball is rolling without slipping in a spherical shallow bowl (radius $R$) as shown in the figure and is executing simple harmonic motion. If the radius of the ball is doubled, the period of oscillation

• A. Increases slightly
• B. Is reduced by a factor of $1/2$
• C. Is increased by a factor of $2$
• D. Decreases slightly

Answer 1

The correct option is D Decreases slightly

$a=\frac{gsin\theta }{1+\frac{I}{m{r}^2}}=\frac{5gsin\theta }{7}$

$\alpha =\frac{5gsin\theta }{7r}$

now we will calculate the angular velocity about center of sphere

${\omega }^{\prime }=\frac{\omega r}{R-r}$

${\alpha }^{\prime }=\frac{\alpha r}{R-r}$

${\alpha }^{\prime }=\frac{5gsin\theta }{7(R-r)}$

$T=2\pi \sqrt{\frac{7(R-r)}{5g}}$

hence wih increase in r, T will decrease.