# A hollow sphere is tied to string and the system acts like a simple pendulum . Let the time period of the system be T . By doing which of the following , the time period of the system increases . (Do not consider extension of the string due to mass ) .

$(a)\;By\;filling\;it\;completely\;with\;sand\qquad(b)\;By\;filling\;half\;of\;it\;with\;sand\qquad(c)\;By\;filling\;it\;completely\;with\;water\qquad(d)\;None\;of\;these$

Answer : (b) By filling half of it with sand
Explanation :
Time period (T) = $\;2 \pi \sqrt{\large\frac{l}{g}}$
l - effective length , i . e . it is the length total length till centre of mass of the sphere .
That is , $\; l= l_{0}+r\;$ (for hollow sphere)
$l_{0}\;$- length of the string
$r\;$ - radius of the sphere as the centre of mass lies of the centre
For a sphere fully filled with sand or anything else , the centre of mass lies at the centre only . But for a sphere partially filled with something , the centre of mass shifts downwards and the effect length is $\;l_{0}+r+K\;$ where K is the shift .
The time period increases in the case of option (b)