The string of a simple pendulum replaced by a uniform rod of length `L` and mass `M` while the bob has a mass `m`. It is allowed to make small oscillation. Its time period is

Options

A. `2pisqrt(((2M)/(3m))(L)/(g))`
B. `2pisqrt((2(M+3m)L)/(3(M+2m)g))`
C. `2pisqrt(((M+m)/(M+3m))(L)/(g))`
D. `2pisqrt(((2m+M)/(3(M+2m)))(L)/(g))`

Answer

B

Explanation

When the rod is replaced by the string the simple pendulum will act as compound pendulum for which time period is given by
`T=2pisqrt((I_0)/(Mgd))` ..(i)
`I_0=`moment of inertia about of suspension
`=(ML^2)/(3)+mL^2=((M+3m)L^2)/(3)`
`M'=`total mass`=(M+m)`
`d=`separation between point of suspension and centre of mass of the pendulum
`=(M((L)/(2))+m(L))/((M+m))=((M+2m)L)/(2(M+m))`
Substituting the values in Eq. (i) we get
`impliesT=2pisqrt((2(M+3m)L)/(3(M+2m)g))`