A uniform semicurcular ring having mass m and radius `r` is hanging at one of its ends freely as shown if Fig. The ring is slightly disturbed so the it oscillates in its own plane. The time period of oscillation of the ring is

Options

A. `2pisqrt((r)/(g(1+(1)/(pi^2))))`
B. `2pisqrt((r)/(g(1+(4)/(pi^2))^((1)/(2))))`
C. `2pisqrt((r)/(g(1-(2)/(pi^2))^((1)/(2))))`
D. `2pisqrt((2r)/(g(1+(4)/(pi))^((1)/(2))))`

Answer

D

Explanation

`T=2pisqrt((I_0)/(mgd))`,`I_0=I_c+mr'^(2)impliesmr^2=I_C+mr'^(2)`
`impliesI_C=m(r^2-r'^(2))`
`I_0=I_C+md^2m(r^2-r'^(2))+m(r^2+r'^(2))=2mr^2`
`d=sqrt(r^2((2r)/(pi))^2)=rsqrt(1+(4)/(pi^2))`
`T=2pisqrt((2r)/(g(1+(4)/(pi))^((1)/(2))))`