A person of mass M is setting on a swing of length L and swinging with an angular amplitude `theta_(0)` if the person stands up when the swing passes through its lowest point, the work done by him, assuming that his centre of mass moves by a distance `l(lt lt L)`

Options

A. Mgl
B. `Mgl(1+(theta_(0)^(2))/(2))`
C. `Mgl(1-theta_(0)^(2))`
D. `Mgl(1+theta_(0)^(2))`

Answer

D

Explanation

When man is passing through lowest position its angular momentum is conserved
MvL=Mv(L-l)
`v=((vL)/(L-l)),v=A omega=(L theta_(0))(sqrt((g)/(L)))`
Work energy theorem at lowest position :
`W_(m)+W_(g)=KE_(f)-KE_(i)`
`W_(m)-Mgl=(1)/(2)M(v)^(2)-(1)/(2)Mv^(2)`
`W_(m)-Mgl=(1)/(2)ML^(2)theta_(0)^(2)(g)/(L)[((L)/(L-l))^(2)-1]`
`W_(m)=Mgl+(1)/(2)M theta_(0)^(2)gL[1+2(l)/(L)-1]=Mgl[1+theta_(0)^(2)]`