What is the value of `tan^(-1)(1/(√3))−sin^(−1) 1+cos^(−1) (1/2)` is ________
a) 2π
b) π/2
c) π
d) 0

Explanation

Take the particle plus the sphere as the system
a. Using conservation of linear momentum, the inear speed o the combined system v is given by
`mv_0=(M+m)v or, v=(mv_0)/(M+m)` ……..i
b. Next we shall use conservation angular momentum about the centre of mass, which is to be taken at the centre of the sphere `(Mgtgtm)` Angular momentum of the particle before collsion is `mv_0(h-R).` If the system rotates with angular speed `omega` after collision, the angular momentum of the system becomes
`(2/5MR^2+mR^2)omega`
Hence, `mv_0(h-R)=(2/5 M+m)R^2omega`
or, ` omega=(mv_0(h-R))/(2/5 M+m))R^2`
c. The sphere will start roling just after the colision if
`v=omegaR i.e. (mv_0)/(M+m)=(mv_0(h-R))/(2/5M+m)R)`
giving`h=((7/5M+2m)/(M+m))R=7/5R`.