A uniform round object of mass `m`, radius `R` and moment of inertia about its centre of mas `I_(cm)` is thrown with speed `v`. (Without any rotation) on a rough horizontal surface of coefficient of fricition `mu`. Find (take `(I_(cm))/(mR^(2))=k`)
`(a)` Time after which slipping stops
`(b)` Speed of the round object after slipping stops
`(c )` Angular speed of the round object after slipping stops

Explanation

Force eqution of translatory motion
`f=ma`
`f=muN=mu mg`
`a=mu g`
Equation of motion
`v=v_(0)-at`
`v=v_(0)-mu g t`
Torque equation of rotatory motion
`fR=I_(cm)alpha`
`alpha=(mumgR)/(I_(cm))`
`omega=alphat`
`omega=(mu mg Rt)/(I_(cm))`
For pure rolling
`v=Romega`
`impliesv_(0)-mu g t=(Rmu mg Rt)/(I_(cm))`
`impliesv_(0)=t mu g[1+(1)/(k)]` `[:'(I_(cm))/(mR^(2))=k]`
`impliest=(v_(0)k)/(mu g(1+k))impliesv=v_(0)-mu g(v_(0)k)/((1+k)mu g)`
`impliesv=v_(0)[1+k-k)/(1+k)]impliesv=(v_(0))/(1+k)`
`impliesomega=(v_(0))/(R(1+k))`