A solid sphere of mass `M` is placed on the top of a plank of the same mass, after giving an angular velocity `omega_(0)` at `t= 0`. Find the velocity of the plank and the sphere when the sphere starts rolling:,

Explanation

Step 1: Free body diagram

Step 2: Equation of motion,
`muMg=MAimpliesA=mug` (backward)
(A will be positive)
sphere
`muMg=MAimplies a=mug` (forward)
(a will be positive)
Step: Torque equation
`muMgR=(2/5MR^(2))alpha`
`alpha=5/2(mug)/R` (anticlockwise)
(it will be negative )
Step 4: constraint relation

`V_(P)=V=omegaR=v`
`(0+mugt)=(omega_(0)-5/2(mug)/Rt)R-(0+mug t)`
`(2mu"gt"+5/2mu"gt")=omega_(0)Rimplies(9mut)/2=omega_(0)R`
`implies t-(2omega_(0)R)/(9mug)`
Velocity of the plane when rolling starts
`V=0+mu"gt" `
`V_("plank")=mugxx(2omega_(0)R)/(9mug)=2/9omega_(0)R`
velocity of the sphere when rolling starts
`V_("sphere")=0+mug t=2/9omega_(0)R`