Consider a cylinder of mass `M` and radius `R` lying on a rough horizontal plane. It has a plank lying on its top as shown in figure. A force F is applied on the plank such that the plank moves and causes the cylinder to roll the plank always remains horizontal. there is no slipping at any point of contact. Calculate the acceleration of the cylinder and the frictional forces at the two contact.

Explanation

Since, there is no slip at any contact. Therefore, net work done by friction `=0`
In time `t`
Work done by the applied force
`=` kinetic energy of plank and cyliner `(Fcostheta)` (displacement of plank)
`=(1)/(2)m("velocity plank")^(2)`
`+(1)/(2)(1+(1)/(2))M" (velocity of cylinder")^(2)`
`therefore(Fcostheta)((1)/(2)xx2axxt^(2))=(1)/(2)xxm(2at)^(2)`
`+(3)/(4)xxMxx(at)^(2)`
solving we get
`a=(4Fcostheta)/(3M+8m)`
Equation of plank gives
`Fcostheta-f_(1)=m(2a)`
`thereforef_(1)=Fcostheta-2ma`
`=F-(8mFcostheta)/(3M+8m)`
`(3MFcostheta)/(3M+8m)`
Equation of cylinder gives
`f_(1)-f_(2)=M.a`
`thereforef_(2)=f_(1)-Ma`
`=(3MFcostheta)/(3M+9m)-(4MFcostheta)/(3M+8m)`
or `|f_(2)|=(MFcostheta)/(3M+8m)`