Question

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 ana 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 contacts.

Answer 1

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 cylinder $\left(F\cos \theta \right)$
(displacement of plank)
$=\frac{1}{2}m$ $(velocityofplank{)}^2$
$+\frac{1}{2}(1+\frac{1}{2})M$
$(velocityofcylinder{)}^2$
$\therefore \left(F\cos \theta \right)(\frac{1}{2}\times 2a\times t^2)=\frac{1}{2}\times m(2at{)}^2+\frac{3}{4}\times M\times (at{)}^2$
Solving, we get
$a=\frac{4F\cos \theta }{3M+8m}$
Equation of plank gives,
$F\cos \theta -f_1=m\left(2a\right)$
$\therefore f_1=F\cos \theta -2ma$
$=F-\frac{8mF\cos \theta }{3M+8m}$
$=\frac{3MF\cos \theta }{3M+8m}$
Equation of cylinder gives
$f_1-f_2=M\cdot a$
$\therefore f_2=f_1-Ma=\frac{3MF\cos \theta }{3M+8m}-\frac{4MF\cos \theta }{3M+8m}$
or $\left|f_2\right|=\frac{MF\cos \theta }{3M+8m}$