Question

A cylinder is sandwiched between two planks. Two constant horizontal forces $F$ and $2F$ are applied on the planks as shown. If there is no slipping at the top and bottom of cylinder, then

• A. acceleration of the centre of mass of cylinder is $\frac{21F}{26M}$
• B. acceleration of the centre of mass of cylinder is $\frac{13F}{26M}$
• C. acceleration of the top plank is $\frac{F}{26M}$
• D. acceleration of the top plank is $\frac{F}{13M}$

Answer 1

Answer: (A), (C)

The correct options are
A acceleration of the centre of mass of cylinder is $\frac{21F}{26M}$
C acceleration of the top plank is $\frac{F}{26M}$

Equations of motion are,
$F+f_1=M{a}_1...\left(i\right)$
$f_1+f_2=M{a}_2...\left(ii\right)$
$2F-f_2=2M{a}_3...\left(iii\right)$
$\alpha =\frac{(f_1-f_2)R}{\frac{1}{2}M{R}_2}$
or $\alpha =\frac{2(f_1-f_2)}{MR}....\left(iv\right)$
For no slipping condition. $a_2+R\alpha =-a_1...\left(v\right)$ and
$a_2-R\alpha =a_3...\left(vi\right)$
We have six unknowns,$f_1,f_2,a_1,a_2,a_3$ and $\alpha$ Solving the above six equations. we get
$a_1=\frac{21F}{26M}$ and $a_2=\frac{F}{26M}$