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Question

Find the acceleration of a system consisting of a solid cylinder of mass m and radius R and a plank of mass M placed on a smooth surface if it is pulled with a force F as shown in the figure. Given that sufficient friction is present between the cylinder and the plank surface to prevent sliding of the cylinder.


A
Fm+M
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B
F2m+M
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C
Fm+2M
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D
F3m+M
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Solution

The correct option is D F3m+M
Let the acceleration of the system be a. Free body diagram of the given system is


From FBD,
FTf=Ma(i)
Tf=ma(ii)
Torque experienced by the cylinder is,
f×R=Iα
f×R=12mR2α(iii)
Also, αRa=a (condition for pure rolling)
2a=αR
a=αR2(iv)
Now, f=mRα2 (from equation (iii))
αR=2fm
On adding equation (i) and (ii)
F2f=a(m+M)
Fa(m+M)=2f
f=Fa(m+M)2
Substituting f and α in the equation (iii), we get,
Fa(m+M)2×R=12mR2×2aR
Fa(m+M)=2ma
a=F3m+M

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