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Question

A plank is kept over a cylinder as shown in figure and there is no slipping at any contact. 15 N is the horizontal force applied on the plank. Then, identify the correct statement(s):


A
Acceleration of plank is 607 m/s2
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B
Acceleration of centre of mass of the cylinder is 307 m/s2
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C
Friction force between plank and cylinder is 457 N
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D
Friction force between cylinder and ground is 157 N
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Solution

The correct option is D Friction force between cylinder and ground is 157 N
Let the acceleration of the centre of mass (acom) of the cylinder be a.
Since the tangential acceleration (at=rα) at the top-most point and acceleration of the centre of mass of the cylinder (acom) are in the same direction,
Acceleration of topmost point is,
a=acom+at ...(i)
From condition of no slipping on ground,
acom=rα=a
From Eq.(i),
a=rα+rα=2rα
a=2a ...(ii)
Since, slipping is not taking place at any contact, therefore acceleration of plank (aP) will be equal to the acceleration of the point of contact with the cylinder i.e aP=2a.
Friction(f) will act in forward direction on cylinder and in backward direction on the plank to prevent slipping (at top-most position).


For plank, on applying Newton's second law along its direction of acceleration,
15f=1×2a=2a ...(1)
For cylinder, along its direction of acceleration of COM
f+f=2a ...(2)

[For cylinder, the point of contact with the ground will tend to slip backwards on application of 15 N, hence friction f will act along forward direction.]

On taking instantaneous axis of rotation (I.A.R) at the point of contact of cylinder and ground (because point of contact for cylinder with ground is always at rest)
Applying equation of torque for cylinder about I.A.R,
τ=Iα
Normal reaction from ground, f, mg will pass through I.A.R, their torque will vanish.
τf=Iα
2rf=3mr22.ar
[I=I0+md2,I0=mr22, d=r and a=αr for pure rolling]
f=3ma4
f=3×2×a4=3a2 ...(3)
On putting Eq.(3) in Eq.(1),
153a2=2a
a=307 m/s2 is the acceleration of COM of cylinder.
Acceleration of plank is,
aP=2a=607 m/s2
On putting the value of a in Eq.(3), we get friction between plank and cylinder
f=457 N
On putting f in Eq. (2), we get friction between cylinder and ground
f=157 N
Hence options (a), (b), (c), (d) are correct.

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