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Question

A solid cylinder of mass m is kept on the edge of a plank of mass 3m and length 15 m which in turn is kept on smooth ground. The coefficient of friction between the plank and the cylinder is 0.1. The cylinder is given an impulse which imparts it a velocity of 6 m/s but no angular velocity. Find the time after which the cylinder will start pure rolling.


A
1.8 s
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B
3.8 s
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C
4.8 s
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D
2.8 s
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Solution

The correct option is A 1.8 s
The cylinder will slip on the plank and after some time, it will start pure rolling. The necessary torque will be provided by the friction from the plank.
FBD of cylinder and plank-


For cylinder,
N=mg and
f=μN=μmg=ma
μmg=ma
a=μg
a=0.1×10
a=1 m/s2
Let us suppose the time after which the cylinder starts pure rolling is t.
For the bottom most point (point of contact) of the cylinder, velocity due to pure translation,
v1=u1+at
v1=6t ..............(1)
For the cylinder,
fR=Iα
α=fRI
α=μmgRmR2/2
α=2μgR=2×0.1×10R=2R
Now, ωf=ωi+αt
ωf=0+2tR=2tR
For the bottom most point (point of contact) of the cylinder, velocity due to pure rotation,
v2=ωfR=2t ...........(2)
So, net velocity of bottom most point,
v=v2v1=6t2t=63t .....(3)
[from (1) and (2)]
For plank, f=3ma
μmg=3ma
a=μg3=0.1×103=13 m/s2
So, v=u+at
v=t3.........(4)

For pure rolling, the plank should have the same velocity as the velocity of the point of contact.
From (3) and (4),
63t=t3
t=1.8 s

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