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Question

A horizontal plank of mass m is lying on a smooth horizontal surface. A sphere of same mass and radius r is spinned to an angular frequency ω0 and gently placed on the plank with its axis horizontal as shown in the figure. If coefficient of friction between the plank and sphere is μ. The distance moved by the plank till the sphere starts pure rolling on the plank. The plank is long enough.

734237_ef2276ff8ca3422bb7065c8663708f50.JPG

A
2ω20r281μg
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B
2ω20r291μg
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C
ω20r29μg
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D
ω20r2μg
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Solution

The correct option is A 2ω20r281μg
Normal reaction between plank and sphere is: N=mg
Friction acting is: Fr=μN=μmg
Acceleration of sphere is: a=Frm=μg
In plank's frame of reference, acceleration of sphere is a=2a=2μg

From first equation of motion,
v=u+at
v=2μgt.................(i)

Torque acting on the sphere is:
τ=Frr=μmgr
Iα=μmgr
25mr2α=μmgr
α=5μg2r

From first equation of motion for rotational motion,
ω=ωoαt
ωr=ωor5μg2rrt
At pure rolling, v=rω
v=ωor5μgt/2...........(ii)

From (i) & (ii),
t=2ωor9μg

Distance moved by plank is:
s=ut+.5at2
s=.5×μg×4ω2or281μ2g2
s=2ω2or281μg

773517_734237_ans_3730c523cbcf41d79fe24bee31353965.jpg

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