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Question

In the system shown in figure above, the masses of the bodies are known to be m1 and m2, the coefficient of friction between the body m1 and the horizontal plane is equal to k, and a pulley of mass m is assumed to be a uniform disc. The thread does not slip over the pulley. At the moment t=0 the body m2 starts descending. Assuming the mass of the thread and the friction in the axle of the pulley to be negligible, find the work performed by the friction forces acting on the body m1 over the first t seconds after the beginning of motion.
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A
A=km1(m1km1)g2t2m+2(m1+m2)
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B
A=km1(m1+km1)g2t2m+2(m1m2)
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C
A=km1(m1+km1)g2t2m+2(m1+m2)
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D
None of these
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Solution

The correct option is D A=km1(m1km1)g2t2m+2(m1+m2)
As the system (m+m1+m2) is under constant forces, the acceleration of body m1 and m2 is constant. In addition to it the velocities and accelerations of bodies m1 and m2 are equal in magnitude (say v and w) because the length of the thread is constant.
From the equation of increment of mechanical energy i.e.
ΔT+ΔU=Afr, at time t, when block m1 is distance h below from initial position corresponding to t=0, 12(m1+m2)v2+12(mR22)v2R2m2gh=km1gh (1)
(an angular velocity ω=vR for no slipping of thread.)
But v2=2wh
So using it in (1), we get
w=2(m2km1)gm+2(m1+m2) (2)
Thus the work done by the friction force on m1
Afr=km1gh=km1g(12wt2)
=km1(m1km1)g2t2m+2(m1+m2)

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