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Question

A particle of mass m, moves along the uniformly rough track PQR as shown in the figure. The coefficient of friction (μs=μk=μ), between the particle and the rough track is equal throughout the track. The particle is released from rest, from the point P and it comes to rest again at point R. The energy lost by the particle, over the parts PQ and QR of the track are equal and no energy is lost when the particle changes direction from PQ to QR. Then, values of the distance x=QR and coefficient of friction μ are respectively close to:


A
x=3 m
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B
μ=13
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C
x=23 m
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D
μ=123
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Solution

The correct options are
C x=23 m
D μ=123

From the figure of the track,
sin30=hd=2d
d=4 m (hypotenuse)
cos30=basehypotenuse=32
Base =23 m
FBD of the particle:


Energy lost over path PQ=work done by friction=fs×d
Energy lost over path PQ=μN×d
=μmgcosθ×d
=μmgcosθ×4 ...(i)

Similarly, energy lost over path QR =fs×x
μN×x
Energy lost over path QR=μmgx ...(ii)


Given Energy lost over path PQ=energy lost over path QR
μmgcosθ×4=μmgx
x=4cos30
x=23 m

At point R, particle comes into rest, hence total energy lost by the particle
=Total energy at PTotal energy at R
=mgh0=mgh

along PQ & QR, the particle lost the same energy
Total energy lost=2×(Energy lost in QR)
mgh=2μmgx
or h=2μx
or 2=2×μ×23
μ=123

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