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Question:

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

0.2 and 3.5 m

0.29 and 6.5 m

0.2 and 6.5 m

0.29 and 3.5 m

Solution:

All the potential energy is lost by dissipation due to work done by frictional force.
P.E = Work done by friction
Work done by friction from P to Q = μmg(cosθ)PQ = 2√3μmg
Work done by friction from Q to R = μmg × QR
2mg = 2√3μmg + μmg × QR
2 = 2√3μ + μQR - (1)
Since, equal energies are lost along PQ and QR,
Work done by friction is the same on both path lengths,
μmgcosθPQ = μmgQR
PQcosθ = QR - (2)
From (1) and (2) we get,
μ = 0.29, QR = 3.5