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 $\mu$. 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 $\mu$ and the distance $x(=QR)$, are respectively close to

• A. $0.2$ and $6.5m$
• B. $0.2$ and $3.5m$
• C. $0.29$ and $3.5m$
• D. $0.29$ and $6.5m$

Answer 1

The correct option is C $0.29$ and $3.5m$

All the potential energy is lost by dissipation due to work done by frictional force.
$P.Elost=$ Work done by friction

Work done by friction from P to Q $=\mu mg\left(\cos \theta \right)PQ=2\sqrt{3}\mu mg$
Work done by friction from Q to R $=\mu mg\times QR$
$2mg=2\sqrt{3}\mu mg+\mu mg\times QR$
$2=2\sqrt{3}\mu +\mu QR$ --- (1)
Since equal energies are lost along $PQ$ and $QR$,
Work done by friction is the same on both path lengths.
$\mu mg\cos \theta \times PQ=\mu mg\times QR$
$PQ\cos \theta =QR$ --- (2)
From (1) and (2) we get,
$\mu =0.29,QR=3.5$