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 particle changes direction from PQ to QR. The distance $x$(=QR) is equal to

• A. $3.5\text{m}$
• B. $2.5\text{m}$
• C. $6.5\text{m}$
• D. $4.5\text{m}$

Answer 1

The correct option is A $3.5\text{m}$

Let $W_f$ denote the work done due to friction and $W_g$ denote the work done due to gravity
$(W_f{)}_{PQ}=(W_f{)}_{QR}$
$-\mu mglcos{30}^{\circ }=-\mu mg\times cos{0}^{\circ }\times x$
here sin $30=\frac{2}{l}\Rightarrow l=4\text{m}$
$-\mu mg\times 4\times \frac{\sqrt{3}}{2}=-\mu mgx$
$x=2\sqrt{3}\text{m}=3.5\text{m}$