Question

A uniform cylinder of mass $m$ and radius $R$ rolls without slipping down a slope of angle $\theta$ with the horizontal. The cylinder is connected to a spring having spring constant $K$ while the other end of the spring is connected to a rigid support at $P$. The cylinder is released when the spring is unstretched. The maximum distance that the cylinder travel along the plane is

• A. $\frac{3mg\sin \theta }{4K}$
• B. $\frac{mg\sin \theta }{K}$
• C. $\frac{2mg\sin \theta }{K}$
• D. $\frac{4mg\sin \theta }{3K}$

Answer 1

The correct option is C $\frac{2mg\sin \theta }{K}$

Let us suppose the maximum distance traveled by cylinder is $x$ then this will be maximum extension in spring.
F.B.D of cylinder in equilibrium position is shown in figure.


Work done by spring force, $W_s=-\frac{1}{2}K{x}^2$,
Work done by gravity, $W_g=mg\sin \theta x$
From work energy theorem,
Work done by all forces $=$ change in kinetic energy
$\Rightarrow W_s+W_g=K{E}_f-K{E}_i$
$\Rightarrow -\frac{1}{2}K{x}^2+mg\sin \theta x=0-0$
[ cylinder will be in rest in initial and final position ]
$\Rightarrow x(-\frac{1}{2}Kx+mg\sin \theta )=0$
$\Rightarrow x=\frac{2mg\sin \theta }{K}$ or $x=0$