Question

Two identical balls A and B of mass $m$ kg are attached to two identical massless springs. The spring mass system is constrained to move inside a rigid pipe bent in the form of a circle as shown in figure. The pipe is fixed in a horizontal plane. The centres of the balls can move in a circle of radius $R$ metre. Each spring has a natural length of $R\pi$ metre and spring constant $K$. Initially, both the balls are displaced by an angle $\theta$ radian with respect to diameter PQ of the circles and released from rest. The speed of ball A when A and B are at the two ends of diameter PQ is

• A. $R\theta \sqrt{\frac{m}{K}}$
• B. $2R\theta \sqrt{\frac{K}{m}}$
• C. $2R\theta \sqrt{\frac{m}{K}}$
• D. $R\theta \sqrt{\frac{K}{m}}$

Answer 1

The correct option is B $2R\theta \sqrt{\frac{K}{m}}$

In stretched position of spring,
system P.E. $=2\times \frac{1}{2}k\times {\left(2x\right)}^2=4k{x}^2$

In mean position, both balls have kinetic energy only;

$K.E.=2\left[\frac{1}{2}m{v}^2\right]=m{v}^2$

but P.E $=$ K.E.

$\therefore 4K{x}^2=m{v}^2$

$x=R\theta$

or $v=2x\sqrt{\frac{K}{m}}=2R\theta \sqrt{\frac{K}{m}}$