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 rπmetre and spring constant K. Initially, both the balls are displaced by an angle 8 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 dia PQ is

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

Answer 1

Answer: (B)

$2R\theta\sqrt{\frac{K}{m}}$

Answer 2

In stretched position of spring.

system P.E. = 2 × $\frac{1}{2}$k × 2x² = 4kx²

In mean position, both balls have kinetic energy only;

K.E. = 2$\left\lbrack \frac{1}{2}mv^{2} \right\rbrack = mv^{2}$

but P.E. = K.E.

∴ 4Kx² = mv²

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