A smooth semicircular wire track of radius R is fixed in a vertical plane as shown. One end of a light spring of natural length 4 3 R is attached to the lower point O of the wire track. A small bead

Question

A smooth semicircular wire track of radius R is fixed in a vertical plane as shown. One end of a light spring of natural length $\frac{3 R}{4}$ is attached to the lower point $O$ of the wire track. A small bead of mass $m$ which can slide on the track, is attached to the other end of the spring. The bead is held stationary at point P such that spring makes an angle of 60° with the vertical. The spring constant is $k = \frac{mg}{R}$ . The restoring force in the spring at that instant is :-

TestHub · Accepted Answer

From the figure, In △ COP CO = CP = R ∠ COP = ∠ CPO = 6 0 ∘ Now, ∠ COP + ∠ CPO + ∠ OCP = 18 0 ∘ ∠ OCP = 6 0 ∘ Thus, ∠ COP is an equilateral triangle So CO = CP = OP = R Thus OP = R Now, natural length of spring = 4 3 R ​ So elongation in spring: x = R − 4 3 R ​ x = 4 R ​ So restoring force = kx = R mg ​ × 4 R ​ = 4 mg ​ Thus, the restoring force in the spring is mg /4 Hence, option (B)

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