Hard rods form right triangle with base angle α , which rotates at a constant angular velocity ω about a vertical axis AB (see figure). On the rod AC a spring with stiffness k = m ω 2 and unstretched

Question

Hard rods form right triangle with base angle α , which rotates at a constant angular velocity ω about a vertical axis AB (see figure). On the rod AC a spring with stiffness k = m ω 2 and unstretched length ℓ is coiled. One end of the spring is hinged to A and the other end of spring is attached to a sleeve of mass ' m '.

TestHub · Accepted Answer

For the spring to remain undeformed at equilibrium: $m ω^{2} (ℓ sin α) sin α = m g cos α$ .

$ω = \frac{1}{s i n α} \frac{g c o s α}{ℓ}$ - (A)

Displacing the sleeve along the rod changes the required centripetal force; with $k = m ω^{2}$ the spring supplies a net restoring force, so the equilibrium is stable - (B), not (C).

Small oscillations about a stable equilibrium are SHM - (D)

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