A rod of mass m and length L is pivoted at point O in a car whose acceleration towards left is $$a0$$. The rod is free to rotate in vertical plane. In equilibrium state, the rod remains horizontal when other end is suspended by a spring of force constant k. The time period of small oscillations of rod is $$T=2$$$$π$$$$D3$$.The value of D is $$___________$$. $$(Given$$, $$k=20Nm$$−$$1$$,$$a0=10ms$$−$$2$$,$$m=1kg$$,$$L=1m$$

Question

A rod of mass m and length L is pivoted at point O in a car whose acceleration towards left is $$a0$$. The rod is free to rotate in vertical plane. In equilibrium state, the rod remains horizontal when other end is suspended by a spring of force constant k. The time period of small oscillations of rod is $$T=2$$$$π$$$$D3$$.The value of D is $$___________$$. $$(Given$$, $$k=20Nm$$−$$1$$,$$a0=10ms$$−$$2$$,$$m=1kg$$,$$L=1m$$

Infinity Learn · Accepted Answer

In equilibrium, $$mgL2=kyL$$∴$$y=mg2k($$ initial extension of $$spring)$$ During oscillations, consider a situation when the angular displacement of rod is θ as shown in the figure.$$ma0=$$ Pseudo forceln the reference frame of car, restoring torque is.τ$$=$$−$$ma0L2sin$$⁡θ$$+myL2cos$$⁡θ−$$k(y$$−Lθ$$)Lcos$$⁡θ Using Eq. (i), τ$$=$$−$$ma0L2$$θ$$+kL2$$θ $$($$∵θ is small $$)$$ τ$$=$$−$$ma0L2+kL2$$θ∴ Iα$$=$$−$$ma0L2+kL$$⋅Lθα$$=$$−$$ma02+kLmL2/3$$θ$$=$$−$$3a02L+3km$$θUsing the given values, we get α$$=$$−$$75$$θ∴     ω$$=75=53$$⇒$$T=2$$πω$$=2$$$$π$$$$53$$∴ $$D=5$$

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