1 Answers

Aman Bansal
592 Points
9 years ago

Dear Anurag,

Method I: Torque and Angular Acceleration
Choose the origin for calculating torques and angular momentum as the pivot point of thependulum, and let ! be the angular displacement of the pendulum string from thevertical. The only forces on the pendulum are the gravitational force on the pendulumbob (the pointlike object) and the tension in the string. With respect to the pivot, thetension exerts no torque. The moment arm for the gravitational torque is r" = l sin ! , fora net torque
! net = #mgl sin " .
The minus sign in Equation (1.1) is crucial; the torque will act to restore the angle ! toits equilibrium value ! = 0 . If ! > 0 , ! net < 0 and if ! < 0 , ! net > 0 (this assumes thatthe angle ! is restricted to the range #! < " < ! , implied by the small-angleapproximation sin ! 0 ! ! 0 ).
With the assumption of a massless string, the moment of inertia about the pivot point is
I pivot = ml 2 .
The angular acceleration ! is related kinematically to the displacement angle ! by
d 2!.dt 2
The torque, moment of inertia and angular acceleration are related by
! net = I pivot " .
Combining Equations (1.1), (1.2), (1.3) and (1.4) yields the answer.

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Aman Bansal

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