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Question

Pendulum A is a physical pendulum made from a thin, rigid and uniform rod whose length is d. One end of this rod is attached to the ceiling by a frictionless hinge, so that the rod is free to swing back and forth. Pendulum B is a simple pendulum whose length is also d. Obtain the ratio TATB of their periods for small angle oscillations.

A
1:2
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B
2:1
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C
1:1
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D
1:3
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Solution

The correct option is C 1:1
For physical pendulum time period , TA=2πImgl where I= moment of inertia about point O at the end of rod, i.e I=d0y2(m/d)dy=md23 and l=d/2 distance from O to COM.
thus, TA=2πmd2/3mg(d/2)=2π2d3g
For simple pendulum time period . TB=2πlg=2πdg
TATB=(2d/3g)(g/d)=0.821
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