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Question

A uniform metre scale of length 1 m is balanced on a fixed semi – circular cylinder of radius 30 cm as shown in figure. One end of the scale is slightly depressed and released. The time period (in seconds) of the resulting simple harmonic motion is (Takeg=10ms2)


A
π
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B
π2
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C
π3
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D
π4
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Solution

The correct option is C π3
The magnitude of the restoring
Torque = force × perpendicular distance
= mg × AB
= mg × R sin θ
Since θ is small, sin θ here θ is expressed in radian.
The equation of motion of the scale is

Id2θdt2=mgRθ
or,d2θdt2=(mgRI)θ
ω=mgRIor2πT=mgRIorT=2πImgR.Now I=mL212
Hence T=πL3gR
Using the values L = 1 m, g = 10 ms2 and R = 0.3 m, get T=π3 second. Hence the correct choice is (c)


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