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Simple Harmonic Motion — Physics STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 SciencePhysicsSimple Harmonic Motion3 Marks
Question
A rectangular plate of sides a and b is suspended from a ceiling by two parallel strings of length L each figure The separation between the strings is d. The plate is displaced slightly in its plane keeping the strings tight. Show that it will execute simple harmonic motion. Find the time period.
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Answer

Here we have to consider oscillation of centre of mass, Driving force $\text{F}=\text{mg}\sin\theta.$ Acceleration $=\text{a}=\frac{\text{F}}{\text{m}}=\text{g}\sin\theta.$ For small angle $\theta,\ \sin\theta=\theta.$$\therefore\text{a}=\text{g}\theta=\text{g}\Big(\frac{\text{X}}{\text{L}}\Big)$[where g and L are constant]
$\therefore\text{a}\propto\text{x},$
So the motion is simple Harmonic Time period $\text{T}=2\pi\sqrt{\frac{\text{Displacement}}{\text{Acceleration}}}$$=2\pi\sqrt{\frac{\text{x}}{\Big(\frac{\text{gx}}{\text{L}}\Big)}}$
$=2\pi\sqrt{\frac{\text{L}}{\text{g}}}$

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