A Rectangular Plate Having Edges Of Lengths ‘a’ And ‘b’ Respectively, Hangs Vertically From The Edge Of Length ‘a’. (i) Find The Time Period For Small Oscillations And (ii) The Length Of The Equivalent Simple Pendulum.

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Ans.

Let us consider a rectangular plate having edges of lengths \( a \) and \( b \) respectively. This rectangular plate hangs vertically from the edge of length \( a \). Let \( M \) be the total mass of the rectangular plate and \( G \) be the centre of gravity of the plate.

Fig. 1

(i) Time period for small oscillation:

Let \( I \) be the moment of inertia of the plate about the horizontal axis through the point of suspension \( O \) and perpendicular to the plate lying in the vertical plane.

\( \displaystyle{I=\frac{1}{3}Mb^2} \)
[ to know the derivation of the above equation (CLICK HERE) ]

Now the time period of small oscillation is

\( \displaystyle{T=2\pi\sqrt{\frac{I}{Mg\left(OG\right)}}} \)
[ to know the derivation of the above equation (CLICK HERE) ]

where, \( g \) is the acceleration due to gravity.

From Fig.1 we get, \( OG=\frac{b}{2} \)

therefore, \( \displaystyle{T=2\pi\sqrt{\frac{I}{Mg\left(\frac{b}{2}\right)}}} \)

or, \( \displaystyle{T=2\pi\sqrt{\frac{\frac{1}{3}Mb^2}{Mg\left(\frac{b}{2}\right)}}} \)

or, \( \displaystyle{T=2\pi\sqrt{\frac{2b}{3g}}}\tag{1} \)

(ii) Length of the equivalent simple pendulum:

Let, \( L \) be the length of the equivalent simple pendulum, then

\( \displaystyle{T=2\pi\sqrt{\frac{L}{g}}} \)

Comparing with the equation (1), we get

\( \displaystyle{L=\frac{2}{3}b} \)

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