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प्रश्न
A uniform disc of mass m and radius r is suspended through a wire attached to its centre. If the time period of the torsional oscillations be T, what is the torsional constant of the wire?
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उत्तर
It is given that:
Mass of disc = m
Radius of disc = r
The time period of torsional oscillations is T.
Moment of inertia of the disc at the centre, I \[= \frac{m r^2}{2}\]
Time period of torsional pendulum\[\left( T \right)\] is given by,
\[T = 2\pi\sqrt{\frac{I}{k}} \]
where I is the moment of inertia, and
k is the torsional constant.
On substituting the value of moment of inertia in the expression for time period T, we have:
\[T = 2\pi\sqrt{\frac{m r^2}{2k}}\]
\[\text { On squaring both the sides, we get: }\]
\[ T^2 = 4 \pi^2 \frac{m r^2}{2k} = 2 \pi^2 \frac{m r^2}{k}\]
\[ \Rightarrow 2 \pi^2 m r^2 = k T^2 \]
\[ \Rightarrow k = \frac{2 \pi^2 m r^2}{T^2}\]
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