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Question
A string is wrapped over the edge of a uniform disc and the free end is fixed with the ceiling. The disc moves down, unwinding the string. Find the downward acceleration of the disc.
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Solution
Let the radius of the disc be R.
Let the tension in the string be T.
Let the acceleration of the disc be a.
From the free body diagram, we have
\[mg - T = ma ........(1)\]
Torque about the centre of disc,
\[T \times R = I \times \alpha\]
\[\Rightarrow T \times R = \frac{1}{2}m R^2 \times \frac{a}{R}\]
\[ \Rightarrow T = \frac{1}{2}ma ...........(2)\]
Putting this value in equation (1), we get
\[mg - \frac{1}{2}ma = ma\]
\[ \Rightarrow mg = \frac{3}{2}ma\]
\[ \Rightarrow a = \frac{2g}{3}\]
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