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$.$

Answer 1

Let the radius of the disc $=R$

Therefore according to the question & figure

$Mg-T=ma$

The torque about the center

$=T\times R=1\times \alpha$

$\Rightarrow TR=\left(1/2\right)m{R}^2\times a/R$

$\Rightarrow T=\left(1/2\right)ma$

Putting this value in the equation $\left(1\right)$

we get

$\Rightarrow mg-\left(1/2\right)ma=ma$

$\Rightarrow mg=3/2ma\Rightarrow a=2g/3$