Question

A string is wrapped around a uniform solid cylinder of radius r, as shown in (Figure 1). The cylinder can rotate freely about its axis. The loose end of the string is attached to a block. The block and cylinder each have mass m.

Part A. Find the magnitude α of the angular acceleration of the cylinder as the block descends. Express your answer in terms of the cylinder's radius r and the magnitude of the acceleration due to gravity g.

Answer 1

$\textrm{Equation of motion of the object is,}$

$\textup{mg-T=ma}............(1)$

Where, m=mass, T= Tension and a= acceleration

$\textup{Now,Torque, }\tau =Tr=I\alpha \Rightarrow T=\frac{I\alpha }{r}=\frac{Ia}{r^2}$

$\textrm{Putting the expression for the tension T in Eqn.(1),}$

$mg-\frac{Ia}{r^2}=ma$

$\Rightarrow a=\frac{mg}{m+\frac{I}{r^2}}.........(2)$

$\textit{Put I}=\frac{1}{2}mr^2\textrm{ in equation 2}$

$a=\frac{mg}{m+\frac{mr^2}{2r^2}}=\frac{2}{3}g......................(3)$

$\textrm{Therefore, angular acceleration of the cylinder,}$

$\alpha =\frac{a}{r}=\frac{2g}{3r}$