Question

A light non-stretchable thread is wound on a massive fixed pulley of radius $R$. A small body of mass $m$ is tied to the free endof the thread. At a moment $t=0$ the system is released and starts moving. Find its angular momentum relative to the pulley axle as a function of time $t$.

Answer 1

On the given system the weight of the body $m$ is the only force whose moment is effective about the axis of pulley. Let us take the sense of $\overrightarrow{\omega }$ of the pulley at an arbitrary instant as the positive sense of axis of rotation (z-axis)
As $M_z\left(0\right)=0$, so, $\Delta M_z=M_z\left(t\right)=\int N_{z}dt$
So, $M_z\left(t\right)={\int }_0^{t}mgRdt=mgRt$