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A light non-stretchable thread is wound ...

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 end of 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 funciton of time t.

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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 `omega` of the pulley at an arbitrary instant as the positive sense of axis of rotation (z-axis)
As `M_z(0)=0`, so, `DeltaM_z=M_z(t)=intN_zdt`
So, `M_z(t)=underset(0)overset(t)intmgRdt=mgRt`
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