Question

A solid cylinder $C$ and a hollow pipe $P$ of same diameter are in contact when they are released from rest as shown in the figure on a long incline plane. Cylinder $C$ and pipe $P$ roll without slipping. Determine the clear gap (in m) between them after $2\sqrt{3}$ seconds.

Answer 1

$ForanyBodymgsin\theta -f=maandfr=I\alpha anda=\alpha r\Rightarrow mgsin\theta -\frac{I\alpha }{r}=ma\Rightarrow mgsin\theta =\frac{Ia}{r^2}+ma\Rightarrow a=\frac{mgsin\theta }{(\frac{I}{r^2}+m)}\Rightarrow a_{sc}=\frac{2}{3}gsin\theta and{a}_{hc}=\frac{1}{2}gsin\theta relativeacceleration=\frac{2}{3}gsin\theta -\frac{1}{2}gsin\theta =\frac{1}{6}gsin\theta =\frac{g}{12}gapaftertimet=2\sqrt{3}=\frac{1}{2}\frac{g}{12}\times t^2=5s$