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I am tasked with the following problem: suppose I have a hollow drum of radius $R$ and mass $M$ (moment of inertia $MR^2$). The drum has an axle of radius $r < R$. The drum’s axle is placed on frictionless rails, such that it hovers over the ground. I wish to find the point above the centre of the axle at which I should strike it so that it rolls without slipping. So, just like the standard pool-ball problem, I note that $\Delta L = h \Delta p \Rightarrow MR^2 \omega = hMr\omega \Rightarrow h = \frac{R^2}{r}$. However, $\frac{R}{r}>1$ which implies that $h>R$. How is this possible? Does this just mean it is impossible to cause this drum to roll without slipping as long as it has an axle, or is my physics wrong? Thank you!

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  • $\begingroup$ What is $h$, and what is $r$? $\endgroup$ – PeaBrane Mar 19 '18 at 2:51
  • $\begingroup$ @PeaBrane $r$ as states is the radius of the axle, and $h$ is the distance from the axle at which I should strike it $\endgroup$ – user107224 Mar 19 '18 at 11:40
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You’ve omitted the force, hence impulse, hence some $\Delta p$, from the friction at the contact point. When you push above the center, this acts to increase $L$ and $\omega$ while reducing $v$.

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  • $\begingroup$ Ah sorry, in the given problem the rails are frictionless! $\endgroup$ – user107224 Mar 19 '18 at 11:39
  • $\begingroup$ It's not that it "has an axle", it's that you want it to roll on the axle without any friction. So you can't just hit it at R or less; that won't provide enough angular momentum. $\endgroup$ – Bob Jacobsen Mar 19 '18 at 15:10

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