# The rotating cylinder problem

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A cylinder with radius R spins around its axis with an angular speed ω. On its inner surface there lies a small block; the coefficient of friction between the block and the inner sur- face of the cylinder is μ. Find the values of ω for which the block does not slip (stays still with respect to the cylinder). Consider the case where the axis is inclined by angle α with respect to the horizon.

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This is a case of rotation in a circle of diameter AB inclined at angle $\alpha$ to the vertical. In the diagram below, the circle is viewed sideways on, along diameter AB.

The force down the surface of the drum is $W\sin\alpha$. Opposing this is the friction force $F\le \mu N$ up the drum, where $N$ is the normal reaction. The critical point to consider, at which $F$ might be insufficient to prevent the object sliding down the drum, is the top of the circle. at A.

Applying Newton's 2nd Law parallel and perpendicular to the surface of the drum we have
$F=W\sin\alpha$
$N+W\cos\alpha=mr\omega^2$.
Therefore
$\mu N \ge F$
$\mu(mr\omega^2-W\cos\alpha) \ge W\sin\alpha$
$\omega^2 \ge \frac{g}{r} (\frac{1}{\mu} \sin\alpha+\cos\alpha)$.

answered May 29, 2017 by (28,466 points)
edited May 29, 2017