[Physics] How do centripetal forces and gravity work for objects in a rotating cylinder

classical-mechanicshomework-and-exercises

The following is a question from a past exam paper that I'm working on, as I have an exam coming soon. I would appreciate any help.

A fairground ride takes the form of a hollow, cylinder of radius $R$ rotating about
its axis. People stand against the cylinder wall when it is stationary and its axis is
vertical. The rotation is then started, and once the cylinder has reached its operating
angular velocity $\omega$, its axis, and the people, are gradually rotated to lie horizontally,as shown in the figure below.

Figure

A person called Alice who has mass $m$, is indicated by $A$ in the figure at the moment
when she subtends angle $\theta$ relative to the downwards vertical direction, as seen from
the cylinder’s axis. Assuming that Alice does not slide relative to the cylinder wall,
and neglecting her size compared to the radius $R$.

  1. Show that the component of force upon Alice from the cylinder wall normal to
    its surface is given by $N = mg\cos\theta +m\omega^{2}R$.

    • The acceleration is towards the center of the cylinder, hence
      $$\sum{F}=ma \Rightarrow N – mg\cos\theta = ma \Rightarrow N = mg\cos\theta + m\omega^{2}R.$$

  2. Show that the component of force upon Alice from the cylinder wall parallel to
    its surface is given by $F = mg\sin\theta$.

    • This follows from the diagram.

  3. Derive an expression for the minimum $\omega$ required to ensure that Alice remains
    in contact with the cylinder wall at all times.

  4. Show that in order for Alice to avoid sliding along the cylinder wall, $\omega^{2} \geq \frac{g\sin\theta}{\mu_{S}R} – \frac{g\cos\theta}{R}$, where $\mu_{S}$ is the coefficient of static friction between her and the wall.

    • $f_{S} \leq\mu_{S}N \Rightarrow mg\sin\theta \leq \mu_{S}(mg\cos\theta +m\omega^{2}R).$
      Rearranging, we get $$\omega^{2} \geq \frac{g\sin\theta}{\mu_{S}R} – \frac{g\cos\theta}{R}.$$

  5. Use the result of question 4 to show that Alice will not slip for any $\theta$ if $\omega^{2} \geq \frac{g}{R}\sqrt{1 + \frac{1}{\mu_{S}^{2}}}$.

Question

I would like to know if the working for the questions above that I have answered are correct. I'm not sure what concept underlies question 3. For question 5, I put $\theta = \pi$, the angle that subtends when she is at the top but that doesn't work.

I'm looking for hints to be able to answer questions 3 and 5. Thank you for your time.

Best Answer

I can offer a partial answer. Sorry if it's too late.

For question 3, the minimum $\omega$ is when the normal force $N$ is $0$. If $N$ is non-zero toward the center, then the wall is keeping her in. If $N$ is non-zero away from the center, our conditions are broken and little Alice is falling, because no normal force will pull her towards the wall. So set $N=0$ and $\theta=\pi$ and you should find that $\omega_{min}=\sqrt\frac{g}{R}$.

For question 5, I've got nothing. You have to show by some trickery that $$\frac{\sin\theta}{\mu_S}-\cos\theta\le\sqrt{1+\frac{1}{\mu_S^2}}.$$ I tried and I don't know really where to go on that. In any event, your exam is probably past (passed?). If you figured 5 out, please share.

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