Mechanics question
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.
A person called <PERSON> 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 <PERSON> 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 <PERSON> from the cylinder wall normal to its surface is given by $N = mg\cos\theta +m\omega^{2}R$.
2. The acceleration is towards the center of the cylinder, hence $$\sum{F}=ma \Rightarrow N - mg\cos\theta = ma \Rightarrow N = mg\cos\theta + m\omega^{2}R.$$
3. Show that the component of force upon <PERSON> from the cylinder wall parallel to its surface is given by $F = mg\sin\theta$.
4. 1002
This follows from the diagram.
5. Derive an expression for the minimum $\omega$ required to ensure that <PERSON> remains in contact with the cylinder wall at all times.
6. Show that in order for <PERSON> 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.
7. $f_{S} \leq\mu_{S}N \Rightarrow mg\sin\theta \leq \mu_{S}(mg\cos\theta +m\omega^{2}R).$ Rearranging, we get $$\omega^{2} \geq \frac{g\sin\theta}{\mu_{S}R} - \frac{g\cos\theta}{R}.$$
8. Use the result of question 4 to show that <PERSON> 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. 804