A spoon of length \( L \) (modelled as a uniform straight rod) rests within the centre-plane of a hemispherical bowl of radius \( r \), as illustrated below.
The position of the end of the spoon within the bowl (C) is specified by \( \theta \), the angle \( \angle A O C \) made at the centre of curvature of the bowl ( O ) with the horizontal, as illustrated above, and \( d \) is the length of the spoon within the bowl.
If \( \mu \) is the coefficient of static friction between the spoon and the bowl, then the spoon will remain at rest if
\[
\mu \geqslant|f(L, r, \theta)|
\]
where
\[
f(L, r, \theta)=\frac{L(1+\cos (\theta))-4 d \cos (\theta)}{4 d \sin (\theta)-L \sin (\theta)-L \sin (\theta) \cos (\theta)+L \sin ^{2}(\theta)}
\]
with \( d=r \sqrt{2(1+\cos (\theta))} \).
(a) Write a Python function \( \mathrm{f}(\mathrm{L}, \mathrm{r} \), theta) that represents \( f(L, r, \theta) \) and use your function to show that \( f(3,1, \pi / 2)=0.530330 \) (to 6 decimal places).
(b) For the case of \( L=3 \) and \( r=1 \), use Python to plot a graph of \( f(3,1, \theta) \) against \( \theta \).
The remainder of this question considers the case of a spoon of length \( L=3 \) in a bowl of radius \( r=1 \) in more detail.
(c) For the case of a smooth bowl, when \( \mu=0 \), there is only a single position for which the spoon can remain at rest, given by the solution of \( f(3,1, \theta)=0 \).
(i) What method from Units 1 and 2 would you use to solve this equation? Justify your choice.
(ii) What is the value of \( \theta \) that corresponds to the spoon being at rest in the smooth bowl?
(d) When \( \mu>0 \), there is a range of values of \( \theta \) for which the spoon can remain at rest.
(i) Write a Python function theta_min(mu) which returns the minimum value of \( \theta \) for which the spoon can remain at rest for a given coefficient of friction \( \mu \).
(ii) Use your function to calculate this minimum value when \( \mu=\frac{1}{2} \).
(iii) Similarly, write a Python function theta_max (mu) which returns the maximum value of \( \theta \) for which the spoon can remain at rest for a given coefficient of friction \( \mu \).
