# Tutor profile: Alvin S.

## Questions

### Subject: Physics (Newtonian Mechanics)

How does friction act on the wheels of a bicycle while pedaling and braking?

While pedaling, the force to move the bicycle is applied by the rear wheel and hence friction acts forward on the rear wheel. The front wheel, however, is rotated by the friction on the road and hence is acting backwards. While braking, friction acts backwards on the wheel(s) on which the braking force is applied while the wheel that is not under the action of the braking force has friction acting on it in the forward direction to reduce the angular momentum of that wheel.

### Subject: Basic Math

If $$ \cot ^{-1} \{ \sqrt {\cos \alpha} \} - \tan ^{-1} \{ \sqrt {\cos \alpha} \} = x$$, find $$\sin x$$

Given $$ \cot ^{-1} \{ \sqrt {\cos \alpha} \} - \tan ^{-1} \{ \sqrt {\cos \alpha} \} = x$$ $$ \Rightarrow \frac{\pi}{2} - 2\tan ^{-1} \{ \sqrt {\cos \alpha} \} = x $$ $$(\cot ^{-1} \{a\}= \frac{\pi}{2} - \tan ^{-1} \{ a \} )$$ Therefore $$ \sin x = \sin (\frac{\pi}{2} - 2\tan ^{-1} \{ \sqrt {\cos \alpha} \} ) $$ $$ = \cos (2\tan ^{-1} \{ \sqrt {\cos \alpha} \})$$ $$ ( \sin ( \frac{ \pi }{ 2 } - \theta) = \cos (\theta) ) $$ $$ = \frac{1 + \tan^2 (\tan ^{-1} \{ \sqrt {\cos \alpha} \})}{1 - \tan^2 (\tan ^{-1} \{ \sqrt {\cos \alpha} \})}$$ $$ (\cos ^2 \theta = \frac{1 + \tan ^2 \theta}{1 - \tan ^2 \theta}) $$ $$ = \frac{1 + \cos \alpha}{1 - \cos \alpha} $$ $$ = \frac{2\sin ^2 {\frac{\alpha}{2} }}{2\cos ^2 {\frac{\alpha}{2} }}$$ $$ (1 + \cos 2\theta = 2 \cos ^2 \theta, 1- cos 2\theta = 2 \sin ^2 \theta) $$ $$ = 2\tan ^2 \frac{\alpha}{2} $$

### Subject: Aerospace Engineering

Which shape would you prefer as a design for a pressure vessel - cylinder or sphere?

Both cylindrical and spherical pressure vessels have their own merits and demerits. Cylindrical vessels are used in transportation as spherical vessels take up more space as they leave unused space when stacked together. But in situations where space isn't a consideration, spherical vessels are better as the walls of the pressure vessel are under lesser stress and hence the wall thickness required is reduced.

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