Chapter 3.9, Problem 43E

To determine

(a)

To prove:

The expression $L^2={\left(x-r\cos \theta \right)}^2+r^2{\sin }^2\theta$ from the Pythagoras theorem.

To determine

(b)

To prove:

The expression $2\left(x-r\cos \theta \right)\left(\frac{dx}{dt}+r\sin \theta \frac{d\theta }{dt}\right)+2r^2\sin \theta \cos \theta \frac{d\theta }{dt}=0$

by differentiating the equation $L^2={\left(x-r\cos \theta \right)}^2+r^2{\sin }^2\theta$.

To determine

(c)

To calculate:

The speed of the piston at $\theta =\frac{\pi }{2}$.