Chapter 3.10, Problem 43E

To determine

(a)

To prove:

$L^2={\left(x-r\cos \theta \right)}^2+r^2{\sin }^2\theta$

To determine

(b)

To prove:

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

To determine

(c)

Calculate the speed of the piston when $\theta =\frac{\pi }{2}$