A piston (or envelope) is a mechanism that transforms circular motion into longitudinal motion. A point P rotates around a point O. Connected to P there is a rigid rod L so that the other end X of the rod (the piston) moves horizontally along the OX line.

a) Let theta be the angle that forms the OP segment with the OX half-line, r the radius of the circumference which describes P, L the length of the PX rod and x the distance OX.Demonstrate, using Pythagoras theorem, that: (x-r \cos \theta)^{2}+r^{2} \sin ^{2} \theta=L^{2} And, simplifying the last ecuation, check that:

b) Note that in equation (1), r and I are constants, but 0 and x vary (depending on time).Deriving the two sides of the equation (1) from time t, check that: (x-r \cos \theta) \frac{d x}{d t}+r x \sin \theta \frac{d \theta}{d t}=0

c) Suppose that r= 10 cm, L=30cm. Calculate the values of x in the instants that 0=π/3,0=π/2 and 0=π. d) If P is known to rotate to 4 revolutions per minute in the opposite direction toclockwise, use the formula (2) and the previous paragraph to calculate the pistonspeed at the instant in what 0=л/3, 0=π/2 and 0=π.