Calculus - Engine Design Question?

Question

"In an engine, a connecting rod 18 cm long is fastened to a crank of radius 6 cm at point P. The crankshaft rotates counterclockwise at a constant rate of 200 revolutions per minute. The horizontal velocity (cm/min) of point P is:
v = -2400πsinθ
where θ is the central angle of the crankshaft. What values of θ produce a maximum horizontal velocity?"

Answer 1

To find the maximum, take the first derivative of v with respect to theta (q) and set to zero:

dv/dq = -2400*pi*cos(q) = 0

now cos(q) = 0 when q = +/- pi/2, +/- 3*pi/2 +/- 5*pi/2 ...

or q = +/- n*pi/2 where n = 1, 3, 5, ... or n = odd numbers

Now sin(q) = +/- 1 since sin (n*pi/2) = +/-1 for n odd

Maximum speed occurs than for values of q given above. Note that the sign of the velocity changes due to the direction of motion changing. Thus, the speed (magnitude of the velocity) is maximized, but the numerical value of v given by above equation is max only for the + values.

For q > 0, v is maximized when q = 3*pi/2, 7*pi/2,...

For q < 0, v is maximized when q = -pi/2, -5*pi/2, ...