Bicycle problem $(15$ points$)$

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Consider the following $4-$bar mechanism created by a man riding a bicycle. Let's assign some lengths: $L_1=12cm,L_2=30cm,L_3=24cm,$ and $L_4=40cm.$ We'll define angle of hip deflection, $,$ as the angle of thigh deflection off the body's vertical posture, as shown in diagram on the right.

$($a$)$ If bicycle pedal is rotating at a constant rate $,$ is hip also rotating at constant rate? Explain your reasoning.

$($b$)$ The bicycle transforms reciprocating motion into rotary motion. For this particular scenario, what are the maximum and minimum values of $L_4$ for which the pedal rotates and the hip reciprocates? Express in terms of $L_1,L_2,$ and $L_3.$

$($c$)$ When the pedal is at its lowest point, find a coupled set of constraint equations relating pedal angle to hip angular velocity, ${}^{}.$ Define all your variables with labels on the diagram, and highlight them in your equations.

Figure $5$: Bicycle problem