A thin uniform rod of mass mr and length l is fitted with a small collar of mass mc that can slide freely (without friction), as shown in Figure 9.38. A motor at the rod's attachment point O rotates the rod with a constant torque τ. Assume the system is acting in plane (so that gravity can be neglected).
a. Find the equations of motion of the collar.
b. Using matlab, integrate the equations of motion found in part (a). Let τ = 1 N-m, l = 1 m, mc = 1 kg, and mr = 1/3 kg. Start the system from rest with θ = 0 and with the collar 0.1 m from O and integrate for 1 s. Plot the collar's position along the length of rod and the rod's angle θ versus time.
c. Integrate the system in part (b) until the collar leaves the rod and find the value of θ and the collar's velocity at this time.
d. Repeat your integrations for at least two other values of τ between 1 and 3 N-m, and find the value of θ at which the collar leaves the rod in each instance. Explain your findings.