Help the military to launch aerial targets. The proposed device is based on an ancient engine of war - a trebuchet. As shown below, the trebuchet consists of a pipe, pinned at B, a bucket to hold the targets at C, and a counterweight at A. A motor attached to the pin at B will apply a constant torque to rotate the arm back from θ as shown until θ= 0°, at which point a ratchet gear engages to hold the arm horizontal, and the motor disengages. The target is loaded into the bucket, the ratchet is released, and the counterweight causes the arm to swing up until it hits a stop at θ = 75°. The arm stops, but the target flies off, following a projectile path over the shooting range. The target must reach its peak at a height of 40 feet above point B.
The pipe is 16 feet long, weights 880 lbs, and may be treated as a slender rod. The counterweight is a 2-ft diameter cylinder rigidly attached to the pipe at A. The bucket's weight of 30 lbs centers 0.75 feet from the end of the pipe at C. With or without the 65 lb target loaded, the radius of gyration about that center is known to be k = 0.95 ft.
1. How fast must the target be moving when it leaves the bucket so that it it peaks 40 feet above B? How far to the left of B will it reach this peak?
2. What is the moment of inertia of the trebuchet (pipe, cylinder, bucket, and target) about point B? Describe this as a function of the mass of the counterweight.
3. How much must the cylinder weigh in order for the target to reach the necessary speed when θ= 75°?
4. The motor must return the arm to θ= 0° in 4 seconds, starting from rest. What constant torque must the motor supply to accomplish this? At what angular velocity ω will the arm be turning when the ratchet gear engages?
5. Determine the reaction force of the pin supports at B at the instant the motor engages when θ= 75°.