[1.] This problem has five multiple choice questions. Circle the best answer in each case.

Transcription

1 [1.] This problem has five multiple choice questions. Circle the best answer in each case. [1A.] A force given by 1 2 3, acts on a particle positioned at 2 6. What is its torque about the origin? [a] [b] [c] [d] [e] [1B.] A 100 gram tennis ball is served such that it has an initial velocity of 40 m/s in the direction as it passes over the baseline. A tennis racket strikes the ball and applies an average force of 100 to the ball in the direction for What is the velocity of the ball as it leaves the racket? [a] 10 m/s [b] 20 m/s [c] 30 m/s [d] 60 m/s [e] 90 m/s [1C.] The position vector for a particle with mass m is given by, where A, and B, are positive constants. Determine the time when the angular momentum about the origin is zero. [a] 0, [b] 0, [c], [d] 2, 0 [e] 2, [1D.] A uniform wood plank with mass M is set on two sawhorses placed at a distance d from each other so that the plank s center of mass is exactly halfway in between them. The plank is much longer than d. A person with mass M/2 starts walking from the center of the plank toward one end. How far past the sawhorse can he walk before it tips over? [a] 0 [b] d [c] d/2 [d] 2 d [e] 3/2 d [1E.] A uniform disk with mass M, radius R, and moment of inertia, is free to roll on a flat surface. The disk is initially at rest, when a horizontal, constant force perpendicular to the axis of the disk is applied to its center of mass. Find the magnitude of the static friction necessary for the disk to roll without slipping. [a] F [b] 2F [c] F/2 [d] 3F [e] F/3 1

2 [2.] A long, uniform rod with length L, mass M, and moment of inertia (with respect to its center of mass), is hinged at one end so that it can rotate, without friction, around a horizontal axis. The rod is hanging, motionless, in a vertical position when a piece of putty with same mass M is launched horizontally toward the rod and hits it at its center, with horizontal speed, getting stuck to it. Write your results in terms of L, M, and. Remember to check the dimensions/units for each answer. a) Find the moment of inertia, with respect to the hinge, of the rod+putty system after the collision. b) Find the angular speed of the rod+putty right after the collision. c) As the system rotates around the hinge, what is the maximum angle it reaches before it falls back down? Assume the speed is small so that it does not do full loop. a) For the rod: parallel axis theorem For the putty: Check dimensions: dimensions of moment of inertia are just OK b) Conservation of angular momentum with respect to the hinge:, Check dimensions: OK c) Conservation of energy (using the hinge as origin for gravitational potential energy): 2 2 cos 2 2cos 2 2 cos cos cos 1 Check dimensions: the argument of the cosine must be adimensional: OK 2

3 [3.] A hollow cylinder with height h has inner radius and outer radius (see figure). The density of the cylinder decreases with the distance r from its axis and is given by for, with positive and constant with units of /. Write your results in terms of h,,, and. Remember to check the dimensions/units for each answer. a) Find the mass of the hollow cylinder. b) Find its moment of inertia with respect to its axis. c) Find the moment of inertia with respect to an axis parallel to the axis of the cylinder and passing through point P on the inner radius of the cylinder. a) Use a thin cylindrical shell with radius r, thikness dr, height h, and volume Check dimensions: OK b) 2 Check dimensions: OK c) Parallel axis theorem: 2 Check dimensions: OK 3

4 [4.] A chandelier of mass 5M is being installed in the new opera house at a height h above the floor. The chandelier is connected to one end of a steel cable with negligible mass which runs over a pulley of mass M, radius R, and moment of inertia. The other end of the cable is pulled by the installation crew with constant force F (see figure). Write your results in terms of M, h, and R. Remember to check the dimensions/units for each answer. a) Find the angular acceleration of the pulley. b) How long will it take for the pulley to make the first full loop (i.e., starting from rest)? c) What is the force exerted by the cable on the chandelier? a) Newtons 2 nd law for the chandelier (y component only) + torque equation (z component only) + relation between acceleration and angular accelerations: Check dimensions: OK b) Angular acceleration is constant, and initial position and angular speed are zero: 2 Check dimension: we already checked at (a) that dimension of are, therefore OK c) Solve for T from (a): Check dimensions: OK 4

5 [5.] A CSI team is called to investigate a car accident on University Ave. & Bird Rd. A sedan (car A) with mass m, moving west on Bird, collided with an SUV (car B) with mass 2m moving south-west on University (see figure). The skid marks show that, after the collision, the two cars stuck together, moving with speed in a direction forming an angle of 30 with respect to west Bird Rd. Write your results in terms of m and. Remember to check the dimensions/units for each answer. a) Find the initial velocities and of the two cars. b) Find the initial kinetic energies of the two cars. c) How much mechanical energy, if any, was lost in the collision? Where did it go? a) Conservation of momentum in the x and y direction: 2 cos 45 3 cos 30 2 sin 45 3 sin b) c) The lost energy is responsible for the damage to the cars. 5