physics 111 hw23
assigned20 April 2011
1. You might ask yourself the question, “What causes things to topple over?” Well, if it is just sitting on an incline, it will topple over only if the center of mass (actually gravity) of the object lies outside the area of support. For example, in the diagram below, a rectangular block is on an incline. If the center of mass is over the support area, it does not topple over, as in the left picture. However, on the right the C of M is not over the support area and it will topple over.
For a uniform block of height h and base width w, calculate the maximum angle of an incline it can sit on (as in pictures above) without toppling over. Your answer should be in terms of h and w.
2. Two uniform solid disks spin frictionlessly on the same spindle as shown in the diagram at right. The top disk has a radius of 0.050 m and a mass of 1 kg. The bottom disk has a radius of 0.080 m and a mass of 4 kg. Their initial angular velocities are as shown. They collide and stick together. Find the final angular velocity of the two as they rotate together, and show that kinetic energy was lost during the collision.
3. A large turntable rotates about a fixed vertical axis, making one revolution in 6.00 seconds. The moment of inertia of the turntable about this axis is 1200 kg∙m2. A child of mass 40.0 kg, initially standing at the center of the turntable, runs out along a radius. What is the angular speed of the turntable when the child is 2.00 m from the center? Treat the child as a point particle.
4. A solid wood door 1.00 m wide and 2.00 m high is hinged along one side and has a total mass of 40.0 kg. Initially open and at rest, the door is struck at its center by a handful of sticky mud with mass 0.500 kg, traveling perpendicular to the door at 12.0 m/s just before impact. Find the final angular speed of the door. Does the mud make a significant contribution to the moment of inertia?
5. A small 10.0 g bug stands at one end of a thin, uniform bar which is initially at rest on a smooth, horizontal table. The other end of the bar pivots about a nail driven into the table and can rotate freely, without friction. The bar has mass 50.0 g and is 100 cm in length. The bug jumps off in the horizontal direction, perpendicular to the bar, with a speed of 20.0 cm/s relative to the table.
a) What is the angular speed of the bar just after the frisky insect leaps?
b) What is the total kinetic energy of the system just after the bug leaps?
c) Where does this energy come from?
6. A thin, uniform, metal bar, 2.00 m long and weighing 90.0 N, is hanging vertically from the ceiling by a frictionless pivot. Suddenly it is struck 1.50 m below the ceiling by a small 3.00 kg ball, initially traveling horizontally at 10.0 m/s. The ball rebounds in the opposite direction with a speed of 6.00 m/s.
a) Find the angular speed of the bar just after the collision.
b) During the collision, why is the angular momentum conserved, but not the linear momentum?
7. Draw the top view of the spinning gyroscope shown at right.
a) Indicate in your top view diagram the directions of ω, L, τ, and the dL produced by the torque. Determine the sense of the precession (which way it goes around) and indicate this in your diagram.
b) Repeat part a) for the flywheel rotating in the other direction.