91影视

A small mass m attached to the end of a string revolves in a circle on a frictionless table top. The other end of the string passes through a hole in the table (Fig. 8鈥�62). Initially, the mass revolves with a speed\({v_1} = 2.4\;{\rm{m/s}}\)in a circle of radius\({r_1} = 0.80\;{\rm{m}}\). The string is then pulled slowly through the hole so that the radius is reduced to\({r_2} = 0.48\;{\rm{m}}\). What is the speed,\({v_2}\), of the mass now?

A uniform rod of mass M and length l can pivot freely (i.e., we ignore friction) about a hinge attached to a wall, as in Fig. 8鈥�63. The rod is held horizontally and then released. At the moment of release, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of the tip of the rod. Assume that the force of gravity acts at the center of mass of the rod, as shown. [Hint: See Fig. 8鈥�20g.]

Suppose a star the size of our Sun, but with mass 8.0 times as great, were rotating at a speed of 1.0 revolution every 9.0 days. If it were to undergo gravitational collapse to a neutron star of radius 12 km, losing\(\frac{3}{4}\)of its mass in the process, what would its rotation speed be? Assume the star is a uniform sphere at all times. Assume also that the thrown off mass carries off either (a) no angular momentum, or (b) its proportional share\(\left( {\frac{3}{4}} \right)\)of the initial angular momentum.

A large spool of rope rolls on the ground with the end of the rope lying on the top edge of the spool. A person grabs the end of the rope and walks a distance l, holding onto it, Fig. 8鈥�64. The spool rolls behind the person without slipping. What length of rope unwinds from the spool? How far does the spool鈥檚 center of mass move?

The Moon orbits the Earth such that the same side always faces the Earth. Determine the ratio of the Moon鈥檚 spin angular momentum (about its own axis) to its orbital angular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)

A spherical asteroid with radius\(r = 123\;{\rm{m}}\)and mass\(M = 2.25 \times {10^{10}}\;{\rm{kg}}\)rotates about an axis at four revolutions per day. A 鈥渢ug鈥� spaceship attaches itself to the asteroid鈥檚 south pole (as defined by the axis of rotation) and fires its engine, applying a force F tangentially to the asteroid鈥檚 surface as shown in Fig. 8鈥�65. If\(F = 285\;{\rm{N}}\)how long will it take the tug to rotate the asteroid鈥檚 axis of rotation through an angle of 5.0掳 by this method?

I) Water drives a waterwheel (or turbine) of radius R = 3.0 m as shown in Fig. 8鈥�66. The water enters at a speed\({v_1} = 7.0\;{\rm{m/s}}\)and exits from the waterwheel at a speed\({v_2} = 3.8\;{\rm{m/s}}\). (a) If 85 kg of water passes through per second, what is the rate at which the water delivers angular momentum to the waterwheel? (b) What is the torque the water applies to the waterwheel? (c) If the water causes the waterwheel to make one revolution every 5.5 s, how much power is delivered to the wheel?

The radius of the roll of paper shown in Fig. 8鈥�67 is 7.6 cm and its moment of inertia is \(I = 3.3 \times {10^{ - 3}}\;{\rm{kg}} \cdot {{\rm{m}}^2}\). A force of 3.5 N is exerted on the end of the roll for 1.3 s, but the paper does not tear so it begins to unroll. A constant friction torque of \(I = 0.11\;{\rm{m}} \cdot {\rm{N}}\) is exerted on the roll which gradually brings it to a stop. Assuming that the paper鈥檚 thickness is negligible, calculate (a) the length of paper that unrolls during the time that the force is applied (1.3 s) and (b) the length of paper that unrolls from the time the force ends to the time when the roll has stopped moving