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of 3PROBLEM SET 9: Rotation 1. Moment of Inertia A slender rod with length L has a mass per unit length that varies with distance from the left end, where x = 0, according to dm/dx = ɣx, where ɣ has units of kg/m2. (a) Calculate the total mass of the rod in terms of ɣ and L. (b) Use Eq. (9.20) to calculate the moment of inertia of the rod for an axis at the left end, perpendicular to the rod. Use the expression you derived in part (a) to express I in terms of M and L. How does your result compare to that for a uniform rod? Explain this comparison. (c) Repeat part (b) for an axis at the right end of the rod. How do the results for parts (b) and (c) compare? Explain this result.
2. Rotational Energy Engineers are designing a system by which a falling mass m imparts kinetic energy to a rotating uniform drum to which it is attached by thin, very light wire wrapped around the rim of the drum (see figure). There is no appreciable friction in the axle of the drum, and everything starts from rest. This system is being tested on earth, but it is to be used on Mars, where the acceleration due to gravity is 3.71 m/s . In the earth tests, when m is set to 15.0 kg and allowed to fall through 5.00 m, it gives 250.0 J of kineticenergy to the drum. (a) If the system is operated on Mars, through what distance would the 15.0-kg mass have to fall to give the same amount of kinetic energy to the drum? (b) How fast would the 15.0-kg mass be moving on Mars just as the drum gained 250.0 J of kinetic energy?
3. Rotating Cylinder The 5.0 kg, 60-cm-diameter disk pictured rotates on an axle passing through one edge. The axle is parallel to the floor. The cylinder is held with the center of mass at the same height as the axle, then released. (a) What is the cylinder’s initial angular acceleration? (b) What is the cylinder’s angular velocity when it is directly below the axle?
4. A 40 kg, 5.0-m-long beam is supported by, but not attached to, the two posts in the figure. A 20 kg boy starts walking along the beam. How close can he get to the right end of the beam without it falling over?
5. Rolling Sphere A thin-walled, hollow spherical shell of mass m and radius r starts from rest and rolls without slipping down the track shown. Points A and B are on a circular part of the track having radius R. The diameter of the shell is very small compared to ho and R, and the work done by rolling friction is negligible. (a) What is the minimum height ho for which this shell will make a complete loop-the-loop on the circular part of the track? (b) How hard does the track push on the shell at point B, which is at the same level as the center of the circle? (c) Suppose that the track had no friction and the shell was released from the same height ho you found in part (a). Would it make a complete loop- the-loop? How do you know? (d) In part (c), how hard does the track push on the shell at point A, the top of the circle? How hard did it push on the shell in part (a)?
6. A uniform, 8.0-m, 1500-kg beam is hinged to a wall and supported by a thin cable attached 2.0 m from the free end of the beam, shown in the figure. The beam is supported at an angle of 30.0° above the horizontal. (a) Draw a free-body diagram of the beam. (b) Find the tension in the cable. (c) How hard does the beam push inward on the wall?
7. Pulley The pulley in the figure has radius 0.160 m and moment of inertia 0.560 kg m2. The rope does not slip on the pulley rim. Use energy methods to calculate the speed of the 4.00-kg block just before it strikes the floor.
8. Conservation of Angular Momentum Under some circumstances, a star can collapse into an extremely dense object made mostly of neutrons and called a neutron star. The density of a neutron star is roughly 1014 times as great as that of ordinary solid matter. Suppose we represent the star as a uniform, solid, rigid sphere, both before and after the collapse. The star’s initial radius was 7.0 x 105 km (comparable to our sun); its final radius is 16 km. If the original star rotated once in 30 days, find the angular speed of the neutron star.
9. Conservation of Angular Momentum 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?
10. Angular Momentum A particle of mass 0.400 kg is attached to the 100-cm mark of a meterstick of mass 0.100 kg. The meterstick rotates on the surface of a
frictionless, horizontal table with an angular speed of 4.00 rad/s. Calculate the angular momentum of the system when the stick is pivoted about an axis (a) perpendicular to the table through the 50.0-cm mark and (b) perpendicular to the table through the 0-cm mar