21 Ch9: 6, 7, 26, 30, 67, 80, An electric turntable m in diameter is rotating about a fixed axis with an initial angular velocity of rev/s and a constant angular acceleration of rev/s 2. (a) Compute the angular velocity of the turntable after s. (b) Through how many revolutions has the turntable spun in this time interval? (c) What is the tangential speed of a point on the rim of the turntable at t = s? (d) What is the magnitude of the resultant acceleration of a point on the rim at t = s? At t = 3.00 s a point on the rim of a m-radius wheel has a tangential speed of 50.0 m/s as the wheel slows down with a tangential acceleration of constant magnitude 10.0 m/s 2. (a) Calculate the wheel's constant angular acceleration. (b) Calculate the angular velocities at t = 3.00 s and t = O. (c) Through what angle did the wheel turn between t = 0 and t = 3.00 s? (d) At what time will the radial acceleration equal g? *9.67. A disk of radius 25.0 cm is free to turn about an axle perpendicular to it through its center It has very thin but strong string wrapped around its rim, and the string is attached to a ball that is pulled tangentially away from the rim of the disk (Fig. 9.33). The pull increases in magnitude and produces an acceleration of the ball that obeys the equation a(t)=at, where t is in seconds and A is a constant. The cylinder starts from rest, and at the end of the third second, the ball's acceleration is 1.80 mls 2. (a) Find A. (b) Express the angular acceleration of the disk as a function of time. (c) How much time after the disk has begun to turn does it reach an angular speed of 15.0 rad/s? (d) Through what angle has the disk turned just as it reaches 15.0 rad/s? (Hint: See Section 2.6.) A uniform, solid disk with mass m and radius R is pivoted about a horizontal axis through its center. A small object of the same mass m is glued to the rim of the disk. If the disk is released from rest with the small object at the end of a horizontal radius, find the angular speed when the small object is directly below the axis The pulley in Fig has radius R and a moment of inertia I. The rope does not slip over the pulley, and the pulley spins on a frictionless axle. The coefficient of kinetic friction between block A and the tabletop is k. The system is released from rest, and block B descends. Block A has mass m A and block B has mass m B. Use energy methods to calculate the speed of block B as a function of the distance d that it has descended.
22 Ch10: 8, 29, 35, 38, 40, 64, 70, 71, 83, 87, 91, A uniform, 8.40-kg, spherical shell 50.0 cm in diameter has four small 2.00-kg masses attached to its outer surface and equally spaced around it. This combination is spinning about an axis running through the center of the sphere and two of the small masses. What friction torque is needed to reduce its angular speed from 75.0 rpm to 50.0 rpm in 30.0 s? The engine delivers 175 hp to an aircraft propeller at 2400 rev/min. (a) How much torque does the aircraft engine provide? (b) How much work does the engine do in one revolution of the propeller? A 2.00-kg rock has a horizontal velocity of magnitude 12.0 m/s when it is at point P in Fig (a) At this instant, what are the magnitude and direction of its angular momentum relative to point O? (b) If the only force acting on the rock 36.9' is its weight, what is the rate of 0 change (magnitude and direction) of its angular momentum at this instant? A hollow, thin-walled sphere of mass 12.0 kg and diameter 48.0 cm is rotating about an axle through its center. The angle (in radians) through which it turns as a function of time (in seconds) is given by (t) = At 2 + Bt 4, where A has numerical value 1.50 and B has numerical value (a) What are the units of the constants A and B? (b) At the time 3.00 s, find (i) the angular momentum of the sphere and (ii) the net torque on the sphere A small block on a frictionless, horizontal surface has mass of kg. It is attached to a massless cord passing through a hole in the surface (Fig ). The block is originally revolving at a distance of m from the hole with an angular speed of 1.75 rad/s. The cord is then pulled from below, shortening the radius of the circle in which the block revolves to m. Model the block as a particle. (a) Is angular momentum of the block conserved? Why or why not? (b) What is the new angular speed? (c) Find the change in kinetic energy of the block. (d) How much work was done in pulling the cord? A block with mass m = 5.00 kg slides down a surface inclined to the horizontal. The coefficient of kinetic friction is A string attached to the block is wrapped around a flywheel on a fixed axis at O. The flywheel has mass 25.0 kg and moment of inertia kg m 2 with respect to the axis of rotation. The string pulls without slipping at a perpendicular distance of m from that axis. (a) What is the acceleration of the block down the plane? (b) What is the tension in the string? A thin-walled, hollow spherical shell of mass m and radius r starts from rest and rolls without slipping down the track shown in Fig 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 h 0 and R, and rolling friction is negligible. (a) What is the minimum height h 0 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 h 0 you found in part (a). Would it make a complete loop-theloop? 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)?
23 Figure shows three identical yo-yos initially at rest on a horizontal surface. For each yo-yo. the string is pulled in the direction shown. In each case, there is sufficient friction for the yo-yo to roll without slipping. Draw the free-body diagram for each yo-yo. In what direction will each yo-yo rotate? (Try it!) Explain your answers A uniform, solid cylinder with mass M and radius 2R rests on a horizontal tabletop. A string is attached by a yoke to a frictionless axle through the center of the cylinder so that the cylinder can rotate about the axle. The string runs over a disk-shaped pulley with mass M and radius R that is mounted on a frictionless axle through its center. A block of mass M is suspended from the free end of the string (Fig ). The string doesn't slip over the pulley surface, and the cylinder rolls without slipping on the tabletop. Find the magnitude of the acceleration of the block after the system is released from rest A uniform rod of length L rests on a frictionless horizontal surface. The rod pivots about a fixed frictionless axis at one end. The rod is initially at rest. A bullet traveling parallel to the horizontal surface and perpendicular to the rod with speed v strikes the rod at its center and becomes embedded in it. The mass of the bullet is one-fourth the mass of the rod. (a) What is the final angular speed of the rod? (b) What is the ratio of the kinetic energy of the system after the collision to the kinetic energy of the bullet before the collision? A g bird is flying horizontally at 2.25 m/s, not paying much attention, when it suddenly flies into a stationary vertical bar, hitting it 25.0 cm below the top (Fig ). The bar is uniform, m long, has a mass of 1.50 kg, and is hinged at its base. The collision stuns the bird so that it just drops to the ground afterward (but soon recovers to fly happily away). What is the angular velocity of the bar (a) just after it is hit by the bird, and (b) just as it reaches the ground?