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85 and 86 Rotational Dynamics 31. 33. 35. 37. 39. 41. 43. (I) Estimate the moment of inertia of a bicycle wheel 67 cm in diameter. The rim and tire have a combined mass of 1.1 kg. The mass of the hub (at the center) can be ignored (why?). (II) An oxygen molecule consists of two oxygen atoms whose total mass is 5.3 x 10'26 kg and whose moment of inertia about an axis perpendicular to the line joining the two atoms, midway between them, is 1.9 x 10\"\"5 kg - m2. From these data, estimate the effective distance between the atoms. (II) The forearm in Fig. 846 accelerates a 3.6-kg ball at 7.0 m/s2 by means ofthe triceps muscle, as shown. Calculate (a) the torque needed, and (b) the force that must be exerted by the triceps muscle. Ignore the mass of the arm. (II) A softball player swings a bat, accelerating it from rest to 2.6 rev/s in a time of 0.20 5. Approximate the bat as a 0.90-kg uniform rod of length 0.95 m, and compute the torque the player applies to one end of it. (ll) Calculate the moment of inertia of the array of point objects shown in Fig. 847 about (a) the y axis, and (b) the x axis. Assume m = 2.2 kg, M = 3.4 kg, and the objects are wired together by very light, rigid pieces of wire. The array is rectangular and is split through the middle by the x axis. (6) About which axis would it be harder to accelerate this array? (II) A dad pushes tangentially on a small hand-driven merry-go-round and is able to accelerate it from rest to a frequency of 15 rpm in 10.0 5. Assume the merry-go-round is a uniform disk of radius 2.5 m and has a mass of 560 kg, and two children (each with a mass of 25 kg) sit opposite each other on the edge. Calculate the torque required to produce the acceleration, neglecting frictional torque. What force is required at the edge? (II) Let us treat a helicopter rotor blade as a long thin rod, as shown in Fig. 849. (0') If each of the three rotor helicopter blades is 3.75 m long and has a mass of 135 kg, calculate the moment of inertia of the three rotor blades about the axis of rotation. (b) How much torque must the motor apply to bring the blades from rest up to a speed of 6.0 rev/s in 8.0 s? T 1.50 m m m - - iAxis Axis 0.50 m - -- 0.50 m M M FIGURE 8-47 Problem 39.\fR 90 R FIGURE 8-53 Problem 57.87 Rotational Kinetic Energy 49. 51. 53. 55. 57. (I) An automobile engine develops a torque of 265 m 0 N at 3350 rpm. What is the horsepower of the engine? (I) Calculate the translational speed of a cylinder when it reaches the foot of an incline 7.20 m high. Assume it starts from rest and rolls without slipping. (ll) Estimate the kinetic energy of the Earth with respect to the Sun as the sum of two terms, (a) that due to its daily rotation about its axis, and (b) that due to its yearly revolution about the Sun. [Assume the Earth is a uniform sphere with mass = 6.0 x 1024 kg, radius = 6.4 x 106 m, and is 1.5 x 108 km from the Sun.] (II) A merry-go-round has a mass of 1440 kg and a radius of 7.50 m. How much net work is required to accelerate it from rest to a rotation rate of 1.00 revolution per 7.00 5? Assume it is a solid cylinder. (II) A ball of radius r rolls on the inside of a track of radius R (see Fig. 853). If the ball starts from rest at the vertical edge of the track, what will be its speed when it reaches the lowest point of the track, rolling without slipping? 88 Angular Momentum 61. 63. 65. 67. 69. (I) (a) What is the angular momentum of a 2.8-kg uniform cylindrical grinding wheel of radius 28 cm when rotating at 1300 rpm? (b) How much torque is required to stop it in 6.0 5? (II) A nonrotating cylindrical disk of moment of inertia I is dropped onto an identical disk rotating at angular speed a). Assuming no external torques, what is the final common angular speed of the two disks? (II) A figure skater can increase her spin rotation rate from an initial rate of 1.0 rev every 1.5 s to a final rate of 2.5 rev/s. If her initial moment of inertia was 4.6 kg 0 m2, what is her final moment of inertia? How does she physically accomplish this change? (II) A person of mass 75 kg stands at the center of a rotating merry-go-round platform of radius 3.0 m and moment of inertia 820 kg - ml. The platform rotates without friction with angular velocity 0.95 rad/s. The person walks radially to the edge of the platform. (0) Calculate the angular velocity when the person reaches the edge. (b) Calculate the rotational kinetic energy of the system of platform plus person before and after the person's walk. (II) A 4.2-m-diameter merry-goround is rotating freely with an angular velocity of 0.80 rad/s. Its total moment of inertia is 1360 kg 0 m2. Four people standing on the ground, each of mass 65 kg, suddenly step onto the edge of the merry-go-round. (a) What is the angular velocity of the merry-go-round now? (b) What if the people were on it initially and then jumped off in a radial direction (relative to the merry-go-round)