About what axis is the rotational inertia of your body the least? Answer. Vertical Axis

Transcription

1 1 About what axis is the rotational inertia of your body the least? Vertical Axis 5 The figure shows three small spheres that rotate about a vertical axis. The perpendicular distance between the axis and the center of each sphere is given. Rank the three spheres according to their rotational inertia about that axis, greatest first. 2 If two circular disks of the same weight and thickness are made from metals having different densities, which disk, if either, will have the larger rotational inertia about its central axis? For both disks I = 1/2MR 2. The disk with the greater density has the smaller radius and the smaller rotational inertia. All the same 3 Five solids are shown in cross section (shown in the figure). The cross sections have equal heights and equal maximum widths. The solids have equal masses. (a) Which one has the largest rotational inertia about a perpendicular axis through the center of mass? Explain your logic (b) Which the smallest rotational inertia about a perpendicular axis through the center of mass? Explain your logic 6 The figure shows three kg particles that have been glued to a rod of length L = 6.00 cm and negligible mass. The assembly can rotate around a perpendicular axis through point 0 at the left end. We remove one particle (that is, 33% of the mass). (a) By what percentage does the rotational inertia of the assembly around the rotation axis decrease when that removed particle is the innermost one? (b) By what percentage does the rotational inertia of the assembly around the rotation axis decrease when that removed particle is the outermost one? The hoop has the largest rotational inertia since all of the matter is at a large distance form the axis. On the other hand, most of the matter comprising the prism is relatively close to the axis so this object has the smallest rotational inertia. 4 Why is it more difficult to do a sit-up with your hands behind your head than when they are outstretched in front of you? A diagram may help you to answer this. (a) 7.14 % (b) %

2 7 Four particles, each of mass 0.20 kg, are placed at the vertices of a square with sides of length 0.50 m. The particles are connected by rods of negligible mass. This rigid body can rotate in a vertical plane about a horizontal axis A that passes through one of the particles. The body is released from rest with rod AB horizontal, as shown in in the figure. What is the rotational inertia of the body about axis A? 11 An oxygen molecule consists of two oxygen atoms whose total mass is 5.3 x g and whose moment of inertia about an axis perpendicular to the line joining the two atoms, midway between them, is 1.9 x kg.m 2 Estimate, from these data, the effective distance between the atoms x 10-6 m if assume hollow sphere 1.89 x 10-6 m If assume point sources 1.89 x 10-4 m 12 Tons of dust and small particles rain down onto the Earth from space everyday. As a result, does the Earth's moment of inertia increase, decrease, or stay the same? Explain..020 kg.m 2 s 8 A tennis ball has a mass of 57 g and a diameter of 7 cm. Find the moment of inertia about its diameter. Assume the ball is a thin spherical shell. In principle, the Earth's moment of inertia increases because both its mass and radius increase. 13 Why should changing the axis of rotation of an object change its moment of inertia, given that its shape and mass remain the same? 4.65 x 10-5 kg.m 2 s 9 Calculate the moment of inertia of a 14.0 kg solid sphere of radius m when the axis of rotation is through its center. The moment of inertia of an object changes with the position of the axis of rotation because the distance from the axis to all the elements of mass have been changed. It is not just the shape of an object that matters, but the distribution of mass with respect to the axis of rotation x 10-5 kg.m 2 10 Calculate the moment of inertia of a 66.7 cm diameter bicycle wheel. The rim and tire have a combined mass of 1.25 kg. The mass of the hub can be ignored. (Why?) kg.m 2

3 14 The L-shaped object in the figure can be rotated in one of the following three ways: (i) about the x axis; (ii) about the y axis; and (iii) about the z axis (which passes through the origin perpendicular to the plane of the figure.) (a) In which of these cases is the object's moment of inertia greatest? (b) In which case is it least? Explain. 17 The L-shaped object in the figure can be rotated in one of the following three ways: (i) about the x axis; (ii) about the y axis; and (iii) about the z axis (which passes through the origin perpendicular to the plane of the figure.) (a) In which of these cases is the object's moment of inertia greatest? (b) In which case is it least? Explain. s s The moment of inertia of the object is greatest when it is rotated about the z axis. It is least when rotated about the x axis. The moment of inertia of the object is greatest when it is rotated about the z axis. It is least when rotated about the x axis. 15 The minute and hour hands of a clock have a common axis of rotation and equal mass. The minute hand is long and thin; the hour hand is short and thick. Which hand has the greatest moment of inertia? 18 What is the moment of inertia of the wheel of mass of 1 kg and an axle with a mass of 3 kg as seen in the figure. (You can ignore the spokes in this problem) s The long, thin minute hand with mass far from the axis of rotation has the greater moment of inertia. 16 The moment of inertia of a 0.98 kg bicycle wheel rotating about its center is 0.13 kg. m 2. What is the radius of this wheel, assuming the weight of the spokes can be ignored? s 1.06 x 10-2 kgm m

4 19 The pulley shown in the figure is rotated by the falling masses. The diameter of the wheel is 25 cm, has a thickness of 5 cm and has a mass of 15 kg. The spokes of the wheel are each 5 cm wide, 5 cm thick. and has a mass of 2 kg. Determine the moment of inertia of the pulley. 22 A bicycle wheel of radius 0.3 m has a rim of mass 1.0 kg and 50 spokes, each of mass 0.01 kg. What is its moment of inertia about its axis of rotation? kgm 2 23 In the figure the solid cylinder has a mass of 100 grams, a radius of 3 cm and a length of 10 cm. The 20 gram metal plate which bisects the rod is 1 cm thick, and 8 cm by 8 cm. (a) Determine the moment of inertia of the object when it is rotated around the x-axis (b) Determine the moment of inertia of the object when it is rotated around the y-axis kg,m 2 20 Small blocks, each of mass m, are clamped at the ends and at the center of a light rigid rod of length L. Compute the moment of inertia of the system about an axis perpendicular to the rod and passing through a point one quarter of the length from one end. Neglect the moment of inertia of the rod. 11/16 ml 2 (a) x 10-5 kgm 2 (b) x 10-4 kgm 2 21 A thin rectangular sheet of steel is 0.3 m by 0.4 m and has mass 24 kg. (a) Find the moment of inertia about an axis Through the center, parallel to the long sides; (b) Find the moment of inertia about an axis through the center, parallel to the short sides; (c) Find the moment of inertia about an axis through the center, perpendicular to the plane. (d) Do you notice anything strange about your results? 24 A flywheel consists of a solid disk 0.5 m in diameter and 0.02 m thick, and two projecting hubs 0.1 m in diameter and 0.1 m long. If the material of which it is constructed has a density of 6000 kg/m 3, find its moment of inertia about the axis of rotation. (a).18 kg,m 2 (b).32 kg.m 2 (c).5 kg.m 2 (d) The answers to a + b add to c since the thickness is zero kgm 2

5 25 A solid sphere and a pulley are both rotated by a falling mass. The solid cylinder has a diameter of 15 cm and a mass of 135 kg. The Solid metal disk has a diameter of 15 cm a thickness of 10 cm and a mass of 125 kg. Determine the moment of inertia of each object. 28 Calculate the moment of inertia of a 14.0 kg solid sphere of radius m when the axis of rotation is through its center kgm 2 29 Calculate the moment of inertia of a 66.7 cm diameter bicycle wheel. The rim and tire have a combined mass of 1.25 kg. The mass of the hub can be ignored. (Why?).139 kg.m 2 26 I sphere = x 10-1 kgm 2 I disk = x 10-1 kgm 2 Find the moment of inertia of a disk of radius R and mass M about an axis in the plane of the disk that passes through its center. 30 In the figure, two particles, each with mass m = 0.85 kg, are fastened to each other, and to a rotation axis at 0, by two thin rods, each with length d = 5.6 cm and mass M = 1.2 kg. The combination rotates around the rotation axis with angular speed = 0.30 rad/s. What are the combination's rotational inertia about O? kg.m 2 1/4 mr 2 + 1/12 mr 2 27 A uniform rectangular plate has mass m and sides a and b. (a) What is the moment of inertia about an axis that is perpendicular to the plate and passes through its center of mass? (b) Show by integration that the moment of inertia of the plate about an axis that is perpendicular to the plate and passes through one corner is (1/3)m(a 2 + b 2 ). (a) (b)

6 31 The figure shows an arrangement of 15 identical disks that have been glued together in a rod-like shape of length L = m and (total) mass M = mg. The arrangement can rotate about a perpendicular axis through its central disk at point 0. (a) What is the rotational inertia of the arrangement about that axis? (b) If`we approximated the arrangement as being a uniform rod of mass M and length L, what percentage error would we make to calculate the rotational inertia? 34 What is the moment of inertia of the wheel (mass of 1 kg) and axle (mass of 3 kg) shown in the figure if it is rotated on an axis parallel to the axle and 15 cm from the center of the axle. You may ignore the spokes in this case. (a) 8.32 x 10-3 kg.m 2 (b) 8.32 x 10-3 kg.m x 10-2 kgm An oxygen molecule consists of two oxygen atoms whose total mass is 5.3 x kg and whose moment of inertia about an axis perpendicular to the line joining the two atoms, midway between them, is 1.9 x kg.m 2. Estimate, from these data, the effective distance between the atoms. 35 Compute the moment of inertia of a baton of dimensions shown in the figure about a vertical axis that is parallel with an axis that runs through the center of gravity (CG) and is located 10 cm from the CG axis x 10-5 m 33 Calculate the rotational inertia of a meter stick, with mass 0.56 kg, about an axis perpendicular to the stick and located at the 20 cm mark. (Treat the stick as a thin rod.) 9.7 x 10-2 kg m slug ft 2 36 A rod 4 cm in diameter and 2 m long has a mass of 8 kg. (a) Find the moment of inertia about an axis perpendicular to the rod and passing through its center (b) Find the moment of inertia about an axis perpendicular to the rod and passing through one end, (c) Find the moment of inertia about a longitudinal axis passing through the center of the rod. (a) kgm 2 (b) kgm 2 (c).0016 kgm 2

7 37 A flywheel consists of a solid disk 0.5 m in diameter and 0.02 m thick, and two projecting hubs 0.1 m in diameter and 0.1 m long. If the material of which it is constructed has a density of 4000 kg/m 3 Find its moment of inertia if it is shifted 20 cm from its normal axis of rotation. 39 In the figure the solid cylinder has a mass of 100 grams, a radius of 3 cm and a length of 10 cm. The 20 gram metal plate which bisects the rod is 1 cm thick, and 8 cm by 8 cm. Determine the moment of inertia of the object when it is rotated on an axis located parallel to the x axis and 20 cm from it kgm 2 38 The four bodies shown in the figure have equal masses m. Body A is a solid cylinder of radius R. Body B is a hollow thin cylinder of radius R. Body C is a solid square with length of side 2R. Body D is the same size as C, but hollow (i.e.) made up of four thin sticks). The bodies have axes of rotation perpendicular to the page and through the center of each body. (a) Calculate the moment of inertia of the solid disk. (b) Calculate the moment of inertia of the ring (c) Calculate the moment of inertia of the solid cube (d) Calculate the moment of inertia of the open box (e) Show mathematically which body has the smallest moment of inertia? (f) Show mathematically which body has the largest moment of inertia? x 10-3 The diagram shows 4 thin walled cylinders arranged concentrically around the y-axis. The central cylinder has a diameter of 20 cm and has a length of 36 cm. Each cylinder increases in diameter by 10 cm. Each cylinder weighs 45 g. Determine the moment inertia of the object if it is rotated on an axis located 45 cm from the Y axis. (a) (mr 2 )/2 (b) mr 2 (c) 2/3 mr 2 (d) (16mR 2 )/3 (e) Disk (f) Open Cube 4.25 x 10-2 kgm 2

8 41 The figure shown consists of a of metal rod, a thick walled cylinder and a thin walled cylinder. Each object is 25 cm in length. The rod has a diameter of 10 cm and weighs 300 grams. The thick walled cylinder has an inside diameter of 30 cm and an outside diameter of 45 cm. it weighs 420 grams. The outer cylinder has a diameter of 66 cm and weighs 100 grams. Determine the moment inertia of the object if it is rotated on an axis parallel to the walls of the cylinder and located 120 cm from the center of the central rod. 43 the following questions: (a) Find the moment of inertia I, for the four-particle system of the figure above about the x-axis, which passes through m 3 and m 4. (b) Find I y for the system about the y-axis, which passes through m 1 and m 4 (c) Find the moment of inertia I,, about the z-axis, which passes through m 4 and is perpendicular to the plane of the figure kgm 2 42 Use the parallel-axis theorem find the moment of inertia of the four-particle system in the figure about an axis that is perpendicular to the plane of the masses and passes through the center of mass of the system. 44 the following questions: (a) Use the parallel-axis theorem to find the moment of inertia about an axis that is parallel to the z axis and passes through the center of mass of the system in the figure. (b) Let x' and y' be axes in the plane of the figure that pass through the center of mass and are parallel to the sides of the rectangle. Compute I x' and I y' and use your results and Equation 8-30 to check your result for part

9 45 Use the parallel-axis theorem to find the moment of inertia of a solid sphere of mass M and radius R about an axis that is tangent to the sphere. 47 The figure shows a book-like object (one side is longer than the other) and four choices of rotation axes, all perpendicular to the face of the object. Rank the choices according to the rotational inertia of the object about the axis, greatest first. 1, 2, 4, 3 46 The figure shows a pair of uniform spheres, each of mass 500 g and radius 5 cm. They are mounted on a uniform rod that has a length L = 30 cm and a mass m = 60 g. (a) Calculate the moment of inertia of this system about an axis perpendicular to the rod through the center of the rod using the approximation that the two spheres can be treated as point particles that are 20 cm from the axis of rotation and that the mass of the rod is negligible. (b) Calculate the moment of inertia exactly and compare your result with your answer for part a. 48 Figure a shows a rigid body consisting of two particles of mass m connected by a rod of length L and negligible mass. (a) What is the rotational inertia I com about an axis through the center of mass, perpendicular to the rod as shown? (b) What is the rotational inertia I of the body about an axis through the left end of the rod and parallel to the first axis (see figure b)? (a) 1/2 ml 2 (b) ml 2

10 49 The figure shows a thin, uniform rod of mass M and length L, on an x axis with the origin at the rod's center. (a) What is the rotational inertia of the rod about the perpendicular rotation axis through the center? (b) What is the rod's rotational inertia I about a new rotation axis that is perpendicular to the rod and through the left end? 52 The uniform solid block in the figure has mass kg and edge lengths a = 3.5 cm, b = 8.4 cm, and c = 1.4 cm. Calculate its rotational inertia about an axis through one corner and perpendicular to the large faces x 10-4 kg.m 2 50 The figure shows a uniform metal plate that had been square before 25% of it was snipped off. Three lettered points are indicated. Rank them according to the rotational inertia of the plate around a perpendicular axis through them, greatest first. c, a, b 53 Four identical particles of mass 0.50 kg each are placed at the vertices of a 2.0 m x 2.0 m square and held there by four massless rods, which form the sides of the square. What is the rotational inertia of this rigid body about an axis that (a) What is the rotational inertia of this rigid body about an axis that passes through the midpoints of opposite sides and lies in the plane of the square? (b) What is the rotational inertia of this rigid body about an axis thatpasses through the midpoint of one of the sides and is perpendicular to the plane of the square? (c) What is the rotational inertia of this rigid body about an axis thatlies in the plane of the square and passes through two diagonally opposite particles? 51 Figure a shows a disk that can rotate about an axis that is perpendicular to its face and at a radial distance h from the center of the disk. figure b gives the rotational inertia I of the disk about the axis as a function of that distance h, from the center out to the edge of the disk. What is the mass of the disk? M = 2.5 kg

11 54 The masses and coordinates of four particles are as follows: 50 g, x = 2.0 cm, y = 2.0 cm; 25 g, x = 0, y = 4.0 cm; 25 g, x = -3.0 cm, y = -3.0 cm; 30 g, x = -2.0 cm, y = 4.0 cm. (a) What is the rotational inertia of this collection about the x axes? (b) What is the rotational inertia of this collection about the y axes? (c) What is the rotational inertia of this collection about the z axes? (d) Suppose the answers to (a) and (b) are A and B, respectively. Then what is the answer to (c) in terms of A and B? 56 Three 0.50 kg particles form an equilateral triangle with 0.60 m sides. The particles are connected by rods of negligible mass. What is the rotational inertia of this rigid body about (a) What is the rotational inertia of this rigid body about an axis that passes through one of the particles and is parallel to the rod connecting the other two, (b) What is the rotational inertia of this rigid body about an axis that passes through the midpoint of one of the sides and is perpendicular to the plane of the triangle, and (c) What is the rotational inertia of this rigid body about an axis that is parallel to one side of the triangle and passes through the midpoints of the other two sides? (a) 1.3 x 10 3 g cm 2 (b) 5.5 x 10 2 g cm 2 (c) 1.9 x 10 2 g cm 2 (d) A + B (a).27 kg m 2 (b).22 kg m 2 (c).10 kg m 2 55 the following questions. (a) Show that the rotational inertia of a solid cylinder of mass M and radius R about its central axis is equal to the rotational inertia of a thin hoop of mass M and radius R/ 2about its central axis. (b) Show that the rotational inertia I of any given body of mass M about any given axis is equal to the rotational inertia of an equivalent hoop about that axis, if the hoop has the same mass M and a radius k given by the equation. The radius k of the equivalent hoop is called the radius of gyration of the given body. 57 In the figure, m = 1.5 kg and M = 3.0 kg. The array is rectangular and it is split through the middle by the horizontal axis. Assume the objects are wired together by very light rigid pieces of wire. (a) Calculate the moment of inertia of the array of point objects shown in the figure about the vertical axis. (b) Calculate the moment of inertia of the array of point objects shown in the figure about the horizontal axis. (c) About which axis would it be harder to accelerate this array?

12 58 Many machines employ cams for various purposes, such as opening and closing valves. In the figure, the cam is a circular disk rotating on a shaft that does not pass through the center of the disk. In the manufacture of the cam, a uniform solid cylinder of radius R is first machined. Then an off-center hole of radius R/2 is drilled, parallel to the axis of the cylinder, and centered at a point a distance R/2 from the center of the cylinder. The cam, of mass M, is then slipped onto the circular shaft and welded into place. Assume R = 10 cm and M = 40 kg. The circular shaft is 20 kg. Calculate the Moment of Inertia of the Cam Shaft. 60 Use the parallel-axis theorem find the moment of inertia of the four-particle system in the figure about an axis that is perpendicular to the plane of the masses and passes through the center of mass of the system kg m 2 Suppose a bicycle wheel is rotated about an axis through its rim and parallel to its axle. Is its moment of inertia greater than, less than, or the same as when it rotates about its axle? Explain. 61 A uniform disk of radius 30 cm, thickness 3 cm, and mass 5 kg rotates at omega ( ) = 10 rad/s about an axis parallel to the symmetry axis but 0.5 cm from that axis. (a) Find the net force on the bearings due to this imbalance. (b) Where should a 100 g mass be placed on the disk to correct this problem? (a).245 nm (b).25 m s The moment of inertia is greater when the axis of rotation is on the rim of the wheel. The reason is that much of the wheel s mass is now at a significantly greater distance from the axis of rotation, compared with the case where the axis is at the center of the wheel. 62 A uniform, 100 kg cylinder of radius 0.60 m is placed flat on some smooth ice. Two skaters wind ropes around the cylinder in the same sense. The skaters then pull on their ropes as they skate away, exerting constant forces of 40 N and 60 N, respectively, for 5 s. (a) Describe the motion of the cylinder. (b) What is its angular acceleration? (c) What is its angular velocity at the end of the 5 seconds? (a) (b) 3.33 rad/s 2 (c) rad/s 2

13 63 A wheel mounted on an axis that is not frictionless is initially at rest. A constant external torque of 50 N.m is applied to the wheel for 20 s. At the end of the 20 s, the wheel has an angular velocity of 600 rev/min. The external torque is then removed and the wheel comes to rest after 120 s more. (a) What is the moment of inertia of the wheel? (b) What is the frictional torque, which is assumed to be constant? 67 A softball player swings a bat, accelerating it from rest to 3.0 rev/s in a time of 0.20 s. Approximate the bat as a 2.2 kg uniform rod of length 0.95 m, and compute the torque the player applies to one end of it. 64 (a) kg m 2 (b) 7.10 nm A 2.0-kg stone is tied to a 0.50 m string and swung around a circle at a constant angular velocity of 12 rad/s. The circle is parallel to the xy-plane and is centered on the z-axis, 0.75 m from the origin. The magnitude of the torque about the origin is: 68 A day-care worker pushes tangentially on a small hand-driven merry-go-round and is able to accelerate it from rest to a spinning rate of 30 rpm in 10.0 s. Assume the merry-go-round is a disk of radius 2.5 m and has a mass of 800 kg, and two children (each with a mass of 25 kg) sit opposite each other on the edge. (a) Calculate the torque required to produce the acceleration, neglecting frictional torque. (b) What force is required?.108 Nm 65 The bolts on the cylinder head of an engine require tightening to a torque of 80 m-n. If a wrench is 30 cm long, what force perpendicular to the wrench must the mechanic exert at its end? If the six-sided bolt head is 15 mm in diameter, estimate the force applied near each of the six points by a socket wrench (see the figure). 69 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 the figure. The rod is held horizontally and then released. See the figure. (a) At the moment of release, determine the angular acceleration of the rod. (b) At the moment of release, determine 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. 66 A small 1.05-kg ball on the end of a light rod is rotated in a horizontal circle of radius m. (a) Calculate the moment of inertia of the system about the axis of rotation. (b) Calculate the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of N on the ball..01

14 70 A wheel of radius m, which is moving initially at 43.0 m/s, rolls to a stop in 225 m. (a) Calculate its linear acceleration. (b) Calculate its angular acceleration. (c) The wheel's rotational inertia is kg.m 2 about its central axis. Calculate the torque exerted by the friction on the wheel, about the central axis. 73 The figure is an overhead view of a horizontal bar that can pivot; two horizontal forces act on the bar, but it is stationary. If the angle between the bar and F 2 is now decreased from 90 0 and the bar is still not to turn, should F 2 be made larger, made smaller, or left the same? 71 The figure shows an overhead view of a meter stick that can pivot about the dot at the position marked 20 (for 20 cm). All five forces on the stick are horizontal and have the same magnitude. Rank the forces according to the magnitude of the torque they produce, greatest first. larger 74 The figure shows an overhead view of a horizontal bar that is rotated about the pivot point by two horizontal forces, F 1 and F 2, with F 2 at angle to the bar, Rank the following values of according to the magnitude of the angular acceleration of the bar, greatest first: 90 0, 70 0, and The figure shows an overhead view of a meter stick that can pivot about the point indicated, which is to the left of the stick's midpoint. Two horizontal forces, F 1 and F 2, are applied to the stick. Only F 1 is shown. Force F 2 is perpendicular to the stick and is applied at the right end. (a) If the stick is not to turn, what should be the direction of F 2? 90, then 70 and 110 tie (b) If the stick is not to turn, should F 2 be greater than, less than, or equal to F 1? (a) F 2 down (b) less

15 75 In the overhead view of the figure, five forces of the same magnitude act on a strange merry-go-round; it is a square that can rotate about point P, at midlength along one of the edges. Rank the forces according to the magnitude of the torque they create about point P, greatest first. 77 The figure shows three flat disks (of the same radius) that can rotate about their centers like merry-go-rounds. Each disk consists of the same two materials, one denser than the other (density is mass per unit volume). In disks 1 and 3, the denser material forms the outer half of the disk area. In disk 2, it forms the inner half of the disk area. Forces with identical magnitudes are applied tangentially to the disk, either at the outer edge or at the interface of the two materials, as shown. (a) Rank the disks according to the torque about the disk center. (b) Rank the disks according to the rotational inertia about the disk center. (c) Rank the disks according to the angular acceleration of the disk, greatest first. F5, F4, F2, F1, F3 (zero) 76 In the figure, two forces F 1 and F 2 act on a disk that turns about its center like a merry-go-round. The forces maintain the indicated angles during the rotation, which is counterclockwise and at a constant rate, However, we are to decrease the angle of F 1 without changing the magnitude of F 1. (a) To keep the angular speed constant, should we increase, decrease, or maintain the magnitude of F 2? (a) 1 and 2 tie, then 3; (b) 1 and 3 tie, then 2; (c) 2, 1, 3 (b) Does force F 1 tend to rotate the disk clockwise or counterclockwise? (c) Does force F 2 tend to rotate the disk clockwise or counterclockwise? 78 The length of a bicycle pedal arm is m, and a downward force of 111 N is applied to the pedal by the rider. (a) What is the magnitude of the torque about the pedal arm's pivot when the arm is at angle 30 0 with the vertical? (b) What is the magnitude of the torque about the pedal arm's pivot when the arm is at angle 90 0 with the vertical? (c) What is the magnitude of the torque about the pedal arm's pivot when the arm is at angle with the vertical? (a) 8.4 N.m (b) 17 N.m (c) 0 (a) decrease; (b) clockwise; (c) counterclockwise

16 79 The body in the figure is pivoted at 0, and two forces act on it as shown. If r 1 = 1.30 m, r 2 = 2.15 m, F 1 = 4.20 N, F 2 = If a 32.0 N.m torque on a wheel causes angular acceleration 25.0 rad/s 2, what is the wheel's rotational inertia? N, 1 = , and 2 = , what is the net torque about the pivot? 1.28 kg m N.m 83 The figure shows a uniform disk that can rotate around its center like a merry-go-round. The disk has a radius of 2.00 cm and a mass of 20.0 grams and is initially at rest. Starting at time t = 0, two forces are to be applied tangentially to the rim as indicated, so that at time t = 1.25 s the disk has an angular velocity of 250 rad/s counterclockwise. Force F 1 has a magnitude of N. What is magnitude F 2? 80 The body in the figure is pivoted at 0. Three forces act on it: F A = 10 N at point A, 8.0 m from 0; F B = 16 N at B 4.0 m from 0; and F C = 19 at C, 3.0 m from 0. What is the net torque about 0? N 84 In the figure, a cylinder having a mass of 2.0 kg can rotate about its central axis through point 0. Forces are applied as shown: F 1 = 6.0 N, F 2 = 4.0 N, F 3 = 2.0 N, and F 4 = 5.0 N. Also, N.m During the launch from a board, a diver's angular speed about her center of mass changes from zero to 6.20 rad/s in 220 ms. Her rotational inertia about her center of mass is 12.0 kg.m 2. During the launch, what is the magnitude of (a) During the launch, what is the magnitude of her average angular acceleration? (b) During the launch, what is the magnitude of the average external torque on her from the board? r = 5.0 cm and R = 12 cm. (a) Find the magnitude of the angular acceleration of the cylinder. (b) Find the direction of the angular acceleration of the cylinder. (During the rotation, the forces maintain their same angles relative to the cylinder.) (a) 28.2 rad/s 2 (b) 338 N.m (a) 9.7 rad/s 2 (b) The direction is counterclockwise (which is the positive sense of rotation)

17 85 The figure shows particles 1 and 2, each of mass m, attached to the ends of a rigid massless rod of length L 1 + L 2, with L 1 = 20 cm and L 2 = 80 cm. The rod is held horizontally on the fulcrum and then released. (a) What is the magnitude of the initial accelerations of particle 1? (b) What is the magnitude of the initial accelerations of particle 2? 87 A pulley, with a rotational inertia of 1.0 x 10-3 kg.m 2 about its axle and a radius of 10 cm, is acted on by a force applied tangentially at its rim. The force magnitude varies in time as F = 0.50t t 2, with F in newtons and t in seconds. The pulley is initially at rest. (a) At t = 3.0 s what is its angular acceleration? (b) At t = 3.0 s what is its angular speed? (a) 4.2 x 10 2 rad/s 2 (b) 5.0 x 10 2 rad/s 86 (a) 1.7 m/s 2 (b) 6.9 m/s 2 In figure a, an irregularly shaped plastic plate with uniform thickness and density (mass per unit volume) is to be rotated around an axle that is perpendicular to the plate face ind through point 0. The rotational inertia of the plate about that axle is measured with the following method. A circular disk of mass kg and radius 2.00 cm is glued to the plate, with its center aligned with point 0 (figure b). A string is wrapped around the edge of the disk the way a string is wrapped around a top. Then the string is pulled for 5.00 s. As a result, the disk and plate are rotated by a constant force of N that is applied by the string tangentially to the edge of the disk. The resulting angular speed is 114 rad/s. What is the rotational inertia of the plate about the axle? 88 Beverage engineering. The pull tab was a major advance in the engineering design of beverage containers. The tab pivots on a central bolt in the can's top. When you pull upward on one end of the tab, the other end presses downward on a portion of the can's top that has been scored. If you pull upward with a 10 N force, approximately what is the magnitude of the force applied to the scored section? (You will need to examine a can with a pull tab.) 25 N 2.51 x kg m 2 89 Two uniform solid spheres have the same mass of 1.65 kg, but one has a radius of m and the other has a radius of m. Each can rotate about an axis through its center. (a) What is the magnitude of the torque required to bring the smaller sphere from rest to an angular speed of 317 rad/s in 15.5 s? (b) What is the magnitude F of the force that must be applied tangentially at the sphere's equator to give that torque? (c) What is the corresponding values of for the larger sphere? (d) What is the corresponding values of F for the larger sphere? (a).689 N.m (b) 3.05 N (c) 9.84 Nm (c) 11.5 N

18 90 A small ball with mass 1.30 kg is mounted on one end of a rod m long and of negligible mass. The system rotates in a horizontal circle about the other end of the rod at 5010 rev/min. (a) Calculate the rotational inertia of the system about the axis of rotation. (b) There is an air drag of 2.30 x 10-2 N on the ball, directed opposite its motion. What torque must be applied to the system to keep it rotating at constant speed? 93 A uniform disk of radius 0.12 m and mass 5 kg is pivoted in such that it rotates freely about its axis. A string wrapped around the disk is pulled with a force of 20 N. (a) What is the torque exerted on the disk? (b) What is the angular acceleration of the disk? (c) If the disk starts from rest, what is its angular velocity after 3 s? (d) Find the total angle the disk turns in the 3 seconds. (a).791 kg m 2 (b) 1.79 x 10-2 N.m 91 A bicyclist of mass 70 kg puts all his mass on each downwardmoving pedal as he pedals up a steep road. Take the diameter of the circle in which the pedals rotate to be 0.40 m, and determine the magnitude of the maximum torque he exerts. 1.4 x 10 2 N.m 92 A uniform, hollow, cylindrical spool has an outside radius R = 12 cm, inside radius R/2, and mass M = 5 kg. It is mounted so that it rotates on a fixed horizontal axle of mass 10 kg Calculate the moment of inertia of the object 94 A thin spherical shell has a radius of 1.90 m. An applied torque of 960 N.m gives the shell an angular acceleration of 6.20 rad/s 2 about an axis through the center of the shell. (a) What is the rotational inertia of the shell about that axis? (b) What is the mass of the shelf? (a) 155 kg.m 2 (b) 64.4 kg 95 A potter's wheel a thick stone disk of radius m and mass 100 kg is freely rotating at 50.0 rev/min. The potter can stop the wheel in 6.00 s by pressing a wet rag against the rim and exerting a radially inward force of 70.0 N. Find the effective coefficient of kinetic friction between wheel and rag.

19 96 Find the net torque on the wheel in the Figure about the axle through O if a = 10.0 cm and b = 25.0 cm. 100 Force F = (2.0 N)i + (3.0 N)k acts on a pebble with postion vector r = (0.50 m)j (2.0 m)k, relative to the origin. (a) What is the resulting torque acting on the the pebble about the origin? (b) What is the resulting torque acting on a point with coordinates (2.0 m, 0, -3.0 m) (a) (-1.5i 4.0j k) N.m (b) (-1.5i 4.0j k) N.m 101 What is the net torque about the origin on a flea located at coordinates (0, -4.0 m, 5.0 m) when forces F 1 = (3.0 N)k and F 2 = (-2.0 N)j act on the flea? 97 Given that r = ix + y j + zk and F = Fxi + Fyj + Fzk, show that the torque = r X F is given by -2.0i N.m = (yfz zfy)i + (zfx x Fz)j + x(fy yfx)k 102 Force F = (-8.0 N)i + (6.0 N) j acts on a partice with position vector r = (3.0 m)i + (4.0 m)j. (a) What is the torque of the particle around the origin? (b) What is the angle between the directions of r and F? 98 Show that, if r and F lie in a given plane, the torque = r X F has no component in that plane (a) 50 k N.m (b) 90 degrees 99 the following questions: (a) What are the magnitude and direction of torque about the origin on a particle located at coordinates (0, -4.0 m, 3.0 m) due to a force F 1 with components Fx 1 = 2.0 N and F 1y = F 1z = 103 At t = 0, a 2.0 kg particle has position vector r = (4.0 m)i (2.0 m)j relative to the origin. Its velocity is given by v = (-6t 4 m/s) i + (3.0 m/s)j. What is the torque acting on the particle about the origin and for t > (b) What are the magnitude and direction of torque about the origin on a particle located at coordinates (0, -4.0 m, 3.0 m) due toa force F 2 with components F 2x = 0, F 2y = 2.0 N, and F 2z = 4.0 N? t 3 k N.m At t = 0, a 2.0 kg particle has position vector r = (4.0 m)i (2.0 m)j relative to the origin. Its velocity is given by v = (-6t 4 m/s) i + (3.0 m/s)j. What is the torque acting on the particle at coordinates (-2.0 m. 3.0 m, 0) and for t > 0. (a) 10 N.m parallel to the yz plane at 53 degrees to the +y axis. (b) 22 N.m, -x 48t 3 k N.m

20 105 The position vector r of a particle relative to a certain point has a magnitude of 3 m, and the force F on the particle has a magnitude of 4 N. (a) What is the angle between the directions of r and F if the magnitude of the associated torque equals zero? (b) What is the angle between the directions of r and F if the magnitude of the associated torque equals 2 N.m? 108 the following questions. (a) What is the torque about the origin on a jar of jalapeno peppers located at coordinates (3.0 m, -2.0 m, 4.0 m) due to a force F 1 = (3.0 N)i (4.0 N)j (5.0 N)k (b) What is the torque about the origin on a jar of jalapeno peppers located at coordinates (3.0 m, -2.0 m, 4.0 m) due to a force F 2 = (-3.0 N)i (4.0 N)j (5.0 N)k (c) What is the torque about the origin on a jar of jalapeno peppers located at coordinates (3.0 m, -2.0 m, 4.0 m) due to the sum of force F 1 = (3.0 N)i (4.0 N)j (5.0 N)k and F 2 = ( the following questions: (a) What is the torque about the origin on a jar of jalapeno peppers located at coordinates (3.0 m, -2.0 m, 4.0 m) due to a force F 1 = (3.0 N)i - (4.0 Nj + (5.0 N)k, (b) What is the torque about the origin on a jar of jalapeno peppers located at coordinates (3.0 m, -2.0 m, 4.0 m) due to a force F 2 = (-3.0 N)i - (4.0 N)j - (5.0 N)k (c) What is the torque about the origin on a jar of jalapeno peppers located at coordinates (3.0 m, -2.0 m, 4.0 m) due to the vector sum of F 1 and F 2? N)i (4.0 N)j (5.0 N)k. (d) What is the torque about the origin on a jar of jalapeno peppers located at coordinates (3.0 m, 2.0 m, 4.0 m) due to the sum of force F 1 = (3.0 N)i (4.0 N)j (5.0 N)k and F 2 = (-3.0 N)i (4.0 N)j (5.0 N)k. (a) (6.0i 3.0j 6.0k) N.m (b) (26i + 3.0j 18k) N.m (c) (32i 24k) N.m (d) 0 (d) Repeat (c) for a point with coordinates (3.0 m, 2.0 m, 4.0 m) instead of the origin. 109 At a certain time the position vector in meters of a 0.25 kg object is r = 2.0i 2.0k. At that instant, its velocity in meters is v = -5.0i + 5.0k., and the force in newtons action on it is F = 4.0 j. What torque acts on it? 107 The figure shows plots of angular position versus time t for three cases in which a disk is rotated like a merry-go- round. In each ease, the rotation direction changes at a certain angular position change. (a) For each case, determine whether change, is clockwise or counterclockwise from = 0, or whether it is at = 0. (b) For each case, determine whether is zero before, after, or at t = 0? (c) For each case, determine whether is positive, negative, or zero. 110 (8.1i + 8.0k) N.m A 3.0 kg particle is x = 3.0 m, y = 8.0 m with a velocity of v = (5.0 m/s)i + (6.0 m/s)j. It is acted on by a 7.0 N force in the negative x direction. What torque about the origin acts on the particle? +56k N.m 111 A performer, seated on a trapeze, is swinging back and forth with a period of 8.85 s. If she stands up, thus raising the center of mass of the system trapeze + performer by 35.0 cm, what will be the new period of the trapeze? Treat trapeze + performer as a simple pendulum. s My solution 8.77 seconds

21 112 A disk whose radius R is 12.5 cm is suspended, as a physical pendulum, from a point at distance h from its center C (See the figure on the left). Its period T is s when h = R/2. What is the free-fall acceleration g at the location of the pendulum? 115 A pendulum is formed by pivoting a long thin rod of length L and mass m about a point on the rod that is a distance d above the center of the rod. (a) Find the period of this pendulum in terms of d, L, m, and g, assuming that it swings with small amplitude (b) What happens to the period if d is decreased, (c) What happens to the period if L is increased? (d) What happens to the period if m is increased? s 116 A meter stick swinging from one end oscillates with a frequency f 0. What would be the frequency, in terms of f 0, if the bottom half of the stick were cut off? s s 113 The figure at the left shows three physical pendulums consisting of identical uniform spheres of the same mass that are rigidly connected by identical rods of negligible mass. Each pendulum is vertical and can pivot about suspension point 0. Rank the pendulums according to period of oscillation, greatest first. 117 A stick with length L oscillates as a physical pendulum, pivoted about point 0 in the figure at the left. (a) Derive an expression for the period of the pendulum in terms of L and x, the distance from the point of support to the center of mass of the pendulum. (b) For what value of x/l is the period a minimum? (c) Show that if L = 1.00 m and g = 9.80 m/s 2 this minimum is 1.53 s. s 114 A physical pendulum consists of a meter stick that is pivoted at a small hole drilled through the stick a distance x from the 50 cm mark. The period of oscillation is observed to be 2.5 s. Find the distance x. s s

22 118 The center of oscillation of a physical pendulum has this interesting property: if an impulsive force (assumed horizontal and in the plane of oscillation) acts at the center of oscillation, no reaction is felt at the point of support. Baseball players (and players of many other sports) know that unless the ball hits the bat at this point (called the "sweet spot" by athletes), the reaction due to the impact will sting their hands. To prove this property, let the stick in the figure at the left simulate a baseball bat. Suppose that a horizontal force F (due to impact with the ball) acts toward the right at P, the center of oscillation. The batter is assumed to hold the bat at 0, the point of support of the stick. (a) What acceleration does point 0 undergo as a result of F? (b) What angular acceleration is produced by F about the center of mass of the stick? (c) As a result of the angular acceleration in (b), what linear acceleration does point 0 undergo? (d) Considering the magnitudes and directions of the accelerations in (a) and (c), convince yourself that P is indeed the "sweet spot." s