17. Stars originate as large bodies of slowly rotating gas. Because of gravity, these regions of gas slowly decrease in size. What happens to the angular velocity of a star as it shrinks? Explain. 18. Use the principle of conservation of angular momentum to form a hypothesis that explains how a cat can always land on its feet regardless of the position from which it is dropped. 19. Often when a high diver wants to turn a flip in midair, she will draw her legs up against her chest. Why does this make her rotate faster? What should she do when she wants to come out of her flip? 20. As a tether ball winds around a pole, what happens to its angular velocity? Explain. PROBLEMS 293 21. Space colonies have been proposed that consist of large cylinders placed in space. Gravity would be simulated in these cylinders by setting them into rotation about their long axis. Discuss the difficulties that would be encountered in attempting to set the cylinders into rotation. 22. For a particle undergoing uniform circular motion, how are its linear momentum, p and the angular momentum L oriented with respect to each other? 23. f the net force acting on a system is zero, then is it necessarily true that the net torque on it is also zero? 24. Why do tightrope walkers carry a long pole to help balance themselves while walking a tightrope?..! Í :, PROBLEMS Section. Rolling Motion of a Rigid Body l. A cylinder of mass 10 kg rolls without slipping on a rough surface. At the instant its center of niass has a speed of 10 m/s, determine (a) the translational kinetic energy of its center of mass, (b) the rotational kinetic energy about its center of mass, and (e) its total kinetic energy. 2. A solid sphere has a radius of 0.2 manda mass ofl50 kg. How much work is required to get the sphere rolling with an angular speed of 50 rad/son a horizontal surface? (Assume the sphere start's from rest and rolls without slipping.) 3. (a) Determine the acceleration of the center of mass of. a uniform solid disk rolling down an incline and compare this acceleration with that of a uniform hoop: (b) What is the minimum coefficient of friction required to maintain pure rolling motion for the disk? 4. A uniform solid disk and a uniform hoop are placed side by side at the.top of a rough incline of height h. f they are r~leased from rest and roll without slipping, determine their velocities when they reach the bottom. Which object reaches the bottom first? 5. A bowling ball has a mass of 4. O kg, a moment of inertia of 1.6 X 10-2 kg m 2, and a radius of 0.20 m. f it rolls down the lane without slipping at a linear speed of 4.0 m/s, what is its total kinetic energy? 6. The centers of mass of a solid sphere and a solid cylinder, each of mass Mand radius R, are moving with speed v with respect to the floor. Find the ratios of their total kinetic energies. 8. Given M = 6i + 2j - k and-n = 2i - j - 3k, calculate the vector product M X N. 9. Vector A is in the negative y direction, and vector Bis in the negative x direction. What are the directions of (a) A X Band (b) BX A? 1 O. A particle is located at the vector position r = (i + 3j) m, and the force acting on it is F = (3i + 2j) N. What is the torque about (a) the origin and (b) the point having coordinates (O, 6) m? 11. f JA X BJ= A B, what is the angle between A andb? 12. Verify Equation 11.14 for the cross product of any two vectors A and B, and show that the cross product may be written in the following determinant form: i AXB= A% Bx j k Ay AZ By Bz 13. Two forces F 1 and F 2 act along the two sides of an equilateral triangle as shown in Figure i.23. Find a. f Section 11.2 The Vector Product and Torque 7, Two vectors are given by A = - 3i + 4j and B = 2i + 3j. Find (a) A X Band (b) the angle between A and B. Figure 11.23 (Problem 13).
1!! 1, ' '\' i. '1 :'11 ' '. '! ' ' :! 294 CHAPTER 11 ROLLNG MOTON, ANGULAR MOMENTUM, AND TORQUE third force F 3 to be applied at B and along BC which will make the net torque about the point of intersection of the altitudes zero. Will the net torque change if F 3 were to be applied not at B but at any other point along BC? 14. A force F= 2i + 3j (in newtons) is applied to an object that is pivoted about a fixed axis aligned along the z coordinate axis. f the force is applied at the point r = 4i + 5j + Ok (in miters), find (a) the magnitude of the net torque about the z axis and (b) the direction of the torque vector T. Section 11.3 Angular Momentum of a Particle 15. A light rigid rod 1 min length rotates in the xy plane about a pivot through the rod's center. Two particles of mass 4 kg and 3 kg are connected to its ends (Fig. 11.24). Determine the angular momentum of the system about the origin at the instant the speed of each particle is 5 m/s. / / / y -- - Figure 11.24 (Problem 15). 16. At a certain instant the position of a stone in a sling is given by r = 1. 7i m. The linear momentum p of the stone is 12j kg m/s. Calculate its angular momentum L = r X p. 17. The position vector of a particle of mass 2 kg is given as a function of time by r = (6i + 5tj) m. Determine the angular momentum of the particle as a function of time. 18. Two particles move in opposite directions along a straight line (Fig. 11.25). The particle of mass m moves to the right with a speed v while the particle of mass 3m moves to the left with a speed v. What is the total angular momentum of the system relative to (a) the point A, (b) the point O, and (e) the point B? 19. A 1.5-kg particle moves in the xy plane with velocity v = (4.2i - 3.6j) m/s. Determine its angular momentum when its position vector is r = (.5i + 2.2j) m. 20. An airplane of mass 12 000 kg flies level to the ground at an altitude of 10 km with a constant speed of 175 \ 2d AJ Figure 11.25 (Problem 18). m/s relative to the earth. (a) What is the magnitude of the airplane's angular momentum relative to a ground observer who is directly below the airplane? (b) Does this value change as the airplane continues its motion alonga straight line? 21. (a) Calculate the angular momentum of the earth due to its spinning motion about its axis. (b) Calculate the angular momentum of the earth due to its orbital motion about the sun and compare this with (a). (Take the earth-sun distance to be 1.49 X 10 11 m.) 22. A 4-kg mass is attached to a light cord, which is wound around a pulley (Fig. 10.18). The pulley is a uniform solid cylinder of radius 8 cm and mass 2 kg. (a) What is the net torque on the system about the point O?. (b) When the 4-kg mass has a speed v, the pulley has an angular velocity w = v/r. Determin~tal angular momentum of the system about O. (e) Using the fact that T = dl/dt and your result from (b), calculate the acceleration of the 4-kg mass. 1 J3ii A particle of mass mis shot with an initial velocity v 0.J making an angle O with the horizontal as shown in Figure 11.26. The particle moves in the gravitational field of the earth. Find the angular momentum of the particle about the origin when the particle is (a) at the origin, (b) at the highest point of its trajectory, and - (c) just before it hits the ground. (d) What torque causes its angular momentum to change? Figure 11.26 (Problem 23).
PROBLEMS Section 11.4 Rotation of a Rigid Body About a Fixed Axis 24, A uniform solid disk of mass 3 kg and radius 0.2 m rotates about a fixed axis perpendicular to its face. f the angular frequency óf rotation is 6 rad/s, calculate the angular momentum of the disk when the axis of rotation (a) passes through its center of mass and (b) passes through a point midway between the center and the rim. 25. A particle of mass 0.4 kg is attached to the 100-cm mark of a meter stick of mass O.1 kg. The meter stick rotates on a horizontal, smooth table with an angular velocity of 4 rad/s. Calculate the angular momentum of the srstem if the stick is pivoted about an axis (a) perpendicular to the table through the 50-cm mark and (b) perpendicular to the table through the O-cm mark. 26. Two point masses, each 4 kg, are joined by a light rigid rod 0.40 m long. This "dumbbell" system is placed on a horizontal frictionless surface and set into rotation about a vertical axis through the center of mass with an angular speed of 3 rad/s. Find (a) the moment of inertia of the system about the center of mass and (b) the angular momentum about the center of mass. (e) What tangential force, applied to each mass, will produce an angular acceleration of 6 rad/s2? 29. 30. 31. Section 11.5 Conservation of Angular Momentum 27. A cylinder with moment of inertia rotates with angular velocity ru 0 about a vertical, frictionless axle. A second cylinder, with moment of inertia 2 initially not rotating, drops onto the first cylinder (Fig. 11.27). Since the surfaces are rough, the two eventually reach the same angular velocity ro. (a) Calculate to. (b) Show that energy is lost in this situation and calculate the ratio of the) final to the initial kinetic energy. 32. 295 merry-go-round. What is the new angular speed of the merry-go-round (in rpm)? A woman whose mass is 60 kg stands at the rim of a horizontal turntable having a moment of inertia of 500 kg m2 and a radius of 2 m. The system is initially at rest, and the turntable is free to rotate about a frictionless, vertical axle through its center. The woman then starts walking around the rim in a clockwise direction (looking downward) at a constant speed of 1.5 m/s relative to the earth. (a) n what direction and with what angular speed does the turntable rotate? (b) How much work does the woman do to set the system into motion? A uniform rod of mass 100 g and length 50 cm rotates in a horizontal plane about a fixed, vertical, frictionless pin through its center. Two small beads, each of mass 30 g, are mounted on the rod such that they are able to slide without friction along its length. nitially the. beads are held by catches at positions 1 O cm on each side of center, at which time the system rotates at an angular speed of 20 rad/s. Suddenly, the catches are released and the small beads slide outward along the rod. Find (a) the angular speed of the system at the instant the beads reach the ends of the rod, and (b) the angular speed of the rod after the beads fly off the ends. The student in Figure 11.1 7 holds two weights, each of mass 1 O kg. When his arms are extended horizontally, the weights are 1 m from the axis of rotation and he rotates with an angular speed of 2 rad/s. The moment of inertia of the student plus the stool is 8 kg m2 and is assumed to be constant. f the student pulls the weights horizontally to 0.25 m from.the rotation axis, calculate (a) the final angular speed of the system and (b) the change in the mechanical energy of the system. A particle of mass m = 1 O g and speed v0 = 5 m/s collides with and sticks to the edge of a uniform solid sphere of mass M = 1 kg and radius R = 20 cm (Fig. 11.28). f the sphere is initially at rest and is pivoted about a frictionless axle through O perpendicular to the plane of the paper, (a) find the angular velocity of the system after the collision and (b) determine how much energy is lost in the collision. ----m Vo Before After Figure 11.27 (Problem 27). Figure 11.28 (Problem 32). 28. A merry-go-round of radius R = 2 m has a moment of inertia = 250 kg m2, and is rotating at 1 O rpm. A 33. A wooden block of mass M resting on a frictionless horizontal surface is attached to a rigid rod oflength t and of negligible mass (Fig. 11.29). The rod is pivoted child whose mass is 25 kg jumps onto the edge of the ~i ) ', 1
296 CHAPTER 11 ROLLNG MOTON, ANGULAR MOMENTUM, AND TORQUE! ' l; ' l', l:' ',, i i 1 : 1 j at the other end. A bullet of mass m traveling parallel to the horizontal surface and normal to the rod with speed v hits the block and gets embedded in it. (a) What is the angular momentum of the bullet and block system? (b) What fraction of the original kinetic energy is lost in the collision? Figure 11.29 (Problems 33 and 34). 34. Consider the previous problem with f = 2 m, M == 2 kg, m = O g, and v = 200 m/s. n this case the bullet leaves the block with speed v = 25 m/s horizontal to the surface. (a) Determine the angular momentum of the block just after the bullet exits it. (b) Determine the kinetic energy lost in this collision. Neglect the transit time through the block. "Section 11.7 Angular Momentum as a Fundamental Quantity 35. n the Bohr model of the hydrogen atom, the electron moves in a circular orbit of radius 0.529 X 10-10 m around the proton. Assuming the orbital angular momentum of the electron is equal to h, calculate (a) the orbital speed of the electron, (b) the kinetic energy of the electron, and (e) the angular frequency of the electron's motion. ADDTONAL PROBLEMS 36. A uniform solid sphere of radius r is placed on the inside surface of a hemispherical bowl of radius R. The sphere is released from rest at an angle () to the vertical and rolls without slipping (Fig. 11.30). Determine the angular speed of the sphere when it reaches the bottom of the bowl. 37. (a) Compute the kinetic energy of the earth due to its annual orbit around the sun. (b) Compute the rotational kinetic energy of the earth due to its daily rotation about its own axis. (e) Compute thé ratio Korb1JKrotation 38. A thin uniform cylindrical turntable of radius 2 m and mass 30 kg rotates in a horizontal plane with an initial angular velocity, 4n rad/s. The turntable bearing is frictionless. A small clump of clay of mass 0.25 kg is dropped onto the turntable and sticks at a point 1. 8 m from the center of rotation. (a) Find the inal angular velocity of the clay and turntable. (Treat the clay as a point mass.) (b) s mechanical energy conserved in this collision? Explain and use numerical results to verify your answer.. 39. A string is wound around a uniform disk of radius R and mass M. The disk is released from rest with the string vertical and its top end tied to a fixed support (Fig. 11.31). As the disk descends, show that (a) the tension ín the string is one third the weight of the disk, (b) the acceleration of the center of mass is 2g/3, and (e) the velocity of the center of mass is (4gh/3) 1 1 2 Verify your answer to (e) using the energy approach. Figure 11.31 (Problem 39). 40. A constant horizontal force F is applíéd to a lawn roller in the form of a uniform solid cylinder of radius Rand mass M (Fig.. 11.32). f the roller rolls without slipping on the horizontal surface, show that (a) the acceleration of the center of mass is 2F/3M and (b) the minimum coefficient of friction necessary to prevent J. Figure 11.30 (Problem 36). f; Figure 11.32 (Problem 40).
ADDTONAL PROBLEMS 297 slipping is F/3Mg. (Hint: Take the torque with respect to the center of mass.) 41. A light rope passes over a light, frictionless pulley. One end is fastened to a bunch of bananas of mass M, and a monkey of mass M clings to the other end of the rope (Fig. 11.33). The monkey climbs the rope in an attempt to reach the bananas. (a) Treating the system as consisting of the monkey, bananas, rope, and pulley, evaluate the net torque about the pulley axis. (b) Using the results to'(a), determine the total angular momentum about the pulley axis and describe the motion of the system. Will the monkey reach the bananas? 43. This problem describes a method of determining the moment of inertia of an irregularly shaped object such as the payload for a satellite. Figure 11.35 shows one method of determining experimentally. A mass mis suspended by a cord wound around the inner shaft (radius r) of a turntable supporting the object. When the mass is released from rest, it descends uniformly a distance h, acquiring a speed v. Show that the moment of inertia of the equipment (including the turntable) is mr2(2gh/v2-1). Figure 11.35 (Problem 43). ' Figure 11.33 (Problem 41). 42. A small, solid sphere of mass m and radius r rolls without slipping along the track shown in Figure 11.34. f it starts fro/ rest at the top of the track at a height h, where h is large compared to r, (a) what is the minimum value of h (in terms of the radius of the loop R) such that the sphere completes the loop? (b) What are the force components on the sphere at the point P if h= 3R? Figure 1L34 (Problem 42). 44. Consider the problem of the solid sphere rolling down an incline as described in Example 11.1. (a) Choose the axis of the origin for the torque equation as the instantaneous axis through the contact point P and show that the acceleration of the center of mass is given by a = tg sin e. (b) Show that the minimum coefficient of friction such that the sphere will roll without slipping is given by µmin = ftan e. 45. The position vector of a particle of mass 5 kg is given by r = (2 t2i + 3j) m, where tis in seconds. Determine the angular momentum and the torque acting on the particle about the origin. 46. A particle of mass mis located at the vector position r and has a linear momentum p. (a) f rand p both have nonzero x, y, and z components, show that the angular momentum of the particle relative to the origin has components given by L,, = yp, - zpy, Ly= zp - xp,, and L, = xpy - yp,,. (b) f the particle moves only in the xy plane, prove that L,, =u= O and L, == O. 47. A constant torque of 25 N m is applied to a grindstone whose moment of inertia is 0.13 kg m 2. Using energy principles, find the angular speed (in revolutions per second) after the grindstone has made 15 revolutions. (Neglect friction.) 48. t is proposed to power a passenger bus with a massive rotating flywheel that is periodically brought up to its maximum speed (3500 rpm) by means of an electric motor. The flywheel has a mass of 1200 kg, a diameter of 1.8 m, and is in the shape of a solid cylinder. (This is not an efficient shape for a flywheel that is designed to power a vehicle: can you see why?) (a) What is the maximum amount of kinetic energy that can be stored ~. f'. '/ l, 1'1, :
r 298 CHAPTER 11 ROLLNG MOTON, ANGULAR MOMENTUM, AND TORQUE in the flywheel? (b) f the bus requires an average power of 30 hp, how long will the flywheel rotate? 49. A mass mis attached to a cord passing through a small hole in a frictionless, horizontal surface (Fig. 11.36). The mass is initially orbiting in a circle of radius r 0 with velocity v 0 The cord is then slowly pulled from below, decreasing the radius of the circle to r. (a) What is the velocity of the mass when the radius is r? (b) Find the tension in the cord as a function of r. (e) How much work is done in moving m from r 0 tor? (Note: The tension depends on r.) (d) Obtain numerical values for v, T, and W when r = O.1 m, if m = 50 g, r 0 = 0.3 m, and v 0 = 1.5 m/s.... ----,-...:. F. ', ( ~, ro.' ) '- ~,... m... 1-r. Vo vehicle)? (e) Find the values of F,, and F 11 given that W= 1500 N, d = 0.8 m,l= 3m,h= l.5m,arida= -2 m/s 2 ' 52. (a) A thin rod oflength hand mass Mis held vertically up with its lower end resting on a frictionless horizontal surface. The rod is then let go to fall freely. Determine the velocity of its center of mass just before it hits the horizontal surface. (b) Suppose the rod were pivoted at its lower end. Determine the speed of the rod's center of mass just before it hits the surface. 53. Two astronauts (Figure 11.38), each having a mass of 75 kg, are connected by a 10-m rope of negligible mass. They are isolated in space, orbiting their center of mass at speeds of 5 m/s. Calculate (a) the magnitude of the angular momentum of the system by treating the astronauts as particles and (b) the kinetic energy of the system. By pulling in on the rope, the astronauts shorten the distance between them to 5 m. (c) What is the new angular momentum of the system? (d) What are their new speeds? (e) What is the new kinetic energy of the system? (f) How much work is done by the astronauts in shortening the rope? Figure ll.36 (Problem 49). 50. A bowling ball is given an initial speed v 0 on an alley such that it initially slides without rolling. The coefficient of friction between the ball and the alley is µ. Show that at the time pure rolling motion occurs, (a) the velocity of the ball's center of mass is 5v 0 /7 and (b) the distance it has traveled is 12v 02 /49 µg. (Hint: When pure rolling motion occurs, ve = Rw. Since the frictional force provides the deceleration, from Newton's second law it follows that ªe= -µg.) 51. A trailer with loaded weight W is being pulled by a vehicle with a force F, as in Figure 11.37. The trailer is loaded such that its center of gravity is located as shown. Neglect the force of rolling friction and assume the trailer has an acceleration a. (a) Find the vertical component of Fin terms of the given parameters. (b) f a= 2 m/s 2 and h = 1.5 m, what must be the value of din order that F 11 = O (no vertical load on the e \ f:"-----10 m~-/j \ Figure 11.38 (Problem 53). 54. A solid cube of wood of side 2a and mass Mis resting on a horizontal surface. The cube is constrained to rotate about an axis AB (see Figure 11.39). A bullet of mass m and speed v is shot at the face opposite to ABCD at a height of 4a/3. The bullet gets embedded in the block. Find the minimum value of the v required to tip the cube to fall on its face ABCD. Assume m-s M. 1 " A Figure 11.39 (Problem 54). Figure 11.37 (Problem 51). 55. Toppling chimneys often break apart in mid-fall because the mortar between the bricks cannot withstand
ADDffiONAL PROBLEMS 299 much tension force. As the chimney falls, this tension supplies the centripetal forces on the topmost segments that they need to keep them traveling in a circle. Consider a long uniform rod oflength f pivoted at the lower end. The rod starts at rest in a vertical position and falls over under the influence of gravity. What fraction of the length of the rod will have a tangential acceleration greater than the component of gravitational acceleration in the tangential direction? 56. A solid sphere is released from height h from the top ofan incline making an angle O with the horizontal. Calculate the velocity of the sphere when it reaches the bott~m of the incline in the case that (a) it rolls without slipping, and (b) it slips frictionlessly without rolling. Compare the times required to reach the bottom in cases (a) and (b). 57. A spool of wire of mass Mand radius R is unwound under a constant force F (Fig. ll.40). Assuming the spool is a uniform solid cylinder that doesn't slip, show that (a) the acceleration of the center of mass is 4F/3M and (b) the force of friction is to the right and equal to F/3. (e) f the cylinder starts from rest and rolls without slipping, what is the velocity of its center of mass after it has rolled through a distance d? (Assume the force remains constant.) Figure 11.40 (Problem 57). 58. A uniform solid disk is set into rotation about an axis through its center with an angular velocity w 0 The rotating disk is lowered to a rough, horizontal surface with this angular velocity and released as in Figure 11.41. (a) What is the angular velocity of the disk once pure rolling takes place? (b) Find the fractional loss in kinetic energy from the time the disk is released until pure rolling occurs. (Hint: Consider torques about the center of mass.) F Figure 11.41 (Problems 58 and 59). 59. Suppose a solid disk of radius R is given an angular velocity w 0 about an axis through its center and is then lowered to a rough, horizontal surface and released, as in Problem 58 (Fig. 11.41). Furthermore, assume that the coefficient of friction between the disk and surface isµ. (a) Show thatthe time it takes pure rolling motion to occur is given by R Vo/3µg. (b) Show that the distance the disk travels before pure rolling occurs is given by R2Wo2/8 µg. 60. A large, cylindrical roll of tissue paper of initial radius R lies on a long, horizontal surface with the open end of the paper nailed to the surface so that it can unroll easily. The roll is given a slight shove (v 0 "" O) and commences to unroll. (a) Determine the speed of the, center of mass of the roll when its radius has diminished to r. (b) Calculate a numerical value for this speed at r = 1 mm, assuming R = 6 m. (e) What happens to the energy of the system when the paper is completely unrolled? (Hint: Assume the roll has a uniform density and apply energy methods.) 61. A solid cube of side 2a and mass Mis sliding on a frictionless surface with uniform velocity v 0 as in Figure 11.42a. t hits a small obstacle at the end of the table, which causes the cube to tilt as in Figure ll.42b. Find the minimum value of v 0 such that the cube falls off the table. Note that the moment of inertia of the cube about an axis along one of its edges is 8Ma2/3. (Hint: The cube undergoes an inelastic collision at the edge.) Figure 11,42 (Problem 61). (a) (b)