21-41. Derive the scalar form of the rotational equation

21-1. Show that the sum of the moments of inertia of a body, Ixx + /yy + is independent of the orientation of the .v, y, z axes and thus depends only on the location of the origin.

21-2. Determine the moment of inertia of the cone with respect to a vertical y axis passing through the cone's center of mass. What is the moment of inertia about a parallel axisy' that passing through the diameter of the base of the cone? The cone has a mass m.

21-3. Determine the moments of inertia lx and /,. of the paraboloid of revolution. The mass of the paraboloid is m.

*21-4. Determine the radii of gyration kx and ky for the solid formed by revolving the shaded area about the y axis. The density of the material is p.

21-5. Determine by direct integration the product of inertia lxz for the homogeneous prism. The density of the material is p. Express the result in terms of the total mass m of the prism.

Determine by direct integration the product of inertia \(I_{x y}\) for the homogeneous prism. The density of the material is p. Express the result in terms of the total mass m of the prism.

Determine the product of inertia \(I_{x y}\) of the object formed by revolving the shaded area about the line x = 5 ft. Express the result in terms of the density of the material, \(\rho\).

*21-8. Determine the moment of inertia ly of the object formed by revolving the shaded area about the line ,r = 5 ft. Express the result in terms of the density of the material, p.

21-9. Determine the elements of the inertia tensor for the cube with respect to the x.y, z coordinate system. The mass of the cube is m.

21-10. Determine the mass moment of inertia of the homogeneous block with respect to its centroidal x' axis.

2111. Determine the moment of inertia of the cylinder with respect to the a-a axis of the cylinder.The cylinder has a mass m.

*21-12. Determine (he moment of inertia /w of the composite plate assembly.The plates have a specific weight of 6 lb/fr.

21-13. Determine the product of inertia Iyz of the composite plate assembly.The plates have a weight of 6 Ib/ft2.

21-14. Determine the products of inertia fxp Iyz, and 1XV of the thin plate.The material has a density per unit area of 50 kg/m2.

21-15. Determine the products of inertia /vv, /v. and /,. of the solid.The material is steel, which has a specific weight of 490 Ib/ft3.

The bent rod has a mass of 4 kg/m. Determine the moment of inertia of the rod about the Oa axis.

21-17. The bent rod has a weight of 1.5 Ib/ft. Locate the center of gravity G(*. y) and determine the principal moments of inertia /v-, /Y*. and /- of the rod with respect to the x\ y\z' axes.

21-18. Determine the moments of inertia about the x, y, z *21-20. The assembly consists of a 15-lb plate /4,40-lb axes of the rod assembly. The rods have a mass of 0.75 kg/m. plate B. and four 7-lb rods. Determine the moments of inertia of the assembly with respect to the principal x, y, z axes.

21-19. Determine the moment of inertia of the composite body about the aa axis. The cylinder weighs 20 lb. and each hemisphere weighs 10 lb.

21-20. The assembly consists of a 15-lb plate /4,40-lb plate B. and four 7-lb rods. Determine the moments of inertia

21-21. Determine the moment of inertia of the rod-and-thin-ring assembly about the z axis. The rods and ring have a mass per unit length of 2 kg/m.

21-22. It a body contains no planes of symmetry. the principal moments of inertia can be determined mathematically.To show how this is done, consider the rigid body which is spinning with an angular velocity to, directed along one of its principal axes of inertia. If the principal moment of inertia about this axis is /. the angular momentum can be expressed as H = ho = ltox\ + Itoyj + Ito. k. The components of H may also be expressed by Eqs. 21-10, where the inertia tensor is assumed to be known. Equate the i. j, and k components of both expressions for H and consider tox,toy, and to. to be unknown.The solution of these three equations is obtained provided the determinant of the coefficients is zero. Show that this determinant, when expanded, yields the cubic equation 4 - 4 - 4)/ - (Uyyhz - 2/xvW* ~ - /vv4 - Izfy = 0

21-23. Show that if the angular momentum of a body is determined with respect to an arbitrary point A. then H4 can be expressed by Eq. 21-9. This requires substituting Pa = Pc, + Po/a Eq. 21-6 and expanding, noting that J fa dm = 0 by definition of the mass center and VG = V* + OJ X . The three positive roots of I, obtained from the solution of this equation, represent the principal moments of inertia /,. 7V, and /..

*21-24. The 15-kg circular disk spins about its axle with a constant angular velocity of to{ = 10 rad/s. Simultaneously, the yoke is rotating with a constant angular velocity of (o2 = 5 rad/s. Determine the angular momentum of the disk about its center of mass O, and its kinetic energy.

The cone has a mass m and rolls without slipping on the conical surface so that it has an angular velocity about the vertical axis of \(\boldsymbol{\omega}\). Determine the kinetic energy of the cone due to this motion.

21-26. The circular disk has a weight of 15 lb and is mounted on the shaft AB at an angle of 45C1 with the horizontal. Determine the angular velocity of the shaft when / = 3 s if a constant torque M = 2 lb ft is applied to the shaft. The shaft is originally spinning at &>, = 8rad/s when the torque is applied.

21-27. The circular disk has a weight of 15 lb and is mounted on the shaft AB at an angle of 45 with the horizontal. Determine the angular velocity of the shaft when / = 2 s if a torque M = (4e01/) lb* ft. where t is in seconds, is applied to the shaft. The shaft is originally spinning at w, =8 rad/s when the torque is applied.

:5:2128. The space capsule has a mass of 5 Mg and the radii of gyration are kx = kz = 1.30 m and ky 0.45 m. If it travels with a velocity v<; = {400j 4- 200k} m/s, compute its angular velocity just after it is struck by a meteoroid having a mass of 0.80 kg and a velocity v, = {-300i + 200j 150k} m/s. Assume that the meteoroid embeds itself into the capsule at point A and that the capsule initially has no angular velocity.

21-29. The 2-kg gear A rolls on the fixed plate gear C. Determine the angular velocity of rod OB about the z axis after it rotates one revolution about the z axis, starting from rest. The rod is acted upon by the constant moment M = 5 N m. Neglect the mass of rod OB. Assume that gear A is a uniform disk having a radius of 100 mm.

21-30. The rod weighs 3 lb/ft and is suspended from parallel cords at A and B. If the rod has an angular velocity of 2 rad/s about the z axis at the instant shown, determine how high the center of the rod rises at the instant the rod momentarily stops swinging.

21-31. Rod AB has a weight of 6 lb and is attached to two smooth collars at its ends by ball-and-socket joints. If collar A is moving downward with a speed of 8 ft/s when z = 3 ft. determine the speed of A at the instant c = 0. The spring has an unstretched length of 2 ft. Neglect the mass of the collars. Assume the angular velocity of rod AB is perpendicular to its axis.

*21-32. The 5-kg circular disk spins about AB with a constant angular velocity of w, =15 rad/s. Simultaneously, the shaft to which arm OAB is rigidly attached, rotates with a constant angular velocity of a>2 =6 rad/s. Determine the angular momentum of the disk about point O. and its kinetic energy.

21-33. The 20-kg sphere rotates about the axle with a constant angular velocity of oj- = 60 rad/s. If shaft AB is subjected to a torque of M = 50 N-m, causing it to rotate, determine the value of top after the shaft has turned 90 from the position shown. Initially. = 0. Neglect the mass of arm CDE.

21-34. The 200-kg satellite has its center of mass at point G. Its radii of gyration about the z\ x\ y' axes are kz = 300 mm. kx< = ky- = 500 mm. respectively. At the instant shown, the satellite rotates about the .v\ y \ and z! axes with the angular velocity shown, and its center of mass G has a velocity of \G = {2501 + 200j + 120k} m/s. Determine the angular momentum of the satellite about point A at this instant.

21-35. The 200-kg satellite has its center of mass at point G. Its radii of gyration about thez',.r',y' axes are kz> = 300 mm, kx' = ky = 500 mm. respectively. At the instant shown, the satellite rotates about the x\y\ and z' axes with the angular velocity shown, and its center of mass G has a velocity of vc = {2501 + 200j + 120k} m/s. Determine the kinetic energy of the satellite at this instant.

The 15-kg rectangular plate is free to rotate about the y axis because of the bearing supports at A and B. When the plate is balanced in the vertical plane, a 3-g bullet is fired into it. perpendicular to its surface, with a velocity \(\mathbf{v}=\{-2000 \mathbf{i}\} \mathrm{m} / \mathrm{s}\). Compute the angular velocity of the plate at the instant it has rotated \(180^{\circ}\). If the bullet strikes corner D with the same velocity v. instead of at C, does the angular velocity remain the same? Why or why not?

21-37. The circular plate has a weight of 19 lb and a diameter of 1.5 ft. If it is released from rest and falls horizontally 2.5 ft onto the hook at S, which provides a permanent connection, determine the velocity of the mass center of the plate just after the connection with the hook is made.

21-38. The 10-kg disk rolls on the horizontal plane without slipping. Determine the magnitude of its angular momentum when it is spinning about the y axis at 2 rad/s.

If arm OA is subjected to a torque of \(M=5 \mathrm{~N} \cdot \mathrm{m}\), determine the spin angular velocity of the 10-kg disk after the arm has turned 2 rev. starting from rest. The disk rolls on the horizontal plane without slipping. Neglect the mass of the arm.

21-40. Derive the scalar form of the rotational equation of motion about the x axis if ft # to and the moments and products of inertia of the body arc not constant with respect to time.

21-41. Derive the scalar form of the rotational equation of motion about the x axis if ft = to and the moments and products of inertia of the body are constant with respect to time.

Derive the Euler equations of motion for \(\Omega \neq \omega\), i.e.. Eqs. 21-26.

21-43. The 4-lb bar rests along the smooth corners of an open box. At the instant shown, the box has a velocity v = {3j} ft/s and an acceleration a = {-6j } ft/s2. Determine the x, y, z. components of force which the corners exert on the bar.

*21-44. The uniform plate has a mass of m = 2 kg and is given a rotation of = 4 rad/s about its bearings at A and B. Tf a = 0.2 m and c = 0.3 m. determine the vertical reactions at the instant shown. Use the x, y, z axes shown and note that /

21-45. If the shaft AB is rotating with a constant angular velocity of w = 30 rad/s, determine the X, Y, Z components of reaction at the thrust bearing A and journal bearing B at the instant shown. The disk has a weight of 15 lb. Neglect the weight of the shaft AB.

21-46. The 40-kg flywheel (disk) is mounted 20 mm off its true center at G. If the shaft is rotating at a constant speed a) = 8 rad/s, determine the maximum reactions exerted on the journal bearings at A and B.

The 40-kg flywheel (disk) is mounted 20 mm off its true center at G. If the shaft is rotating at a constant speed \(\omega=8\ \mathrm{rad}/\mathrm{s}\). determine the minimum reactions exerted on the journal bearings at A and B during the motion.

*21-48. The man sits on a swivel chair which is rotating with a constant angular velocity of 3 rad/s. He holds the uniform 5-lb rod AB horizontal. He suddenly gives it an angular acceleration of 2 rad/s2. measured relative to him. as shown. Determine the required force and moment components at the grip. A. necessary to do this. Establish axes at the rods center of mass G, with +z upward, and +y directed along the axis of the rod towards A.

21-49. The 5-kg rod AB is supported by a rotating arm. The support at A is a journal bearing, which develops reactions normal to the rod. The support at B is a thrust bearing, which develops reactions both normal to the rod and along the axis of the rod. Neglecting friction, determine the x, y. z components of reaction at these supports when the frame rotates with a constant angular velocity of o) = 10 rad/s.

21-50. The rod assembly is supported by a ball-and-socket joint at Cand a journal bearing at D, which develops only x and y force reactions. The rods have a mass of 0.75 kg/m. Determine the angular acceleration of the rods and the components of reaction at the supports at the instant o) = 8 rad/s as shown.

21-51. The uniform hatch door, having a mass of 15 kg and a mass center at G. is supported in the horizontal plane by bearings at A and B. If a vertical force F = 300 N is applied to the door as shown, determine the components of reaction at the bearings and the angular acceleration of the door. The bearing at A will resist a component of force in the y direction, whereas the bearing at B will not. For the calculation, assume the door to be a thin plate and neglect the size of each bearing.The door is originally at rest.

*21-52. The conical pendulum consists of a bar of mass m and length L that is supported by the pin at its end A. If the pin is subjected to a rotation to, determine the angle 0 that the bar makes with the vertical as it rotates.

21-54. The rod assembly is supported by journal bearings at A and B. which develop only x and z force reactions on the shaft. If the shaft AB is rotating in the direction shown at to = {5j} rad/s, determine the reactions at the bearings when the assembly is in the position shown. Also, what is the shafts angular acceleration? The mass of each rod is 1.5 kg/m.

21-53. The car travels around the curved road of radius p such that its mass center has a constant speed %. Write the equations of rotational motion with respect to the .v. y, z axes. Assume that the car's six moments and products of inertia with respect to these axes are known.

The 20-kg sphere is rotating with a constant angular speed of \(\omega_1=150\mathrm{\ rad}/\mathrm{s}\) about axle CD. which is mounted on the circular ring. The ring rotates about shaft AB with a constant angular speed of \(\omega_2=50\mathrm{\ rad}/\mathrm{s}\). If shaft AB is supported by a thrust bearing at A and a journal bearing at B, determine the X. Y. Z components of reaction at these bearings at the instant shown. Neglect the mass of the ring and shaft.

*21-56. The rod assembly has a weight of 5 lb/ft. It is supported at B by a smooth journal bearing, which develops x and v force reactions, and at A by a smooth thrust bearing, which develops x, y, and z force reactions. If a 50-lb ft torque is applied along rod AH. determine the components of reaction at the bearings when the assembly has an angular velocity to = 10 rad/s at the instant shown.

21-57. The blades of a wind turbine spin about the shaft S with a constant angular speed of <ox. while the frame precesses about the vertical axis with a constant angular speed of top. Determine the x, y, and z components of moment that the shaft exerts on the blades as a function of 0. Consider each blade as a slender rod of mass m and length /.

21-58. The cylinder has a mass of 30 kg and is mounted on an axle that is supported by bearings at A and B. If the axle is turning at to = { 40j} rad/s, determine the vertical components of force acting at the bearings at this instant.

21-59. The thin rod has a mass of 0.8 kg and a total length of 150 mm. It is rotating about its midpoint at a constant rate t) = 6 rad/s, while the table to which its axle A is fastened is rotating at 2 rad/s. Determine the x, y, z moment components which the axle exerts on the rod when the rod is in any position 0.

21-60. Show that the angular velocity of a body, in terms of Euler angles 4> 0. and <//. can be expressed as a) = (<j> sin 0 sin <// 4- 0 cos i//)i 4- (<f> sin 0 cos 4' ~ sin </0j 4 (4> cos 0 4- <//)k. where i. j. and k are directed along the x. y. z axes as shown in Fig. 2115^/.

21-61. A thin rod is initially coincident with the Z axis when it is given three rotations defined by the Euler angles 4> = 30. 0 = 45. and <// = 60. If these rotations are given in the order stated, determine the coordinate direction angles a, (3. y of the axis of the rod with respect to the X. Y. and Z axes. Are these directions the same for any order of the rotations? Why?

21-62. The turbine on a ship has a mass of 400 kg and is mounted on bearings A and 13 as shown. Its center of mass is at G. its radius of gyration is k. = 0.3 m, and kx = kY = 0.5 m. If it is spinning at 200 rad/s, determine the vertical reactions at the bearings when the ship undergoes each of the following motions: (a) rolling, = 0.2 rad/s, (b) turning, co2 = 0.8 rad/s, (c) pitching, o>3 = 1.4 rad/s.

21-63. The 10-kg disk spins about axle AB at a constant rate of = 100 rad/s. If the supporting arm processes about the vertical axis at a constant rate of tof) = 5 rad/s. determine the internal moment at O caused only by the gyroscopic action.

*21-64. The 10-kg disk spins about axle AB at a constant rate of <as = 250 rad/s. and f) = 30. Determine the rate of precession of arm OA. Neglect the mass of arm OA. axle AB. and the circular ring D.

21-65. When OA precesses at a constant rate of (up = 5 rad/s, when 0 = 90, determine the required spin of the 10-kg disk C. Neglect the mass of arm OA, axle A B, and the circular ring D.

21-66. The car travels at a constant speed of vc = 100 km/h around the horizontal curve having a radius of 80 m. If each wheel has a mass of 16 kg. a radius of gyration kG = 300 mm about its spinning axis, and a radius of 400 mm, determine the difference between the normal forces of the rear wheels, caused by the gyroscopic effect. The distance between the wheels is 1.30 m.

21-67. A wheel of mass m and radius r rolls with constant spin co about a circular path having a radius a. If the angle of inclination is 0, determine the rate of precession. Treat the wheel as a thin ring. No slipping occurs.

*21-68. The top consists of a thin disk that has a weight of 8 lb and a radius of 0.3 ft.The rod has a negligible mass and a length of 0.5 ft. If the top is spinning with an angular velocity (i= 300rad/s. determine the steady-state precessional angular velocity u)f, of the rod when 0 = 40.

21-69. Solve Prob. 21-68 when 0 = 90.

21-70. The top has a mass of 90 g. a center of mass at G, and a radius of gyration k = 18 mm about its axis of symmetry. About any transverse axis acting through point O the radius of gyration is k, = 35 mm. If the top is connected to a ball-and-socket joint at O and the precession is cop = 0.5 rad/s, determine the spin co,.

21-71. The 1-lb top has a center of gravity at point G. If it spins about its axis of symmetry and precesses about the vertical axis at constant rates of cos = 60 rad/'s and <op = lOrad/s, respectively, determine the steady state angle 61.The radius of gyration of the top about the z axis is k. = 1 in., and about the x and y axes it is kx = ky = 4 in.

*21-72. While the rocket is in free flight, it has a spin of 3 rad/s and precesses about an axis measured 10 from the axis of spin. If the ratio of the axial to transverse moments of inertia of the rocket is 1/15, computed about axes which pass through the mass center G\ determine the angle which the resultant angular velocity makes with the spin axis. Construct the body and space cones used to describe the motion. Is the precession regular or retrograde?

21-73. The 0.2-kg football is thrown with a spin co: = 35 rad/s. If the angle 0 is measured as 60. determine the precession about the Z axis. The radius of gyration about the spin axis is k. = 0.05 m, and about a transverse axis it is k, = 0.1 m.

21-74. The projectile shown is subjected to torque-free motion. The transverse and axial moments of inertia are I and /-, respectively. If 0 represents the angle between the precessional axis Z and the axis of symmetry z, and /3 is the angle between the angular velocity oj and the z axis, show that /3 and 0 are related by the equation tan H = (///,) tan fi.

21-75. The 4-kg disk is thrown with a spin m: = 6 rad/s. If the angle 6 is measured as 160, determine the precession about the Z axis.

*21-76. The rocket has a mass of 4 Mg and radii of gyration k. = 0.85 m and ky = 2.3 m. It is initially spinning about the z axis at w: = 0.05 rad/s when a meteoroid M strikes it at A and creates an impulse I = {300i} N-s. Determine the axis of precession after the impact.

The football has a mass of 450 g and radii of gyration about its axis of symmetry (z axis) and its transverse axes (x or y axis) of \(k_{z}=30 \mathrm{\ mm} \text { and } k_{x}=k_{y}=50 \mathrm{\ mm}\), respectively. If the football has an angular momentum of \(H_{G}=0.02 \mathrm{~kg} \cdot \mathrm{m}^{2} / \mathrm{s}\), determine its precession \(\dot{\phi}\) and spin \(\dot{\psi}\). Also, find the angle \(\beta\) that the angular velocity vector makes with the z axis.

21-78. The projectile precesses about the Z axis at a constant rate of <j) = 15 rad/s when it leaves the barrel of a gun. Determine its spin ij/ and the magnitude of its angular momentum H,f.The projectile has a mass of 1.5 kg and radii of gyration about its axis of symmetry (z axis) and about its transverse axes (x and y axes) of kz = 65 mm and kx = ky = 125 mm. respectively.

21-79. The space capsule has a mass of 3.2 Mg. and about axes passing through the mass center G the axial and transverse radii of gyration are kz = 0.90 m and k, = 1.85 m, respectively. If it is spinning at tos = 0.8 rev/s. determine its angular momentum. Precession occurs about the Z axis.