Dynamics · Engineering Mechanics · Open Problem Companion

Chapter Twenty-One
Three-Dimensional Kinetics of a Rigid Body

21.1 Moments & Products of Inertia 23 problems
Concepts
  1. 1.Show that the sum of the moments of inertia of a body, \(I_{xx} + I_{yy} + I_{zz}\), is independent of the orientation of the \(x,y,z\) axes and thus depends only on the location of the origin.
  2. 2.Determine the moment of inertia of the cone with respect to a vertical \(\overline{y}\) axis passing through the cone's center of mass. What is the moment of inertia about a parallel axis \(y'\) that passes through the diameter of the base of the cone? The cone has a mass \(m\). figure — add your own
  3. 3.Determine the moments of inertia \(I_x\) and \(I_y\) of the paraboloid of revolution. The mass of the paraboloid is \(m\). figure — add your own
  4. 4.Determine by direct integration the product of inertia \(I_{yz}\) for the homogeneous prism. The density of the material is \(\rho\). Express the result in terms of the total mass \(m\) of the prism. figure — add your own
  5. 5.Determine by direct integration the product of inertia \(I_{xy}\) for the homogeneous prism. The density of the material is \(\rho\). Express the result in terms of the total mass \(m\) of the prism. figure — add your own
  6. 6.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 \(m\). figure — add your own
  7. 7.Determine the products of inertia \(I_{xy}, I_{yz}\), and \(I_{xz}\) of the homogeneous solid. The material has a density of \(7.85 \text{ Mg/m}^3\). figure — add your own
  8. 8.The assembly consists of two square plates \(A\) and \(B\) which have a mass of \(3 \text{ kg}\) each and a rectangular plate \(C\) which has a mass of \(4.5 \text{ kg}\). Determine the moments of inertia \(I_x, I_y\) and \(I_z\). figure — add your own
  9. 9.Determine the moment of inertia of the cone about the \(z'\) axis. The mass of the cone is \(15 \text{ kg}\), the height is \(h = 1.5 \text{ m}\), and the radius is \(r = 0.5 \text{ m}\). figure — add your own
  10. 10.Determine the moment of inertia about the \(z\) axis of the assembly which consists of the \(1.5\text{-kg}\) rod \(CD\) and the \(7\text{-kg}\) disk. figure — add your own
  11. 11.Determine the moment of inertia of the cylinder with respect to the \(a-a\) axis of the cylinder. The cylinder has a mass \(m\). figure — add your own
  12. 12.Determine the product of inertia \(I_{xy}\) for the bent rod. The rod has a mass per unit length of \(2 \text{ kg/m}\). figure — add your own
  13. 13.Determine the moments of inertia \(I_{xx}, I_{yy}, I_{zz}\) for the bent rod. The rod has a mass per unit length of \(2 \text{ kg/m}\). figure — add your own
  14. 14.Determine the products of inertia \(I_{xy}, I_{yz}\) and \(I_{xz}\) of the thin plate. The material has a mass per unit area of \(50 \text{ kg/m}^2\). figure — add your own
  15. 15.Determine the moment of inertia of both the \(1.5 \text{ kg}\) rod and \(4 \text{ kg}\) disk about the \(z'\) axis. figure — add your own
  16. 16.The bent rod has a mass per unit length of \(3 \text{ kg/m}\). Determine the moment of inertia of the rod about the \(O-a\) axis. figure — add your own
  17. 17.The bent rod has a mass per unit length of \(8.49 \text{ kg/m}\). Locate the center of gravity \(G(\overline{x}, \overline{y})\) and determine the principal moments of inertia \(I_x, I_y,\) and \(I_z\) of the rod with respect to the \(x', y', z'\) axes. figure — add your own
  18. 18.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 \(2 \text{ kg/m}\). figure — add your own
  19. 19.The slender rod has a mass per unit length of \(6 \text{ kg/m}\). Determine its moments and products of inertia with respect to the \(x,y,z\) axes. figure — add your own
  20. 20.Determine the moment of inertia of the disk about the axis of shaft \(AB\). The disk has a mass of \(15 \text{ kg}\). figure — add your own
  21. 21.Determine the moments of inertia about the \(x,y,z\) axes of the rod assembly. The rods have a mass of \(0.75 \text{ kg/m}\). figure — add your own
  22. 22.If 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 \(\omega\), directed along one of its principal axes of inertia. If the principal moment of inertia about this axis is \(I\), the angular momentum can be expressed as \(\mathbf{H} = I\omega = I\omega_x\mathbf{i} + I\omega_y\mathbf{j} + I\omega_z\mathbf{k}\). The components of \(\mathbf{H}\) may also be expressed by Eqs. 21-10, where the inertia tensor is assumed to be known. Equate the \(\mathbf{i}, \mathbf{j}\), and \(\mathbf{k}\) components of both expressions for \(\mathbf{H}\) and consider \(\omega_x, \omega_y,\) and \(\omega_z\) 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 \[ I^3 - (I_{xx} + I_{yy} + I_{zz})I^2 + (I_{xx}I_{yy} + I_{yy}I_{zz} + I_{zz}I_{xx} - I_{xy}^2 - I_{yz}^2 - I_{zx}^2)I - (I_{xx}I_{yy}I_{zz} - 2I_{xy}I_{yz}I_{zx} - I_{xz}I_{yz}^2 - I_{yz}I_{zx}^2 - I_{zx}I_{xy}^2) = 0 \] The three positive roots of \(I\) obtained from the solution of this equation represent the principal moments of inertia \(I_x, I_y,\) and \(I_z\). figure — add your own
  23. 23.Show that if the angular momentum of a body is determined with respect to an arbitrary point \(A\), then \(\mathbf{H}_A\) can be expressed by Eq. 21-9. This requires substituting \(\boldsymbol{\rho}_A = \boldsymbol{\rho}_G + \boldsymbol{\rho}_{G/A}\) into Eq. 21-6 and expanding, noting that \(\int \boldsymbol{\rho}_G \, dm = \mathbf{0}\) by definition of the mass center and \(\mathbf{v}_G = \mathbf{v}_A + \boldsymbol{\omega} \times \boldsymbol{\rho}_{G/A}\). figure — add your own
21.2 Principal Axes & Inertia Tensor 11 problems
Skills
  1. 24.The 15-kg circular disk spins about its axle with a constant angular velocity of \(\omega_1 = 10 \text{ rad/s}\). Simultaneously, the yoke is rotating with a constant angular velocity of \(\omega_2 = 5 \text{ rad/s}\). Determine the angular momentum of the disk about its center of mass \(O\), and its kinetic energy. figure — add your own
  2. 25.The assembly consists of a \(4\text{-kg}\) rod \(AB\) which is connected to link \(OA\) and the collar at \(B\) by ball-and-socket joints. When \(\theta = 0^\circ\), \(y = 600 \text{ mm}\), the system is at rest, the spring is unstretched, and a couple moment \(M = 7 \text{ N}\cdot\text{m}\) is applied to the link at \(O\). Determine the angular velocity of the link at the instant \(\theta = 90^\circ\). Neglect the mass of the link. figure — add your own
  3. 26.The assembly consists of a \(4\text{-kg}\) rod \(AB\) which is connected to link \(OA\) and the collar at \(B\) by ball-and-socket joints. When \(\theta = 0^\circ\), \(y = 600 \text{ mm}\), the system is at rest, the spring is unstretched, and a couple moment \(M = (4\theta + 2) \text{ N}\cdot\text{m}\), where \(\theta\) is in radians, is applied to the link at \(O\). Determine the angular velocity of the link at the instant \(\theta = 90^\circ\). Neglect the mass of the link. figure — add your own
  4. 27.The \(4\text{-kg}\) rod \(AB\) is attached to the \(1\text{-kg}\) collar at \(A\) and a \(2\text{-kg}\) link \(BC\) using ball-and-socket joints. If the rod is released from rest in the position shown, determine the angular velocity of the link after it has rotated \(180^\circ\). figure — add your own
  5. 28.The large gear has a mass of \(5 \text{ kg}\) and a radius of gyration of \(k_z = 75 \text{ mm}\). Gears \(B\) and \(C\) each have a mass of \(200 \text{ g}\) and a radius of gyration about the axis of their connecting shaft of \(15 \text{ mm}\). If the gears are in mesh and \(C\) has an angular velocity of \(\omega_c = \{15\mathbf{j}\} \text{ rad/s}\), determine the total angular momentum for the system of three gears about point \(A\). figure — add your own
  6. 29.The rod assembly is supported at \(G\) by a ball-and-socket joint. Each segment has a mass per unit length of \(0.5 \text{ kg/m}\). If the assembly is originally at rest and an impulse of \(\mathbf{I} = \{-8\mathbf{k}\} \text{ N}\cdot\text{s}\) is applied at \(D\), determine the angular velocity of the assembly just after the impact. figure — add your own
  7. 30.The \(20\text{-kg}\) sphere rotates about the axle with a constant angular velocity of \(\omega_s = 60 \text{ rad/s}\). If shaft \(AB\) is subjected to a torque of \(M = 50 \text{ N}\cdot\text{m}\), causing it to rotate, determine the value of \(\omega_p\) after the shaft has turned \(90^\circ\) from the position shown. Initially, \(\omega_p = 0\). Neglect the mass of arm \(CDE\). figure — add your own
  8. 31.The space capsule has a mass of \(5 \text{ Mg}\) and the radii of gyration are \(k_x = k_z = 1.30 \text{ m}\) and \(k_y = 0.45 \text{ m}\). If it travels with a velocity \(\mathbf{v}_G = [400\mathbf{j} + 200\mathbf{k}] \text{ m/s}\), compute its angular velocity just after it is struck by a meteoroid having a mass of \(0.80 \text{ kg}\) and a velocity \(\mathbf{v}_m = [-300\mathbf{i} + 200\mathbf{j} - 150\mathbf{k}] \text{ m/s}\). Assume that the meteoroid embeds itself into the capsule at point \(A\) and that the capsule initially has no angular velocity. figure — add your own
  9. 32.The \(5\text{-kg}\) thin plate is suspended at \(O\) using a ball-and-socket joint. It is rotating with a constant angular velocity \(\omega = \{2\mathbf{k}\} \text{ rad/s}\) when the corner \(A\) strikes the hook at \(S\), which provides a permanent connection. Determine the angular velocity of the plate immediately after impact. figure — add your own
  10. 33.The \(2\text{-kg}\) thin disk is connected to the slender rod which is fixed to the ball-and-socket joint at \(A\). If it is released from rest in the position shown, determine the spin of the disk about the rod when the disk reaches its lowest position. Neglect the mass of the rod. The disk rolls without slipping. figure — add your own
  11. 34.The 5-kg circular disk spins about \(AB\) with a constant angular velocity of \(\omega_1 = 15 \text{ rad/s}\). Simultaneously, the shaft to which arm \(OAB\) is rigidly attached rotates with a constant angular velocity of \(\omega_2 = 6 \text{ rad/s}\). Determine the angular momentum of the disk about point \(O\) and its kinetic energy. figure — add your own
21.3 Angular Momentum & Kinetic Energy 12 problems
Skills
  1. 35.The \(200\text{-kg}\) satellite has its center of mass at point \(G\). Its radii of gyration about the \(z', x', y'\) axes are \(k_{z'} = 300 \text{ mm}\), \(k_{x'} = k_{y'} = 500 \text{ mm}\), respectively. At the instant shown, the satellite has a velocity of \(\mathbf{v}_G = \{-250\mathbf{i} + 200\mathbf{j} + 120\mathbf{k}\} \text{ m/s}\). Determine the angular momentum of the satellite about point \(A\) at this instant. figure — add your own
  2. 36.The \(200\text{-kg}\) satellite has its center of mass at point \(G\). Its radii of gyration about the \(z', x', y'\) axes are \(k_{z'} = 300 \text{ mm}\), \(k_{x'} = k_{y'} = 500 \text{ mm}\), respectively. At the instant shown, the satellite rotates about the \(x', y'\) and \(z'\) axes with the angular velocities shown, and its center of mass \(G\) has a velocity of \(\mathbf{v}_G = \{-250\mathbf{i} + 200\mathbf{j} + 120\mathbf{k}\} \text{ m/s}\). Determine the kinetic energy of the satellite at this instant. figure — add your own
  3. 37.Determine the kinetic energy of the 7-kg disk and 1.5-kg rod when the assembly is rotating about the \(z\) axis at \(\omega = 5 \text{ rad/s}\). figure — add your own
  4. 38.Determine the angular momentum \(\mathbf{H}_z\) of the 7-kg disk and 1.5-kg rod when the assembly is rotating about the \(z\) axis at \(\omega = 5 \text{ rad/s}\). figure — add your own
  5. 39.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 \(\mathbf{v} = \{-2000\mathbf{i}\} \text{ m/s}\). 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 \(\mathbf{v}\), instead of at \(C\), does the angular velocity remain the same? Why or why not? figure — add your own
  6. 40.Derive the scalar form of the rotational equation of motion about the \(x\) axis if \(\boldsymbol{\Omega} \neq \boldsymbol{\omega}\) and the moments and products of inertia of the body are not constant with respect to time.
  7. 41.Derive the scalar form of the rotational equation of motion about the \(x\) axis if \(\boldsymbol{\Omega} \neq \boldsymbol{\omega}\) and the moments and products of inertia of the body are constant with respect to time.
  8. 42.Derive the Euler equations of motion for \(\boldsymbol{\Omega} \neq \boldsymbol{\omega}\), i.e., Eqs. 21-26.
  9. 43.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 \(\omega\), determine the angle \(\theta\) that the bar makes with the vertical as it rotates. figure — add your own
  10. 44.[Problem 44 is omitted in the source.]
  11. 45.The \(40\text{-kg}\) flywheel (disk) is mounted \(20 \text{ mm}\) off its true center at \(G\). If the shaft is rotating at a constant speed \(\omega = 8 \text{ rad/s}\), determine the maximum reactions exerted on the journal bearings at \(A\) and \(B\). figure — add your own
  12. 46.The assembly is supported by journal bearings at \(A\) and \(B\), which develop only \(y\) and \(z\) force reactions on the shaft. If the shaft is rotating in the direction shown at \(\boldsymbol{\omega} = \{2\mathbf{i}\} \text{ rad/s}\), determine the reactions at the bearings when the assembly is in the position shown. Also, what is the shaft's angular acceleration? The mass per unit length of each rod is \(5 \text{ kg/m}\). figure — add your own
  13. 47.The assembly is supported by journal bearings at \(A\) and \(B\), which develop only \(y\) and \(z\) force reactions on the shaft. If the shaft \(A\) is subjected to a couple moment \(\mathbf{M} = \{40\mathbf{i}\} \text{ N}\cdot\text{m}\), and at the instant shown the shaft has an angular velocity of \(\boldsymbol{\omega} = \{2\mathbf{i}\} \text{ rad/s}\), determine the reactions at the bearings of the assembly at this instant. Also, what is the shaft's angular acceleration? The mass per unit length of each rod is \(5 \text{ kg/m}\). figure — add your own
21.4 Equations of Motion 15 problems
Applications
  1. 48.The uniform hatch door, having a mass of \(15 \text{ kg}\) and a mass center at \(G\), is supported in the horizontal plane by bearings at \(A\) and \(B\). If a vertical force \(F = 300 \text{ N}\) 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. figure — add your own
  2. 49.The \(20 \text{ kg}\) sphere is rotating with a constant angular speed of \(\omega_1 = 150 \text{ rad/s}\) about axle \(CD\) which is mounted on the circular ring. The ring rotates about shaft \(AB\) with a constant angular speed of \(\omega_2 = 50 \text{ rad/s}\). 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. figure — add your own
  3. 50.The rod assembly is supported by a ball-and-socket joint at \(C\) and a journal bearing at \(D\) which develops only \(x\) and \(y\) force reactions. The rods have a mass per unit length of \(0.75 \text{ kg/m}\). Determine the angular acceleration of the rods and the components of reaction at the supports at the instant \(\omega = 8 \text{ rad/s}\) as shown. figure — add your own
  4. 51.The \(5\text{-kg}\) 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 \(\omega = 10 \text{ rad/s}\). figure — add your own
  5. 52.The car travels around the curved road of radius \(\rho\) such that its mass center has a constant speed \(v_G\). Write the equations of rotational motion with respect to the \(x,y,z\) axes. Assume that the car's six moments and products of inertia with respect to these axes are known. figure — add your own
  6. 53.The blades of a wind turbine spin about the shaft \(S\) with a constant angular speed of \(\omega_p\) while the frame precesses about the vertical axis with a constant angular speed of \(\omega_p\). Determine the \(x, y\) and \(z\) components of moment that the shaft exerts on the blades as a function of \(\theta\). Consider each blade as a slender rod of mass \(m\) and length \(l\). figure — add your own
  7. 54.The \(4\text{-kg}\) slender rod \(AB\) is pinned at \(A\) and held at \(B\) by a cord. The axle \(CD\) is supported at its ends by ball-and-socket joints and is rotating with a constant angular velocity of \(2 \text{ rad/s}\). Determine the tension developed in the cord and the magnitude of force developed at the pin \(A\). figure — add your own
  8. 55.The thin rod has a mass of \(0.8 \text{ kg}\) and a total length of \(150 \text{ mm}\). It is rotating about its midpoint at a constant rate \(\dot{\theta} = 6 \text{ rad/s}\) while the table to which its axle \(A\) is fastened is rotating at \(2 \text{ rad/s}\). Determine the \(x,y,z\) moment components which the axle exerts on the rod when the rod is in any position \(\theta\). figure — add your own
  9. 56.The cylinder has a mass of \(30 \text{ kg}\) and is mounted on an axle that is supported by bearings at \(A\) and \(B\). If the axle is subjected to a couple moment \(\mathbf{M} = \{-30\mathbf{j}\} \text{ N}\cdot\text{m}\) and at the instant shown has an angular velocity \(\omega = \{-40\mathbf{j}\} \text{ rad/s}\), determine the vertical components of force acting at the bearings at this instant. figure — add your own
  10. 57.The \(5\text{-kg}\) circular disk is mounted off center on a shaft which is supported by bearings at \(A\) and \(B\). If the shaft is rotating at a constant rate of \(\omega = 10 \text{ rad/s}\), determine the vertical reactions at the bearings when the disk is in the position shown. figure — add your own
  11. 58.The \(20\text{-kg}\) disk is spinning on its axle at \(\omega_s = 30 \text{ rad/s}\), while the forked rod is turning at \(\omega_1 = 6 \text{ rad/s}\). Determine the \(x\) and \(z\) moment components the axle exerts on the disk during the motion. figure — add your own
  12. 59.The \(10\text{-kg}\) disk turns around the shaft \(AB\) while the shaft rotates about \(BC\) at a constant rate of \(\omega_s = 5 \text{ rad/s}\). If the disk does not slip, determine the normal and frictional force it exerts on the ground. Neglect the mass of shaft \(AB\). figure — add your own
  13. 60.Show that the angular velocity of a body, in terms of Euler angles \(\phi, \theta,\) and \(\psi\), can be expressed as \[ \boldsymbol{\omega} = (\dot{\phi}\sin\theta\sin\psi + \dot{\theta}\cos\psi)\mathbf{i} + (\dot{\phi}\sin\theta\cos\psi - \dot{\theta}\sin\psi)\mathbf{j} + (\dot{\phi}\cos\theta + \dot{\psi})\mathbf{k}, \] where \(\mathbf{i}, \mathbf{j},\) and \(\mathbf{k}\) are directed along the \(x,y,z\) axes as shown in Fig. 21-15d.
  14. 61.A thin rod is initially coincident with the \(Z\) axis when it is given three rotations defined by the Euler angles \(\phi = 30^\circ, \theta = 45^\circ,\) and \(\psi = 60^\circ\). If these rotations are given in the order stated, determine the coordinate direction angles \(\alpha, \beta, \gamma\) 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? figure — add your own
  15. 62.The gyroscope consists of a uniform 450-g disk \(D\) which is attached to the axle \(AB\) of negligible mass. The supporting frame has a mass of \(180 \text{ g}\) and a center of mass at \(G\). If the disk is rotating about the axle at \(\omega_D = 90 \text{ rad/s}\), determine the constant angular velocity \(\omega_p\) at which the frame precesses about the pivot point \(O\). The frame moves in the horizontal plane. figure — add your own
21.5 Gyroscopic Motion & Torque-Free Motion 24 problems
Applications
  1. 63.The toy gyroscope consists of a rotor \(R\) which is attached to the frame of negligible mass. If it is observed that the frame is precessing about the pivot point \(O\) at \(\omega_p = 2 \text{ rad/s}\), determine the angular velocity \(\omega_R\) of the rotor. The stem \(OA\) moves in the horizontal plane. The rotor has a mass of \(200 \text{ g}\) and a radius of gyration \(k_{OA} = 20 \text{ mm}\) about \(OA\). figure — add your own
  2. 64.The propeller on a single-engine airplane has a mass of \(15 \text{ kg}\) and a centroidal radius of gyration of \(0.3 \text{ m}\) computed about the axis of spin. When viewed from the front of the airplane, the propeller is turning clockwise at \(350 \text{ rad/s}\) about the spin axis. If the airplane enters a vertical curve having a radius of \(80 \text{ m}\) and is traveling at \(200 \text{ km/h}\), determine the gyroscopic bending moment which the propeller exerts on the bearings of the engine when the airplane is in its lowest position. figure — add your own
  3. 65.The 10-kg disk spins about axle \(AB\) at a constant rate of \(\omega_s = 250 \text{ rad/s}\), and \(\theta = 30^\circ\). Determine the rate of precession of arm \(OA\). Neglect the mass of arm \(OA\), axle \(AB\), and the circular ring \(D\). figure — add your own
  4. 66.When \(OA\) precesses at a constant rate of \(\omega_p = 5 \text{ rad/s}\), when \(\theta = 90^\circ\), determine the required spin of the 10-kg disk \(C\). Neglect the mass of arm \(OA\), axle \(AB\), and the circular ring \(D\). figure — add your own
  5. 67.A wheel of mass \(m\) and radius \(r\) rolls with constant spin \(\omega\) about a circular path having a radius \(a\). If the angle of inclination is \(\theta\), determine the rate of precession. Treat the wheel as a thin ring. No slipping occurs. figure — add your own
  6. 68.The conical top has a mass of \(0.8 \text{ kg}\) and the moments of inertia are \(I_x = I_y = 3.5(10^{-3}) \text{ kg}\cdot\text{m}^2\) and \(I_z = 0.8(10^{-3}) \text{ kg}\cdot\text{m}^2\). If it spins freely in the ball-and-socket joint at \(A\) with an angular velocity \(\omega_s = 750 \text{ rad/s}\), compute the precession of the top about the axis of the shaft \(AB\). figure — add your own
  7. 69.The top has a mass of \(90 \text{ g}\), a center of mass at \(G\), and a radius of gyration \(k = 18 \text{ mm}\) about its axis of symmetry. About any transverse axis acting through point \(O\), the radius of gyration is \(k_t = 35 \text{ mm}\). If the top is connected to a ball-and-socket joint at \(O\) and the precession is \(\omega_p = 0.5 \text{ rad/s}\), determine the spin \(\omega_s\). figure — add your own
  8. 70.The satellite has a mass of \(1.8 \text{ Mg}\). The axial and transverse radii of gyration about axes passing through the mass center \(G\) are \(k_z = 0.8 \text{ m}\) and \(k_t = 1.2 \text{ m}\), respectively. If it is spinning at \(\omega_s = 6 \text{ rad/s}\) when it is launched, determine its angular momentum. Precession occurs about the \(Z\) axis. figure — add your own
  9. 71.The space capsule has a mass of \(2 \text{ Mg}\), center of mass at \(G\), and radii of gyration about its axis of symmetry (\(z\) axis) and its transverse axes (\(x\) or \(y\) axis) of \(k_z = 2.75 \text{ m}\) and \(k_x = k_y = 5.5 \text{ m}\), respectively. If the capsule has the angular velocity shown, determine its precession \(\dot{\phi}\) and spin \(\dot{\psi}\). Indicate whether the precession is regular or retrograde. Also, draw the space cone and body cone for the motion. figure — add your own
  10. 72.The \(0.25 \text{ kg}\) football is spinning at \(\omega_z = 15 \text{ rad/s}\) as shown. If \(\theta = 40^\circ\), determine the precession about the \(z\) axis. The radius of gyration about the spin axis is \(k_z = 0.042 \text{ m}\) and about a transverse axis is \(k_y = 0.13 \text{ m}\). figure — add your own
  11. 73.The projectile shown is subjected to torque-free motion. The transverse and axial moments of inertia are \(I\) and \(I_z\), respectively. If \(\theta\) represents the angle between the precessional axis \(Z\) and the axis of symmetry \(z\), and \(\beta\) is the angle between the angular velocity \(\boldsymbol{\omega}\) and the \(z\) axis, show that \(\beta\) and \(\theta\) are related by the equation \(\tan\theta = (I/I_z)\tan\beta\). figure — add your own
  12. 74.The radius of gyration about an axis passing through the axis of symmetry of the \(1.6\text{-Mg}\) space capsule is \(k_z = 1.2 \text{ m}\) and about any transverse axis passing through the center of mass \(G\), \(k_t = 1.8 \text{ m}\). If the capsule has a known steady-state precession of two revolutions per hour about the \(Z\) axis, determine the rate of spin about the \(z\) axis. figure — add your own
  13. 75.The rocket has a mass of \(4 \text{ Mg}\) and radii of gyration \(k_z = 0.85 \text{ m}\) and \(k_x = k_y = 2.3 \text{ m}\). It is initially spinning about the \(z\) axis at \(\omega_z = 0.05 \text{ rad/s}\) when a meteoroid \(M\) strikes it at \(A\) and creates an impulse \(\mathbf{I} = \{300\mathbf{i}\} \text{ N}\cdot\text{s}\). Determine the axis of precession after the impact. figure — add your own
  14. 76.The football has a mass of \(450 \text{ g}\) and radii of gyration about its axis of symmetry (\(z\) axis) and its transverse axes (\(x\) or \(y\) axis) of \(k_z = 30 \text{ mm}\) and \(k_x = k_y = 50 \text{ mm}\), respectively. If the football has an angular momentum of \(H_G = 0.02 \text{ kg}\cdot\text{m}^2/\text{s}\), determine its precession \(\dot{\phi}\) and spin \(\dot{\psi}\). Also, find the angle \(\beta\) that the angular velocity vector makes with the \(z\) axis. figure — add your own
  15. 77.[Problem 77 is omitted in the source.]
  16. 78.The radius of gyration about an axis passing through the axis of symmetry of the \(1.2\text{-Mg}\) satellite is \(k_z = 1.4 \text{ m}\) and about any transverse axis passing through the center of mass \(G\), \(k_t = 2.20 \text{ m}\). If the satellite has a known spin of \(2700 \text{ rev/h}\) about the \(z\) axis, determine the steady-state precession about the \(Z\) axis. figure — add your own
  17. 79.[Problem 79 is omitted in the source.]
  18. 80.[Problem 80 is omitted in the source.]
  19. 81.[Problem 81 is omitted in the source.]
  20. 82.[Problem 82 is omitted in the source.]
  21. 83.[Problem 83 is omitted in the source.]
  22. 84.[Problem 84 is omitted in the source.]
  23. 85.[Problem 85 is omitted in the source.]