Engineering Mechanics · Dynamics · Open Problem Companion

Chapter Twenty
Three-Dimensional Kinematics
of a Rigid Body

20‑1 Rolling and Rotational Motion 7 problems
Concepts & Skills
  1. 20‑1The conical spool rolls on the plane without slipping. If the axle has an angular velocity of \(\omega_{1} = 3\) rad/s and an angular acceleration of \(\alpha_{1} = 2\) rad/s\(^2\) at the instant shown, determine the angular velocity and angular acceleration of the spool at this instant. figure — add your own
  2. 20‑2At a given instant, the antenna has an angular motion \(\omega_{1} = 3\) rad/s and \(\dot{\omega}_{1} = 2\) rad/s\(^2\) about the \(z\) axis. At this same instant \(\theta = 30^{\circ}\), the angular motion about the \(x\) axis is \(\omega_{2} = 1.5\) rad/s, and \(\dot{\omega}_{2} = 4\) rad/s\(^2\). Determine the velocity and acceleration of the signal horn \(A\) at this instant. The distance from \(O\) to \(A\) is \(d = 1\) m. figure — add your own
  3. 20‑3The propeller of an airplane is rotating at a constant speed \(\omega_{x}\), while the plane is undergoing a turn at a constant rate \(\omega_{x}\). Determine the angular acceleration of the propeller if (a) the turn is horizontal, i.e. \(\omega_{x} \mathbf{k}\), and (b) the turn is vertical, downward, i.e. \(\omega_{t} \mathbf{j}\). figure — add your own
  4. 20‑4The disk rotates about the \(z\) axis at a constant rate \(\omega_{z} = 0.5\) rad/s without slipping on the horizontal plane. Determine the velocity and the acceleration of point \(A\) on the disk. figure — add your own
  5. 20‑5At the instant \(\theta = 30^{\circ}\), the satellite's body is rotating with an angular velocity of \(\omega_{1} = 20\) rad/s, and it has an angular acceleration of \(\dot{\omega}_{1} = 5\) rad/s\(^2\). Simultaneously, the solar panels rotate with a constant angular velocity of \(\omega_{2} = 5\) rad/s. Determine the velocity and acceleration of point \(B\) located at the end of one of the solar panels at this instant. figure — add your own
  6. 20‑6At the instant \(\theta = 30^{\circ}\), the satellite's body is rotating with an angular velocity of \(\omega_{1} = 20\) rad/s, and it has an angular acceleration of \(\dot{\omega}_{1} = 5\) rad/s\(^2\). At the same instant, the satellite travels in the \(x\) direction with a velocity of \(\mathbf{v}_{O} = [5000\mathbf{i}]\) m/s, and it has an acceleration of \(\mathbf{a}_{O} = [500\mathbf{i}]\) m/s\(^2\). Simultaneously, the solar panels rotate with a constant angular speed of \(\omega_{2} = 5\) rad/s. Determine the velocity and acceleration of point \(B\) located at the end of one of the solar panels at this instant. figure — add your own
  7. 20‑7The right circular cone rotates about the \(z\) axis at a constant rate of \(\omega_{1} = 4\) rad/s without slipping on the horizontal plane. Determine the magnitudes of the velocity and acceleration of points \(B\) and \(C\). figure — add your own
20‑2 Angular Velocity and Acceleration 8 problems
Concepts & Skills
  1. 20‑8The electric fan is mounted on a swivel support such that the fan rotates about the \(\mathbf{z}\) axis at a constant rate of \(\omega_{z} = 1\) rad/s and the fan blade is spinning at a constant rate \(\omega_{s} = 60\) rad/s. If \(\phi = 45^{\circ}\) for the motion, determine the angular velocity and the angular acceleration of the blade. figure — add your own
  2. 20‑9The electric fan is mounted on a swivel support such that the fan rotates about the \(\mathbf{z}\) axis at a constant rate of \(\omega_{z} = 1\) rad/s and the fan blade is spinning at a constant rate \(\omega_{s} = 60\) rad/s. If at the instant \(\phi = 45^{\circ}\), \(\dot{\phi} = 2\) rad/s for the motion, determine the angular velocity and the angular acceleration of the blade. figure — add your own
  3. 20‑10If the plate gears \(A\) and \(B\) are rotating with the angular velocities shown, determine the angular velocity of gear \(C\) about the shaft \(DE\). What is the angular velocity of \(DE\) about the \(y\) axis? figure — add your own
  4. 20‑11The drill pipe \(P\) turns at a constant angular rate \(\omega_{P} = 4\) rad/s. Determine the angular velocity and angular acceleration of the conical rock bit, which rolls without slipping. Also, what are the velocity and acceleration of point \(A\)? figure — add your own
  5. 20‑12The disk rotates about the shaft \(S\) while the shaft is turning about the \(\mathbf{z}\) axis at a rate of \(\omega_{z} = 4\) rad/s, which is increasing at \(2\) rad/s\(^2\). Determine the velocity and acceleration of point \(A\) on the disk at the instant shown. No slipping occurs. figure — add your own
  6. 20‑13The disk rotates about the shaft \(S\) while the shaft is turning about the \(\mathbf{z}\) axis at a rate of \(\omega_{z} = 4\) rad/s, which is increasing at \(2\) rad/s\(^2\). Determine the velocity and acceleration of point \(B\) on the disk at the instant shown. No slipping occurs. figure — add your own
  7. 20‑14The antenna is following the motion of a jet plane. At the instant \(\theta = 25^{\circ}\) and \(\phi = 75^{\circ}\), the constant angular rates of change are \(\dot{\theta} = 0.4\) rad/s and \(\dot{\phi} = 0.6\) rad/s. Determine the velocity and acceleration of the signal horn \(A\) at this instant. The distance \(OA\) is \(0.8\) m. figure — add your own
  8. 20‑15Gear \(B\) is driven by a motor mounted on turntable \(C\). If gear \(A\) is held fixed, and the motor shaft rotates with a constant angular velocity of \(\omega_{y} = 30\) rad/s, determine the angular velocity and angular acceleration of gear \(B\). figure — add your own
  9. 20‑16Gear \(B\) is driven by a motor mounted on turntable \(C\). If gear \(A\) and the motor shaft rotate with constant angular speeds of \(\omega_{A} = [10\mathbf{k}]\) rad/s and \(\omega_{y} = [30\mathbf{j}]\) rad/s, respectively, determine the angular velocity and angular acceleration of gear \(B\). figure — add your own
20‑3 Velocity and Acceleration of Points 9 problems
Concepts & Skills
  1. 20‑17At the instant when \(\theta = 90^{\circ}\), the satellite's body is rotating with an angular velocity of \(\omega_{1} = 15\) rad/s and angular acceleration of \(\dot{\omega}_{1} = 3\) rad/s\(^2\). Simultaneously, the solar panels rotate with an angular velocity of \(\omega_{2} = 6\) rad/s and angular acceleration of \(\dot{\omega}_{2} = 1.5\) rad/s\(^2\). Determine the velocity and acceleration of point \(B\) on the solar panel at this instant. figure — add your own
  2. 20‑18At the instant when \(\theta = 90^{\circ}\), the satellite's body travels in the \(x\) direction with a velocity of \(\mathbf{v}_{O} = [500\mathbf{i}]\) m/s and acceleration of \(\mathbf{a}_{O} = [50\mathbf{i}]\) m/s\(^2\). Simultaneously, the body also rotates with an angular velocity of \(\omega_{1} = 15\) rad/s and angular acceleration of \(\dot{\omega}_{1} = 3\) rad/s\(^2\). At the same time, the solar panels rotate with an angular velocity of \(\omega_{2} = 6\) rad/s and angular acceleration of \(\dot{\omega}_{2} = 1.5\) rad/s\(^2\). Determine the velocity and acceleration of point \(B\) on the solar panel. figure — add your own
  3. 20‑19Gear \(A\) is fixed while gear \(B\) is free to rotate on the shaft \(S\). If the shaft is turning about the \(z\) axis at \(\omega_{z} = 5\) rad/s, while increasing at \(2\) rad/s\(^2\), determine the velocity and acceleration of point \(P\) at the instant shown. The face of gear \(B\) lies in a vertical plane. figure — add your own
  4. 20‑20The wheel is spinning about shaft \(AB\) with an angular velocity of \(\omega_{s} = 10\) rad/s, which is increasing at a constant rate of \(\dot{\omega}_{s} = 6\) rad/s\(^2\), while the frame precesses about the \(\Xi\) axis with an angular velocity of \(\omega_{p} = 12\) rad/s, which is increasing at a constant rate of \(\dot{\omega}_{p} = 3\) rad/s\(^2\). Determine the velocity and acceleration of point \(C\) located on the rim of the wheel at this instant. figure — add your own
  5. 20‑21Shaft \(BD\) is connected to a ball-and-socket joint at \(B\), and a beveled gear \(A\) is attached to its other end. The gear is in mesh with a fixed gear \(C\). If the shaft and gear \(A\) are spinning with a constant angular velocity \(\omega_{1} = 8\) rad/s, determine the angular velocity and angular acceleration of gear \(A\). figure — add your own
  6. 20‑22Gear \(B\) is connected to the rotating shaft, while the plate gear \(A\) is fixed. If the shaft is turning at a constant rate of \(\omega_{z} = 10\) rad/s about the \(\Xi\) axis, determine the magnitudes of the angular velocity and the angular acceleration of gear \(B\). Also, determine the magnitudes of the velocity and acceleration of point \(P\). figure — add your own
  7. 20‑23The differential of an automobile allows the two rear wheels to rotate at different speeds when the automobile travels along a curve. For operation, the rear axles are attached to the wheels at one end and have beveled gears \(A\) and \(B\) on their other ends. The differential case \(D\) is placed over the left axle but can rotate about \(C\) independent of the axle. The case supports a pinion gear \(E\) on a shaft, which meshes with gears \(A\) and \(B\). Finally, a ring gear \(G\) is fixed to the differential case so that the case rotates with the ring gear when the latter is driven by the drive pinion \(H\). This gear, like the differential case, is free to rotate about the left wheel axle. If the drive pinion is turning at \(\omega_{H} = 100\) rad/s and the pinion gear \(E\) is spinning about its shaft at \(\omega_{E} = 30\) rad/s, determine the angular velocity, \(\omega_{A}\) and \(\omega_{B}\), of each axle. figure — add your own
  8. 20‑24The anemometer located on the ship at \(A\) spins about its own axis at a rate \(\omega_{x}\), while the ship rolls about the \(x\) axis at the rate \(\omega_{x}\) and about the \(y\) axis at the rate \(\omega_{y}\). Determine the angular velocity and angular acceleration of the anemometer at the instant the ship is level as shown. Assume that the magnitudes of all components of angular velocity are constant and that the rolling motion caused by the sea is independent in the \(x\) and \(y\) directions. figure — add your own
  9. 20‑25(The problem statement for 20‑25 is not fully transcribed in the source; it appears to reference an image. Please consult the original course for the complete problem.) figure — add your own
20‑4 Mechanisms and Gear Systems 11 problems
Concepts & Skills
  1. 20‑26The rod is attached to smooth collars \(A\) and \(B\) at its ends using ball-and-socket joints. Determine the speed of \(B\) at the instant shown if \(A\) is moving at \(v_{A} = 8\) m/s. Also, determine the angular velocity of the rod if it is directed perpendicular to the axis of the rod. figure — add your own
  2. 20‑27If the collar \(A\) in Prob. 20‑26 has a deceleration of \(\mathbf{a}_{A} = [-5\mathbf{k}]\) m/s\(^2\) at the instant shown, determine the acceleration of collar \(B\) at this instant. figure — add your own
  3. 20‑28If crank \(BC\) rotates with a constant angular velocity of \(\omega_{BC} = 6\) rad/s, determine the velocity of the collar at \(A\). Assume the angular velocity of \(AB\) is perpendicular to the rod. figure — add your own
  4. 20‑29If crank \(BC\) is rotating with an angular velocity of \(\omega_{BC} = 6\) rad/s and an angular acceleration of \(\alpha_{BC} = 1.5\) rad/s\(^2\), determine the acceleration of collar \(A\) at this instant. Assume the angular velocity and angular acceleration of \(AB\) are perpendicular to the rod. figure — add your own
  5. 20‑30Disk \(A\) rotates at a constant angular velocity of \(10\) rad/s. If rod \(BC\) is joined to the disk and a collar by ball-and-socket joints, determine the velocity of collar \(B\) at the instant shown. Also, what is the rod's angular velocity \(\omega_{BC}\) if it is directed perpendicular to the axis of the rod? figure — add your own
  6. 20‑31Rod \(AB\) is attached to collars at its ends by using ball-and-socket joints. If collar \(A\) moves along the fixed rod at \(v_{A} = 5\) m/s, determine the angular velocity of the rod and the velocity of collar \(B\) at the instant shown. Assume that the rod's angular velocity is directed perpendicular to the axis of the rod. figure — add your own
  7. 20‑32Rod \(AB\) is attached to collars at its ends by using ball-and-socket joints. If collar \(A\) moves along the fixed rod with a velocity of \(v_{A} = 5\) m/s and has an acceleration \(a_{A} = 2\) m/s\(^2\) at the instant shown, determine the angular acceleration of the rod and the acceleration of collar \(B\) at this instant. Assume that the rod's angular velocity and angular acceleration are directed perpendicular to the axis of the rod. figure — add your own
  8. 20‑33Rod \(CD\) is attached to the rotating arms using ball-and-socket joints. If \(AC\) has the motion shown, determine the angular velocity of link \(BD\) at the instant shown. figure — add your own
  9. 20‑34Rod \(CD\) is attached to the rotating arms using ball-and-socket joints. If \(AC\) has the motion shown, determine the angular acceleration of link \(BD\) at this instant. figure — add your own
  10. 20‑35Solve Prob. 20‑30 if the connection at \(B\) consists of a pin as shown in the figure below, rather than a ball-and-socket joint. Hint: The constraint allows rotation of the rod both along the bar (\(\mathbf{j}\) direction) and along the axis of the pin (\(\mathbf{n}\) direction). Since there is no rotational component in the \(\mathbf{u}\) direction, i.e., perpendicular to \(\mathbf{n}\) and \(\mathbf{j}\) where \(\mathbf{u} = \mathbf{j} \times \mathbf{n}\), an additional equation for solution can be obtained from \(\boldsymbol{\omega} \cdot \mathbf{u} = 0\). The vector \(\mathbf{n}\) is in the same direction as \(\mathbf{r}_{B/C} \times \mathbf{r}_{D/C}\). figure — add your own
  11. 20‑36Member \(ABC\) is pin connected at \(A\) and has a ball-and-socket joint at \(B\). If the collar at \(B\) is moving along the inclined rod at \(v_{B} = 8\) m/s, determine the velocity of point \(C\) at the instant shown. Hint: See Prob. 20‑35. figure — add your own
20‑5 Collars and Joints 9 problems
Concepts & Skills
  1. 20‑37Solve Example 20.5 such that the \(x,y,z\) axes move with curvilinear translation, \(\Omega = 0\), in which case the collar appears to have both an angular velocity \(\Omega_{xyz} = \omega_{1} + \omega_{2}\) and radial motion. figure — add your own
  2. 20‑38Solve Example 20.5 by fixing \(x,y,z\) axes to rod \(BD\) so that \(\Omega = \omega_{1} + \omega_{2}\). In this case the collar appears only to move radially outward along \(BD\); hence \(\Omega_{xyz} = 0\). figure — add your own
  3. 20‑39The crane rotates about the \(z\) axis with a constant rate \(\omega_{1} = 0.6\) rad/s, while the boom rotates downward with a constant rate \(\omega_{2} = 0.2\) rad/s. Determine the velocity and acceleration of point \(A\) located at the end of the boom at the instant shown. figure — add your own
  4. 20‑40The crane rotates about the \(z\) axis with a rate of \(\omega_{1} = 0.6\) rad/s, which is increasing at \(\dot{\omega}_{1} = 0.6\) rad/s\(^2\). Also, the boom rotates downward at \(\omega_{2} = 0.2\) rad/s, which is increasing at \(\dot{\omega}_{2} = 0.3\) rad/s\(^2\). Determine the velocity and acceleration of point \(A\) located at the end of the boom at the instant shown. figure — add your own
  5. 20‑41At the given instant, the rod is turning about the \(z\) axis with a constant angular velocity \(\omega_{1} = 3\) rad/s. At this same instant, the disk is spinning at \(\omega_{2} = 6\) rad/s when \(\dot{\omega}_{2} = 4\) rad/s\(^2\), both measured relative to the rod. Determine the velocity and acceleration of point \(P\) on the disk at this instant. figure — add your own
  6. 20‑42At the instant shown, the industrial manipulator is rotating about the \(z\) axis at \(\omega_{1} = 5\) rad/s, and about joint \(B\) at \(\omega_{2} = 2\) rad/s. Determine the velocity and acceleration of the grip \(A\) at this instant, when \(\phi = 30^{\circ}\), \(\theta = 45^{\circ}\), and \(r = 1.6\) m. figure — add your own
  7. 20‑43At the instant shown, the industrial manipulator is rotating about the \(z\) axis at \(\omega_{1} = 5\) rad/s, and \(\dot{\omega}_{1} = 2\) rad/s\(^2\); and about joint \(B\) at \(\omega_{2} = 2\) rad/s and \(\dot{\omega}_{2} = 3\) rad/s\(^2\). Determine the velocity and acceleration of the grip \(A\) at this instant, when \(\phi = 30^{\circ}\), \(\theta = 45^{\circ}\), and \(r = 1.6\) m. figure — add your own
  8. 20‑44At the instant \(\theta = 30^{\circ}\), the frame of the crane and the boom \(AB\) rotate with a constant angular velocity of \(\omega_{1} = 1.5\) rad/s and \(\omega_{2} = 0.5\) rad/s respectively. Determine the velocity and acceleration of point \(B\) at this instant. figure — add your own
  9. 20‑45At the instant \(\theta = 30^{\circ}\), the frame of the crane is rotating with an angular velocity of \(\omega_{1} = 1.5\) rad/s and angular acceleration of \(\dot{\omega}_{1} = 0.5\) rad/s\(^2\), while the boom \(AB\) rotates with an angular velocity of \(\omega_{2} = 0.5\) rad/s and angular acceleration of \(\dot{\omega}_{2} = 0.25\) rad/s\(^2\). Determine the velocity and acceleration of point \(B\) at this instant. figure — add your own
20‑6 Cranes and Industrial Manipulators 6 problems
Concepts & Skills
  1. 20‑46At the instant shown, the arm \(AB\) is rotating about the fixed pin \(A\) with an angular velocity \(\omega_{1} = 4\) rad/s and angular acceleration \(\dot{\omega}_{1} = 3\) rad/s\(^2\). At this same instant, rod \(BD\) is rotating relative to rod \(AB\) with an angular velocity \(\omega_{2} = 5\) rad/s, which is increasing at \(\dot{\omega}_{2} = 7\) rad/s\(^2\). Also, the collar \(C\) is moving along rod \(BD\) with a velocity of \(3\) m/s and an acceleration of \(2\) m/s\(^2\), both measured relative to the rod. Determine the velocity and acceleration of the collar at this instant. figure — add your own
  2. 20‑47The particle \(P\) slides around the circular hoop with a constant angular velocity of \(\dot{\theta} = 6\) rad/s, while the hoop rotates about the \(x\) axis at a constant rate of \(\omega = 4\) rad/s. If at the instant shown the hoop is in the \(x\)-\(y\) plane and the angle \(\theta = 45^{\circ}\), determine the velocity and acceleration of the particle at this instant. figure — add your own
  3. 20‑48At the instant shown, the frame of the brush cutter is traveling forward in the \(x\) direction with a constant velocity of \(1\) m/s, and the cab is rotating about the vertical axis with a constant angular velocity of \(\omega_{1} = 0.5\) rad/s. At the same instant the boom \(AB\) has an angular velocity of \(\dot{\theta} = 0.8\) rad/s in the direction shown. Determine the velocity and acceleration of point \(B\) at the connection to the mower at this instant. figure — add your own
  4. 20‑49During the instant shown the frame of the X-ray camera is rotating about the vertical axis at \(\omega_{z} = 5\) rad/s and \(\dot{\omega}_{z} = 2\) rad/s\(^2\). Relative to the frame the arm is rotating at \(\omega_{\mathrm{rel}} = 2\) rad/s and \(\dot{\omega}_{\mathrm{rel}} = 1\) rad/s\(^2\). Determine the velocity and acceleration of the center of the camera \(C\) at this instant. figure — add your own
  5. 20‑50The boom \(AB\) of the locomotive crane is rotating about the \(z\) axis with an angular velocity \(\omega_{1} = 0.5\) rad/s, which is increasing at \(\dot{\omega}_{1} = 3\) rad/s\(^2\). At this same instant, \(\theta = 30^{\circ}\) and the boom is rotating upward at a constant rate of \(\dot{\theta} = 3\) rad/s. Determine the velocity and acceleration of the tip \(B\) of the boom at this instant. figure — add your own
  6. 20‑51The locomotive crane is traveling to the right at \(2\) m/s and has an acceleration of \(1.5\) m/s\(^2\), while the boom is rotating about the \(z\) axis with an angular velocity \(\omega_{1} = 0.5\) rad/s, which is increasing at \(\dot{\omega}_{1} = 3\) rad/s\(^2\). At this same instant, \(\theta = 30^{\circ}\) and the boom is rotating upward at a constant rate \(\dot{\theta} = 3\) rad/s. Determine the velocity and acceleration of the tip \(B\) of the boom at this instant. figure — add your own
20‑7 Miscellaneous Problems 4 problems
Concepts & Skills
  1. 20‑52At the instant shown, the base of the robotic arm is turning about the \(z\) axis with an angular velocity of \(\omega_{1} = 4\) rad/s, which is increasing at \(\dot{\omega}_{1} = 3\) rad/s\(^2\). Also, the boom segment \(BC\) is rotating at a constant rate of \(\omega_{BC} = 8\) rad/s. Determine the velocity and acceleration of the part \(C\) held in its grip at this instant. figure — add your own
  2. 20‑53At the instant shown, the base of the robotic arm is turning about the \(z\) axis with an angular velocity of \(\omega_{1} = 4\) rad/s, which is increasing at \(\dot{\omega}_{1} = 3\) rad/s\(^2\). Also, the boom segment \(BC\) is rotating at \(\omega_{BC} = 8\) rad/s, which is increasing at \(\dot{\omega}_{BC} = 2\) rad/s\(^2\). Determine the velocity and acceleration of the part \(C\) held in its grip at this instant. figure — add your own
  3. 20‑54The motor rotates about the \(z\) axis with a constant angular velocity of \(\omega_{1} = 3\) rad/s. Simultaneously, shaft \(OA\) rotates with a constant angular velocity of \(\omega_{2} = 6\) rad/s. Also, collar \(C\) slides along rod \(AB\) with a velocity and acceleration of \(6\) m/s and \(3\) m/s\(^2\) (both relative to the rod). Determine the velocity and acceleration of collar \(C\) at the instant shown. figure — add your own
  4. 20‑55(The problem statement for 20‑55 is not fully transcribed in the source; it appears to be a continuation or a missing problem. Please consult the original course for the complete problem.) figure — add your own