Engineering Mechanics: Dynamics · Chapter 18

Chapter Eighteen
Planar Kinetics of a Rigid Body: Work and Energy

18.4 Principle of Work and Energy 1 problem
Concepts
  1. P18–1.Determine the kinetic energy of the 100‑kg object in configurations (a) through (e). figure — add your own
F18 Fundamental Problems 12 problems
Fundamentals
  1. F18–1.The \(80\,\mathrm{kg}\) wheel has a radius of gyration about its mass center \(O\) of \(k_O = 400\,\mathrm{mm}\). Determine its angular velocity after it has rotated 20 revolutions starting from rest. figure — add your own
  2. F18–2.The uniform \(25\,\mathrm{kg}\) slender rod is subjected to a couple moment \(M = 150\,\mathrm{N\cdot m}\). If the rod is at rest when \(\theta = 0^\circ\), determine its angular velocity when \(\theta = 90^\circ\). figure — add your own
  3. F18–3.The uniform \(50\,\mathrm{kg}\) slender rod is at rest in the position shown when \(P = 600\,\mathrm{N}\) is applied. Determine the angular velocity of the rod when it reaches the vertical position. figure — add your own
  4. F18–4.The \(50\,\mathrm{kg}\) wheel is subjected to a force of \(50\,\mathrm{N}\). If the wheel starts from rest and rolls without slipping, determine its angular velocity after it has rotated 10 revolutions. The radius of gyration about \(G\) is \(k_G = 0.3\,\mathrm{m}\). figure — add your own
  5. F18–5.If the uniform \(30\,\mathrm{kg}\) slender rod starts from rest at the position shown, determine its angular velocity after it has rotated 4 revolutions. The forces remain perpendicular to the rod. figure — add your own
  6. F18–6.The \(20\,\mathrm{kg}\) wheel has a radius of gyration about \(G\) of \(k_G = 300\,\mathrm{mm}\). When subjected to a couple moment \(M = 50\,\mathrm{N\cdot m}\), it rolls without slipping. Determine its angular velocity after its center \(G\) has traveled \(s_G = 20\,\mathrm{m}\), starting from rest. figure — add your own
  7. F18–7.If the \(30\,\mathrm{kg}\) disk is released from rest when \(\theta = 0^\circ\), determine its angular velocity when \(\theta = 90^\circ\). figure — add your own
  8. F18–8.The \(50\,\mathrm{kg}\) reel has a radius of gyration about \(O\) of \(k_O = 300\,\mathrm{mm}\). If released from rest, determine its angular velocity when its center \(O\) has traveled \(6\,\mathrm{m}\) down the smooth inclined plane. figure — add your own
  9. F18–9.The \(60\,\mathrm{kg}\) rod \(OA\) is released from rest when \(\theta = 0^\circ\). Determine its angular velocity when \(\theta = 45^\circ\). The spring remains vertical during motion and is unstretched at \(\theta = 0^\circ\). figure — add your own
  10. F18–10.The \(30\,\mathrm{kg}\) rod is released from rest when \(\theta = 0^\circ\). Determine its angular velocity when \(\theta = 90^\circ\). The spring is unstretched at \(\theta = 0^\circ\). figure — add your own
  11. F18–11.The \(30\,\mathrm{kg}\) rod is released from rest when \(\theta = 45^\circ\). Determine its angular velocity when \(\theta = 0^\circ\). The spring is unstretched at \(\theta = 45^\circ\). figure — add your own
  12. F18–12.The \(20\,\mathrm{kg}\) rod is released from rest when \(\theta = 0^\circ\). Determine its angular velocity when \(\theta = 90^\circ\). The spring has an unstretched length of \(0.5\,\mathrm{m}\). figure — add your own
18–1 Kinetic Energy & Work 15 problems
Concepts
  1. 18–1.At a given instant, a body of mass \(m\) has angular velocity \(\omega\) and mass-center velocity \(\mathbf{v}_G\). Show that its kinetic energy can be represented as \(T = \frac{1}{2} I_{IC} \omega^2\), where \(I_{IC}\) is the moment of inertia about the instantaneous axis of zero velocity, located a distance \(r_{G/IC}\) from the mass center. figure — add your own
  2. 18–2.A force \(P = 20\,\mathrm{N}\) is applied to the cable, causing the 175‑kg reel to turn on two rollers \(A\) and \(B\). Determine the angular velocity after two revolutions from rest. Neglect rollers and cable mass. \(k_G = 0.42\,\mathrm{m}\). figure — add your own
  3. 18–3.Same as 18–2, but the reel rolls without slipping on the rollers. Each roller is an 18‑kg cylinder of radius \(0.1\,\mathrm{m}\). Determine the angular velocity of the reel. Neglect cable mass. figure — add your own
  4. 18–4.The wheel is made from a 5‑kg thin ring and two 2‑kg slender rods. A torsional spring at the center has stiffness \(k = 2\,\mathrm{N\cdot m/rad}\) and is rotated until the torque \(M = 25\,\mathrm{N\cdot m}\) is developed. Determine the maximum angular velocity if released from rest. figure — add your own
  5. 18–5.Same wheel as 18–4, with spring torque \(M = (2\theta)\,\mathrm{N\cdot m}\) (\(\theta\) in rad). Determine the maximum angular velocity if rotated two revolutions and released from rest. figure — add your own
  6. 18–6.A force \(P = 60\,\mathrm{N}\) is applied to the cable, causing the 200‑kg reel to turn on rollers \(A\) and \(B\). Determine the angular velocity after two revolutions from rest. Neglect rollers and cable. \(k_O = 0.6\,\mathrm{m}\). figure — add your own
  7. 18–7.The wheel and reel have combined mass \(25\,\mathrm{kg}\), \(k_A = 150\,\mathrm{mm}\). Pulley \(B\) is subjected to a torque \(M = 60(2 - e^{-0.1\theta})\,\mathrm{N\cdot m}\) (\(\theta\) in rad). Determine the velocity of the 100‑kg crate after it moves up \(1.5\,\mathrm{m}\) from rest. Neglect pulley \(B\). figure — add your own
  8. 18–8.Same system as 18–7, but \(M = 75\,\mathrm{N\cdot m}\) (constant). Determine the crate velocity after pulley \(B\) turns 5 revolutions. figure — add your own
  9. 18–9.The pendulum consists of a \(10\,\mathrm{kg}\) uniform disk and a \(3\,\mathrm{kg}\) uniform slender rod. Released from rest in the shown position, determine its angular velocity when it rotates clockwise \(90^\circ\). figure — add your own
  10. 18–10.A motor supplies constant torque \(M = 6\,\mathrm{kN\cdot m}\) to the winding drum that operates the elevator. Elevator mass \(900\,\mathrm{kg}\), counterweight \(C\) mass \(200\,\mathrm{kg}\), drum mass \(600\,\mathrm{kg}\), \(k = 0.6\,\mathrm{m}\). Determine the elevator speed after it rises \(5\,\mathrm{m}\) from rest. Neglect pulley masses. figure — add your own
  11. 18–11.The \(20\,\mathrm{kg}\) disk is subjected to a couple moment \(M = (2\theta + 4)\,\mathrm{N\cdot m}\) (\(\theta\) in rad). If it starts from rest, determine its angular velocity after two revolutions. figure — add your own
  12. 18–12.The \(10\,\mathrm{kg}\) uniform slender rod is suspended at rest when force \(F = 150\,\mathrm{N}\) is applied perpendicular to its end. Determine its angular velocity when it has rotated \(90^\circ\) clockwise. figure — add your own
  13. 18–13.Same rod and force as 18–12, determine angular velocity when it has rotated \(180^\circ\) clockwise. figure — add your own
  14. 18–14.The spool has mass \(40\,\mathrm{kg}\), \(k_O = 0.3\,\mathrm{m}\). The 10‑kg block is released from rest. Determine the distance the block must fall for the spool to have \(\omega = 15\,\mathrm{rad/s}\). Also find the tension in the cord. figure — add your own
  15. 18–15.A force \(T = 20\,\mathrm{N}\) is applied to the cord. Determine the angular velocity of the 20‑kg wheel after 4 revolutions from rest. \(k_O = 0.3\,\mathrm{m}\). figure — add your own
18–10 Rotational Work & Energy 10 problems
Skills
  1. 18–16.Determine the velocity of the 50‑kg cylinder after it has descended \(2\,\mathrm{m}\). Initially at rest. Reel mass \(25\,\mathrm{kg}\), \(k_A = 125\,\mathrm{mm}\). figure — add your own
  2. 18–17.The uniform door has mass \(20\,\mathrm{kg}\) and is treated as a thin plate. It is connected to a torsional spring at \(A\) with \(k = 80\,\mathrm{N\cdot m/rad}\). Determine the required initial twist so that the door has angular velocity \(12\,\mathrm{rad/s}\) when it closes at \(\theta = 0^\circ\) after being opened at \(\theta = 90^\circ\) and released. figure — add your own
  3. 18–18.The 30‑kg disk is at rest, spring unstretched. A couple \(M = 80\,\mathrm{N\cdot m}\) is applied. Determine its angular velocity when its mass center \(G\) has moved \(0.5\,\mathrm{m}\) along the plane. Rolls without slipping. figure — add your own
  4. 18–19.Same disk as 18–18. Determine how far the center travels along the plane before it momentarily stops. figure — add your own
  5. 18–20.Gear \(B\) is rigidly attached to drum \(A\) and supported by rollers at \(E\) and \(D\). Gear \(B\) meshes with gear \(C\) and is subjected to torque \(M = 50\,\mathrm{N\cdot m}\). Determine the angular velocity of the drum after \(C\) has rotated 10 revolutions from rest. Gear \(B\) + drum: \(100\,\mathrm{kg}\), \(k = 250\,\mathrm{mm}\). Gear \(C\): \(30\,\mathrm{kg}\), \(k = 125\,\mathrm{mm}\). figure — add your own
  6. 18–21.The center \(O\) of the thin ring of mass \(m\) is given an initial angular velocity \(\omega_0\). If the ring rolls without slipping, determine its angular velocity after it has traveled a distance \(s\) down the incline. Neglect thickness. figure — add your own
  7. 18–22.The hand winch is used to lift the 50‑kg load. Determine the work required to rotate the handle five revolutions, starting and ending at rest. Gear at \(A\) has radius \(20\,\mathrm{mm}\). figure — add your own
  8. 18–23.The rotary screen \(S\) has mass \(800\,\mathrm{kg}\), \(k_G = 1.75\,\mathrm{m}\). Rotation is achieved by applying torque \(M = 280\,\mathrm{N\cdot m}\) about drive wheel \(A\). If no slipping at \(A\) and supporting wheel \(B\) is free to roll, determine the angular velocity of the screen after 5 revolutions. Neglect mass of \(A\) and \(B\). figure — add your own
  9. 18–24.The wheel has mass \(100\,\mathrm{kg}\), \(k_O = 0.2\,\mathrm{m}\). Motor supplies torque \(M = (40\theta + 900)\,\mathrm{N\cdot m}\) (\(\theta\) in rad) about shaft \(O\). Determine the speed of the 300‑kg loading car after it travels \(s = 4\,\mathrm{m}\). Initially at rest when \(s = 0\), \(\theta = 0^\circ\). Neglect cable and car wheels. figure — add your own
  10. 18–25.If \(P = 200\,\mathrm{N}\) and the \(15\,\mathrm{kg}\) uniform slender rod starts from rest at \(\theta = 0^\circ\), determine the rod's angular velocity just before \(\theta = 45^\circ\). figure — add your own
18–20 Gears, Pulleys & Systems 18 problems
Skills
  1. 18–26.Corner \(A\) of the 60‑kg plate is subjected to a vertical force \(P = 500\,\mathrm{N}\) and released from rest when \(\theta = 0^\circ\). Determine the angular velocity of the plate when \(\theta = 45^\circ\). figure — add your own
  2. 18–27.The linkage consists of two 6‑kg rods \(AB\) and \(CD\) and a 20‑kg bar \(BD\). When \(\theta = 0^\circ\), rod \(AB\) rotates with \(\omega = 2\,\mathrm{rad/s}\). Rod \(CD\) is subjected to a couple \(M = 30\,\mathrm{N\cdot m}\). Determine \(\omega_{AB}\) at \(\theta = 90^\circ\). figure — add your own
  3. 18–28.Same linkage as 18–27, determine \(\omega\) at \(\theta = 45^\circ\). figure — add your own
  4. 18–29.Motor \(M\) exerts a constant force \(P = 750\,\mathrm{N}\) on the rope. The 100‑kg post is at rest when \(\theta = 0^\circ\). Determine its angular velocity at \(\theta = 60^\circ\). Neglect pulley and size; treat post as slender rod. figure — add your own
  5. 18–30.Link \(AB\) is subjected to a couple \(M = 40\,\mathrm{N\cdot m}\). The ring gear \(C\) is fixed. Determine the angular velocity of the 15‑kg inner gear when the link has made two revolutions from rest. Neglect link mass; inner gear is a disk. Motion in vertical plane. figure — add your own
  6. 18–31.The 6‑kg lid is held in equilibrium by the torsional spring at \(\theta = 60^\circ\). If forced closed (\(\theta = 0^\circ\)) and released, determine its angular velocity at \(\theta = 45^\circ\). figure — add your own
  7. 18–32.Two 2‑kg gears \(A\) and \(B\) are attached to the ends of a 3‑kg slender bar. Gears roll within fixed ring gear \(C\) in the horizontal plane. A \(10\,\mathrm{N\cdot m}\) torque is applied to the bar center. Determine the number of revolutions the bar must rotate from rest to have \(\omega_{AB} = 20\,\mathrm{rad/s}\). Assume gears are thin disks. What if the gears lie in the vertical plane? figure — add your own
  8. 18–33.The 10‑kg rod \(AB\) is pin-connected at \(A\) and subjected to \(M = 15\,\mathrm{N\cdot m}\). Released from rest when the spring is unstretched at \(\theta = 30^\circ\), determine its angular velocity at \(\theta = 60^\circ\). The spring remains horizontal due to the roller support at \(C\). figure — add your own
  9. 18–34.The uniform bar has mass \(m\) and length \(l\). Released from rest when \(\theta = 0^\circ\), determine its angular velocity as a function of \(\theta\) before it slips. figure — add your own
  10. 18–35.Same bar as 18–34, released from rest at \(\theta = 0^\circ\). Determine the angle \(\theta\) at which it first begins to slip. The coefficient of static friction at \(O\) is \(\mu_s = 0.3\). figure — add your own
  11. 18–36.The spool has mass \(20\,\mathrm{kg}\), \(k_O = 160\,\mathrm{mm}\). The 15‑kg block \(A\) is released from rest. Determine the distance the block must fall for the spool to have \(\omega = 8\,\mathrm{rad/s}\). Also find the tension in the cord. figure — add your own
  12. 18–37.Same spool as 18–36, determine the velocity of the block when it descends \(600\,\mathrm{mm}\). figure — add your own
  13. 18–38.An automobile tire has mass \(7\,\mathrm{kg}\), \(k_G = 0.3\,\mathrm{m}\). Released from rest at \(A\) on the incline, determine its angular velocity when it reaches the horizontal plane. Rolls without slipping. figure — add your own
  14. 18–39.The spool has mass \(50\,\mathrm{kg}\), \(k_O = 0.280\,\mathrm{m}\). The 20‑kg block \(A\) is released from rest. Determine the distance the block must fall for the spool to have \(\omega = 5\,\mathrm{rad/s}\). Also find the tension. figure — add your own
  15. 18–40.Same spool as 18–39, determine the velocity of the block when it descends \(0.5\,\mathrm{m}\). figure — add your own
  16. 18–41.The assembly consists of a 3‑kg pulley \(A\) and 10‑kg pulley \(B\). A 2‑kg block is suspended from the cord. Determine the block's speed after it descends \(0.5\,\mathrm{m}\) from rest. Treat pulleys as thin disks; no slipping. figure — add your own
  17. 18–42.Same assembly as 18–41, determine the distance the block must descend from rest to cause \(B\) to have angular velocity \(6\,\mathrm{rad/s}\). figure — add your own
  18. 18–43.The 15‑kg semicircular segment is released from rest in the position shown. Determine the velocity of point \(A\) when it has rotated counterclockwise \(90^\circ\). Segment rolls without slipping; \(I_G = 0.25\,\mathrm{kg\cdot m^2}\). figure — add your own
18–30 Springs & Conservative Forces 24 problems
Applications
  1. 18–44.The compound disk pulley has mass \(3\,\mathrm{kg}\), \(k_G = 45\,\mathrm{mm}\). Determine the speed of block \(A\) after it descends \(0.2\,\mathrm{m}\) from rest. Blocks \(A\) and \(B\) each have mass \(2\,\mathrm{kg}\). Neglect cord mass. figure — add your own
  2. 18–45.Determine the speed of the 50‑kg cylinder after it has descended \(2\,\mathrm{m}\) from rest. Gear \(A\): \(10\,\mathrm{kg}\), \(k = 125\,\mathrm{mm}\). Gear \(B\) and drum \(C\): combined mass \(30\,\mathrm{kg}\), \(k = 150\,\mathrm{mm}\). figure — add your own
  3. 18–46.The assembly consists of two 10‑kg bars pin-connected. Released from rest when \(\theta = 60^\circ\), determine their angular velocities at \(\theta = 0^\circ\). The 5‑kg disk at \(C\) has radius \(0.5\,\mathrm{m}\) and rolls without slipping. figure — add your own
  4. 18–47.Same assembly as 18–46, determine angular velocities at \(\theta = 30^\circ\). figure — add your own
  5. 18–48.The two 12‑kg slender rods are pin-connected and released from rest at \(\theta = 60^\circ\). The spring has an unstretched length of \(1.5\,\mathrm{m}\). Determine the angular velocity of rod \(BC\) when \(\theta = 0^\circ\). Neglect mass of roller at \(C\). figure — add your own
  6. 18–49.The uniform garage door has mass \(150\,\mathrm{kg}\) and is guided along smooth tracks. Lifting is done using two springs, each attached to the anchor bracket at \(A\) and to the counterbalance shaft at \(B\) and \(C\). Each spring provides torsional moment \(M = (0.7\theta)\,\mathrm{N\cdot m}\) (\(\theta\) in rad). Determine the angle \(\theta_0\) at which both springs should be attached so that the door is completely balanced (when vertical and given a slight upward force, springs lift the door to horizontal with no final angular velocity). \(k = 0.7\,\mathrm{N\cdot m/rad}\). figure — add your own
  7. 18–50.The 40‑kg wheel has \(k_G = 250\,\mathrm{mm}\). If it rolls without slipping, determine its angular velocity when it has rotated clockwise \(90^\circ\) from the position shown. Spring \(AB\) has \(k = 100\,\mathrm{N/m}\) and unstretched length \(500\,\mathrm{mm}\). Released from rest. figure — add your own
  8. 18–51.The 12‑kg slender rod is attached to a spring with unstretched length \(2\,\mathrm{m}\). Released from rest when \(\theta = 30^\circ\), determine its angular velocity at \(\theta = 90^\circ\). figure — add your own
  9. 18–52.Same rod as 18–51, determine its angular velocity at the instant the spring becomes unstretched. figure — add your own
  10. 18–53.The 6‑kg rod \(ABC\) is connected to the 3‑kg rod \(CD\). Released from rest when \(\theta = 0^\circ\), determine the angular velocity of rod \(ABC\) at the instant it becomes horizontal. figure — add your own
  11. 18–54.The slender 6‑kg bar \(AB\) is horizontal and at rest, spring unstretched. Determine the stiffness \(k\) of the spring so that the motion of the bar is momentarily stopped when it has rotated clockwise \(90^\circ\) after being released. figure — add your own
  12. 18–55.The torsional spring at \(A\) has stiffness \(k = 2000\,\mathrm{N\cdot m/rad}\) and is uncoiled at \(\theta = 0^\circ\). Determine the angular velocity of bars \(AB\) and \(BC\) when \(\theta = 0^\circ\), if released from rest at \(\theta = 90^\circ\). Bars have mass per unit length \(20\,\mathrm{kg/m}\). figure — add your own
  13. 18–56.The slender 6‑kg bar \(AB\) is horizontal and at rest, spring unstretched. Determine its angular velocity when it has rotated clockwise \(45^\circ\) after being released. Spring stiffness \(k = 12\,\mathrm{N/m}\). figure — add your own
  14. 18–57.A spring with \(k = 300\,\mathrm{N/m}\) is attached to the end of the 15‑kg rod, unstretched at \(\theta = 0^\circ\). Released from rest at \(\theta = 0^\circ\), determine its angular velocity at \(\theta = 30^\circ\). Motion in vertical plane. figure — add your own
  15. 18–58.The slender 15‑kg bar is initially at rest and standing vertically when end \(A\) is displaced slightly to the right. If the track is smooth, determine the speed at which end \(A\) strikes corner \(D\). Bar constrained to move in vertical plane; neglect mass of cord \(BC\). figure — add your own
  16. 18–59.The pendulum consists of a 6‑kg slender rod fixed to a 15‑kg disk. The spring has unstretched length \(0.2\,\mathrm{m}\). Determine the angular velocity of the pendulum when released from rest and rotated clockwise \(90^\circ\). Roller at \(C\) keeps spring vertical. figure — add your own
  17. 18–60.The two 12‑kg slender rods are pin-connected and released from rest at \(\theta = 60^\circ\). The spring has unstretched length \(1.5\,\mathrm{m}\). Determine the angular velocity of rod \(BC\) when \(\theta = 30^\circ\). figure — add your own
  18. 18–61.If the 40‑kg gear \(B\) is released from rest at \(\theta = 0^\circ\), determine the angular velocity of the 20‑kg gear \(A\) at \(\theta = 90^\circ\). Radii of gyration: \(k_A = 125\,\mathrm{mm}\), \(k_B = 175\,\mathrm{mm}\). Outer gear ring \(P\) is fixed. figure — add your own
  19. 18–62.The 30‑kg pendulum has mass center at \(G\) and \(k_G = 300\,\mathrm{mm}\). Released from rest when \(\theta = 0^\circ\), determine its angular velocity at \(\theta = 90^\circ\). Spring \(AB\) has \(k = 300\,\mathrm{N/m}\) and is unstretched at \(\theta = 0^\circ\). figure — add your own
  20. 18–63.The spool has mass \(50\,\mathrm{kg}\), \(k_O = 0.280\,\mathrm{m}\). The 20‑kg block \(A\) is released from rest. Determine the distance the block must fall for the spool to have \(\omega = 5\,\mathrm{rad/s}\). Also find the tension. figure — add your own
  21. 18–64.The \(500\,\mathrm{g}\) rod \(AB\) rests along the smooth inner surface of a hemispherical bowl. Released from rest from the position shown, determine its angular velocity at the instant it swings downward and becomes horizontal. figure — add your own
  22. 18–65.The motion of the uniform \(40\,\mathrm{kg}\) garage door is guided at its ends by the track. Determine the required initial stretch in the spring when the door is open (\(\theta = 0^\circ\)) so that when it falls freely it comes to rest when it just reaches the fully closed position (\(\theta = 90^\circ\)). Door treated as thin plate; spring/pulley system on each side. figure — add your own
  23. 18–66.Same garage door as 18–65, released from rest at \(\theta = 0^\circ\). Determine the door's angular velocity at \(\theta = 30^\circ\). The spring is originally stretched \(0.3\,\mathrm{m}\) when held open (\(\theta = 0^\circ\)). figure — add your own
  24. 18–67.The system consists of a 30‑kg disk, a 12‑kg slender rod \(BA\), and a 5‑kg smooth collar \(A\). Disk rolls without slipping. Determine the velocity of the collar at \(\theta = 0^\circ\). Released from rest at \(\theta = 45^\circ\). figure — add your own
  25. 18–68.Same system as 18–67, determine the velocity of the collar at \(\theta = 30^\circ\). Released from rest at \(\theta = 45^\circ\). figure — add your own
R18 Review Problems 8 problems
Review
  1. R18–1.The pendulum of the Charpy impact machine has mass \(50\,\mathrm{kg}\), \(k_A = 1.75\,\mathrm{m}\). Released from rest when \(\theta = 0^\circ\), determine its angular velocity just before it strikes the specimen \(S\) at \(\theta = 90^\circ\). figure — add your own
  2. R18–2.The \(50\,\mathrm{kg}\) flywheel has \(k_O = 200\,\mathrm{mm}\) about its center. Subjected to torque \(M = (9\theta^{1/2} + 1)\,\mathrm{N\cdot m}\) (\(\theta\) in rad), determine its angular velocity after it has rotated 5 revolutions, starting from rest. figure — add your own
  3. R18–3.The drum has mass \(50\,\mathrm{kg}\), \(k_O = 0.23\,\mathrm{m}\) about the pin at \(O\). Starting from rest, the suspended \(15\,\mathrm{kg}\) block \(B\) is allowed to fall \(3\,\mathrm{m}\) without applying brake \(ACD\). Determine the speed of the block at this instant. If \(\mu_k = 0.5\) at the brake pad \(C\), determine the force \(P\) that must be applied at the brake handle to stop the block after it descends another \(3\,\mathrm{m}\). Neglect handle thickness. figure — add your own
  4. R18–4.The spool has mass \(60\,\mathrm{kg}\), \(k_G = 0.3\,\mathrm{m}\). Released from rest, determine how far its center descends down the smooth plane before it attains \(\omega = 6\,\mathrm{rad/s}\). Neglect mass of cord wound around central core. Coefficient of kinetic friction between spool and plane at \(A\) is \(\mu_k = 0.2\). figure — add your own
  5. R18–5.The gear rack has mass \(6\,\mathrm{kg}\); each gear has mass \(4\,\mathrm{kg}\) and radius of gyration \(k = 30\,\mathrm{mm}\) at its center. If the rack is originally moving downward at \(2\,\mathrm{m/s}\) when \(s = 0\), determine its speed when \(s = 600\,\mathrm{mm}\). Gears are free to turn about their centers \(A\) and \(B\). figure — add your own
  6. R18–6.At the instant shown, the \(25\,\mathrm{kg}\) bar rotates clockwise at \(2\,\mathrm{rad/s}\). The spring attached to its end remains vertical due to the roller guide at \(C\). If the spring has unstretched length \(1\,\mathrm{m}\) and stiffness \(k = 90\,\mathrm{N/m}\), determine the angular velocity of the bar the instant it has rotated \(30^\circ\) clockwise. figure — add your own
  7. R18–7.The system consists of a \(10\,\mathrm{kg}\) disk \(A\), a \(2\,\mathrm{kg}\) slender rod \(BC\), and a \(0.5\,\mathrm{kg}\) smooth collar \(C\). Disk rolls without slipping. Determine the velocity of the collar at the instant the rod becomes horizontal (\(\theta = 0^\circ\)). Released from rest at \(\theta = 45^\circ\). figure — add your own
  8. R18–8.At the instant the spring becomes undeformed, the center of the \(40\,\mathrm{kg}\) disk has speed \(4\,\mathrm{m/s}\). From this point determine the distance \(d\) the disk moves down the plane before momentarily stopping. The disk rolls without slipping. figure — add your own