Engineering Mechanics: Dynamics · Open Problem Companion

Chapter Seventeen
Planar Kinetics of a Rigid Body

17.1 Mass Moment of Inertia 23 problems
Concepts & Skills
  1. 1.Determine the moment of inertia \(I_{y}\) for the slender rod. The rod's density \(\rho\) and cross-sectional area \(A\) are constant. Express the result in terms of the rod's total mass \(m\). figure — add your own
  2. 2.The paraboloid is formed by revolving the shaded area around the \(x\) axis. Determine the radius of gyration \(k_{x}\). The density of the material is \(\rho = 5\,\mathrm{Mg/m^3}\). figure — add your own
  3. 3.The solid cylinder has an outer radius \(R\), height \(h\), and is made from a material having a density that varies from its center as \(\rho = k + a r^{2}\), where \(k\) and \(a\) are constants. Determine the mass of the cylinder and its moment of inertia about the \(z\) axis. figure — add your own
  4. 4.Determine the moment of inertia of the thin ring about the \(z\) axis. The ring has a mass \(m\). figure — add your own
  5. 5.The hemisphere is formed by rotating the shaded area around the \(y\) axis. Determine the moment of inertia \(I_{y}\) and express the result in terms of the total mass \(m\) of the hemisphere. The material has a constant density \(\rho\). figure — add your own
  6. 6.The frustum is formed by rotating the shaded area around the \(x\) axis. Determine the moment of inertia \(I_{x}\) and express the result in terms of the total mass \(m\) of the frustum. The frustum has a constant density \(\rho\). figure — add your own
  7. 7.The sphere is formed by revolving the shaded area around the \(x\) axis. Determine the moment of inertia \(I_{x}\) and express the result in terms of the total mass \(m\) of the sphere. The material has a constant density \(\rho\). figure — add your own
  8. 8.Determine the mass moment of inertia \(I_{z}\) of the cone formed by revolving the shaded area around the \(z\) axis. The density of the material is \(\rho\). Express the result in terms of the mass \(m\) of the cone. figure — add your own
  9. 9.Determine the moment of inertia \(I_{z}\) of the torus. The mass of the torus is \(m\) and the density \(\rho\) is constant. Suggestion: Use a shell element. figure — add your own
  10. 10.Determine the location \(\overline{y}\) of the center of mass \(G\) of the assembly and then calculate the moment of inertia about an axis perpendicular to the page and passing through \(G\). The block has a mass of \(3\,\mathrm{kg}\) and the semicylinder has a mass of \(5\,\mathrm{kg}\). figure — add your own
  11. 11.Determine the moment of inertia of the assembly about an axis perpendicular to the page and passing through point \(O\). The block has a mass of \(3\,\mathrm{kg}\) and the semicylinder has a mass of \(5\,\mathrm{kg}\). figure — add your own
  12. 12.Determine the mass moment of inertia of the thin plate about an axis perpendicular to the page and passing through point \(O\). The material has a mass per unit area of \(20\,\mathrm{kg/m^2}\). figure — add your own
  13. 13.Determine the moment of inertia of the homogeneous triangular prism with respect to the \(y\) axis. Express the result in terms of the mass \(m\) of the prism. Hint: For integration, use thin plate elements parallel to the \(x-y\) plane and having a thickness \(dz\). figure — add your own
  14. 14.The wheel consists of a thin ring having a mass of \(10\,\mathrm{kg}\) and four spokes made from slender rods and each having a mass of \(2\,\mathrm{kg}\). Determine the wheel's moment of inertia about an axis perpendicular to the page and passing through point \(A\). figure — add your own
  15. 15.The assembly is made of the slender rods that have a mass per unit length of \(3\,\mathrm{kg/m}\). Determine the mass moment of inertia of the assembly about an axis perpendicular to the page and passing through point \(O\). figure — add your own
  16. 16.The assembly consists of a disk having a mass of \(6\,\mathrm{kg}\) and slender rods \(AB\) and \(DC\) which have a mass per unit length of \(2\,\mathrm{kg/m}\). Determine the length \(L\) of \(DC\) so that the center of mass is at the bearing \(O\). What is the moment of inertia of the assembly about an axis perpendicular to the page and passing through \(O\)? figure — add your own
  17. 17.The pendulum consists of a \(4\,\mathrm{kg}\) circular plate and a \(2\,\mathrm{kg}\) slender rod. Determine the radius of gyration of the pendulum about an axis perpendicular to the page and passing through point \(O\). figure — add your own
  18. 18.Determine the moment of inertia about an axis perpendicular to the page and passing through the pin at \(O\). The thin plate has a hole in its center. Its thickness is \(50\,\mathrm{mm}\) and the material has a density \(\rho = 50\,\mathrm{kg/m^3}\). figure — add your own
  19. 19.The pendulum consists of two slender rods \(AB\) and \(OC\) which have a mass per unit length of \(3\,\mathrm{kg/m}\). The thin circular plate has a mass per unit area of \(12\,\mathrm{kg/m^2}\). Determine the location \(\overline{y}\) of the center of mass \(G\) of the pendulum, then calculate the moment of inertia of the pendulum about an axis perpendicular to the page and passing through \(G\). figure — add your own
  20. 20.The pendulum consists of two slender rods \(AB\) and \(OC\) which have a mass per unit length of \(3\,\mathrm{kg/m}\). The thin circular plate has a mass per unit area of \(12\,\mathrm{kg/m^2}\). Determine the moment of inertia of the pendulum about an axis perpendicular to the page and passing through the pin at \(O\). figure — add your own
  21. 21.The pendulum consists of the \(3\,\mathrm{kg}\) slender rod and the \(5\,\mathrm{kg}\) thin plate. Determine the location \(\overline{y}\) of the center of mass \(G\) of the pendulum; then calculate the moment of inertia of the pendulum about an axis perpendicular to the page and passing through \(G\). figure — add your own
  22. 22.Determine the moment of inertia of the overhung crank about the \(x\) axis. The material is steel having a density of \(\rho = 7.85\,\mathrm{Mg/m^3}\). figure — add your own
  23. 23.Determine the moment of inertia of the overhung crank about the \(x^{\prime}\) axis. The material is steel having a density of \(\rho = 7.85\,\mathrm{Mg/m^3}\). figure — add your own
17.2 Equations of Motion: Translation 32 problems
Preliminary Problems
  1. P1.Draw the free-body and kinetic diagrams of the object \(AB\). figure — add your own
  2. P2.Draw the free-body and kinetic diagrams of the 100-kg object. figure — add your own
Fundamental Problems
  1. F1.The cart and its load have a total mass of \(100\,\mathrm{kg}\). Determine the acceleration of the cart and the normal reactions on the pair of wheels at \(A\) and \(B\). Neglect the mass of the wheels. figure — add your own
  2. F2.If the \(80\,\mathrm{kg}\) cabinet is allowed to roll down the inclined plane, determine the acceleration of the cabinet and the normal reactions on the pair of rollers at \(A\) and \(B\) that have negligible mass. figure — add your own
  3. F3.The \(10\,\mathrm{kg}\) link \(AB\) is pinned to a moving frame at \(A\) and held in a vertical position by means of a string \(BC\) which can support a maximum tension of \(50\,\mathrm{N}\). Determine the maximum acceleration of the frame without breaking the string. What are the corresponding components of reaction at the pin \(A\)? figure — add your own
  4. F4.Determine the maximum acceleration of the truck without causing the assembly to move relative to the truck. Also what is the corresponding normal reaction on legs \(A\) and \(B\)? The \(100\,\mathrm{kg}\) table has a mass center at \(G\) and the coefficient of static friction between the legs of the table and the bed of the truck is \(\mu_{s} = 0.2\). figure — add your own
  5. F5.At the instant shown both rods of negligible mass swing with a counterclockwise angular velocity of \(\omega = 5\,\mathrm{rad/s}\), while the \(50\,\mathrm{kg}\) bar is subjected to the \(100\,\mathrm{N}\) horizontal force. Determine the tension developed in the rods and the angular acceleration of the rods at this instant. figure — add your own
  6. F6.At the instant shown, link \(CD\) rotates with an angular velocity of \(\omega = 6\,\mathrm{rad/s}\). If it is subjected to a couple moment \(M = 450\,\mathrm{N\cdot m}\), determine the force developed in link \(AB\), the horizontal and vertical component of reaction on pin \(D\), and the angular acceleration of link \(CD\) at this instant. The block has a mass of \(50\,\mathrm{kg}\) and center of mass at \(G\). Neglect the mass of links \(AB\) and \(CD\). figure — add your own
Skills
  1. 24.The jet aircraft has a total mass of \(22\,\mathrm{Mg}\) and a center of mass at \(G\). Initially at take-off the engines provide a thrust \(2T = 4\,\mathrm{kN}\) and \(T^{\prime} = 1.5\,\mathrm{kN}\). Determine the acceleration of the plane and the normal reactions on the nose wheel at \(A\) and each of the two wing wheels located at \(B\). Neglect the mass of the wheels and, due to low velocity, neglect any lift caused by the wings. figure — add your own
  2. 25.The 4-Mg uniform canister contains nuclear waste material encased in concrete. If the mass of the spreader beam \(BD\) is \(50\,\mathrm{kg}\), determine the force in each of the links \(AB\), \(CD\), \(EF\), and \(GH\) when the system is lifted with an acceleration of \(a = 2\,\mathrm{m/s^2}\) for a short period of time. figure — add your own
  3. 26.The 4-Mg uniform canister contains nuclear waste material encased in concrete. If the mass of the spreader beam \(BD\) is \(50\,\mathrm{kg}\), determine the largest vertical acceleration \(a\) of the system so that each of the links \(AB\) and \(CD\) are not subjected to a force greater than \(30\,\mathrm{kN}\) and links \(EF\) and \(GH\) are not subjected to a force greater than \(34\,\mathrm{kN}\). figure — add your own
  4. 27.The assembly has a mass of \(8\,\mathrm{Mg}\) and is hoisted using the boom and pulley system. If the winch at \(B\) draws in the cable with an acceleration of \(2\,\mathrm{m/s^2}\), determine the compressive force in the hydraulic cylinder needed to support the boom. The boom has a mass of \(2\,\mathrm{Mg}\) and mass center at \(G\). figure — add your own
  5. 28.The assembly has a mass of \(4\,\mathrm{Mg}\) and is hoisted using the winch at \(B\). Determine the greatest acceleration of the assembly so that the compressive force in the hydraulic cylinder supporting the boom does not exceed \(180\,\mathrm{kN}\). What is the tension in the supporting cable? The boom has a mass of \(2\,\mathrm{Mg}\) and mass center at \(G\). figure — add your own
  6. 29.Determine the shortest time possible for the rear-wheel drive, 2-Mg truck to achieve a speed of \(16\,\mathrm{m/s}\) with a constant acceleration starting from rest. The coefficient of static friction between the wheels and the road surface is \(\mu_{s} = 0.8\). The front wheels are free to roll. Neglect the mass of the wheels. figure — add your own
  7. 30.The 150-kg uniform crate rests on the 10-kg cart. Determine the maximum force \(P\) that can be applied to the handle without causing the crate to slip or tip on the cart. The coefficient of static friction between the crate and cart is \(\mu_{s} = 0.2\). figure — add your own
  8. 31.The 150-kg uniform crate rests on the 10-kg cart. Determine the maximum force \(P\) that can be applied to the handle without causing the crate to tip on the cart. Slipping does not occur. figure — add your own
  9. 32.The pipe has a mass of \(460\,\mathrm{kg}\) and is held in place on the truck bed using the two boards \(A\) and \(B\). Determine the greatest acceleration of the truck so that the pipe begins to lose contact at \(A\) and the bed of the truck and starts to pivot about \(B\). Assume board \(B\) will not slip on the bed of the truck, and the pipe is smooth. Also, what force does board \(B\) exert on the pipe during the acceleration? figure — add your own
  10. 33.The uniform girder \(AB\) has a mass of \(8\,\mathrm{Mg}\). Determine the internal axial, shear, and bending-moment loadings at the center of the girder if a crane gives it an upward acceleration of \(3\,\mathrm{m/s^2}\). figure — add your own
  11. 34.The mountain bike has a mass of \(40\,\mathrm{kg}\) with center of mass at point \(G_1\), while the rider has a mass of \(60\,\mathrm{kg}\) with center of mass at point \(G_2\). Determine the maximum deceleration when the brake is applied to the front wheel, without causing the rear wheel \(B\) to leave the road. Assume that the front wheel does not slip. Neglect the mass of all the wheels. figure — add your own
  12. 35.The mountain bike has a mass of \(40\,\mathrm{kg}\) with center of mass at point \(G_1\), while the rider has a mass of \(60\,\mathrm{kg}\) with center of mass at point \(G_2\). When the brake is applied to the front wheel, it causes the bike to decelerate at a constant rate of \(3\,\mathrm{m/s^2}\). Determine the normal reaction the road exerts on the front and rear wheels. Assume that the rear wheel is free to roll. Neglect the mass of all the wheels. figure — add your own
  13. 36.The trailer with its load has a mass of \(150\,\mathrm{kg}\) and a center of mass at \(G\). If it is subjected to a horizontal force of \(P = 600\,\mathrm{N}\), determine the trailer's acceleration and the normal force on the pair of wheels at \(A\) and at \(B\). The wheels are free to roll and have negligible mass. figure — add your own
  14. 37.A force of \(P = 300\,\mathrm{N}\) is applied to the \(60\,\mathrm{kg}\) cart. Determine the reactions at both the wheels at \(A\) and both the wheels at \(B\). Also, what is the acceleration of the cart? The mass center of the cart is at \(G\). figure — add your own
  15. 38.Determine the largest force \(\mathbf{P}\) that can be applied to the \(60\,\mathrm{kg}\) cart, without causing one of the wheel reactions, either at \(A\) or at \(B\), to be zero. Also, what is the acceleration of the cart? The mass center of the cart is at \(G\). figure — add your own
  16. 39.If the cart's mass is \(30\,\mathrm{kg}\) and it is subjected to a horizontal force of \(P = 90\,\mathrm{N}\), determine the tension in cord \(AB\) and the horizontal and vertical components of reaction on end \(C\) of the uniform \(15\,\mathrm{kg}\) rod \(BC\). figure — add your own
  17. 40.If the cart's mass is \(30\,\mathrm{kg}\), determine the horizontal force \(P\) that should be applied to the cart so that the cord \(AB\) just becomes slack. The uniform rod \(BC\) has a mass of \(15\,\mathrm{kg}\). figure — add your own
  18. 41.The uniform bar of mass \(m\) is pin connected to the collar, which slides along the smooth horizontal rod. If the collar is given a constant acceleration of \(a\), determine the bar's inclination angle \(\theta\). Neglect the collar's mass. figure — add your own
  19. 42.The dragster has a mass of \(1500\,\mathrm{kg}\) and a center of mass at \(G\). If the coefficient of kinetic friction between the rear wheels and the pavement is \(\mu_{k} = 0.6\), determine if it is possible for the driver to lift the front wheels, \(A\), off the ground while the rear drive wheels are slipping. Neglect the mass of the wheels and assume that the front wheels are free to roll. figure — add your own
  20. 43.The dragster has a mass of \(1500\,\mathrm{kg}\) and a center of mass at \(G\). If no slipping occurs, determine the frictional force \(\mathbf{F}_B\) which must be developed at each of the rear drive wheels \(B\) in order to create an acceleration of \(a = 6\,\mathrm{m/s^2}\). What are the normal reactions of each wheel on the ground? Neglect the mass of the wheels and assume that the front wheels are free to roll. figure — add your own
  21. 44.The pipe has a length of \(3\,\mathrm{m}\) and a mass of \(500\,\mathrm{kg}\). It is attached to the back of the truck using a \(0.6\,\mathrm{m}\)-long chain \(AB\). If the coefficient of kinetic friction at \(C\) is \(\mu_{k} = 0.4\), determine the acceleration of the truck if the angle \(\theta = 10^{\circ}\) with the road as shown. figure — add your own
  22. 45.The lift truck has a mass of \(70\,\mathrm{kg}\) and mass center at \(G\). If it lifts the \(120\,\mathrm{kg}\) spool with an acceleration of \(3\,\mathrm{m/s^2}\), determine the reactions of each of the four wheels. The loading is symmetric. Neglect the mass of the movable arm \(CD\). figure — add your own
  23. 46.The lift truck has a mass of \(70\,\mathrm{kg}\) and mass center at \(G\). Determine the largest upward acceleration of the \(120\,\mathrm{kg}\) spool so that no reaction of the wheels on the ground exceeds \(600\,\mathrm{N}\). figure — add your own
  24. 47.The uniform crate has a mass of \(50\,\mathrm{kg}\) and rests on the cart having an inclined surface. Determine the smallest acceleration that will cause the crate either to tip or slip relative to the cart. What is the magnitude of this acceleration? The coefficient of static friction between the crate and cart is \(\mu_{s} = 0.5\). figure — add your own
  25. 48.The drop gate at the end of the trailer has a mass of \(1.25\,\mathrm{Mg}\) and mass center at \(G\). If it is supported by the cable \(AB\) and hinge at \(C\), determine the tension in the cable when the truck begins to accelerate at \(5\,\mathrm{m/s^2}\). Also, what are the horizontal and vertical components of reaction at the hinge \(C\)? figure — add your own
  26. 49.The drop gate at the end of the trailer has a mass of \(1.25\,\mathrm{Mg}\) and mass center at \(G\). If it is supported by the cable \(AB\) and hinge at \(C\), determine the maximum deceleration of the truck so that the gate does not begin to rotate forward. What are the horizontal and vertical components of reaction at the hinge \(C\)? figure — add your own
  27. 50.The bar has a weight per length \(w\) and is supported by the smooth collar. If it is released from rest, determine the internal normal force, shear force, and bending moment in the bar as a function of \(x\). figure — add your own
  28. 51.The pipe has a mass of \(800\,\mathrm{kg}\) and is being towed behind the truck. If the acceleration of the truck is \(a_{t} = 0.5\,\mathrm{m/s^2}\), determine the angle \(\theta\) and the tension in the cable. The coefficient of kinetic friction between the pipe and the ground is \(\mu_{k} = 0.1\). figure — add your own
  29. 52.The pipe has a mass of \(800\,\mathrm{kg}\) and is being towed behind a truck. If the angle \(\theta = 30^{\circ}\), determine the acceleration of the truck and the tension in the cable. The coefficient of kinetic friction between the pipe and the ground is \(\mu_{k} = 0.1\). figure — add your own
  30. 53.The \(100\,\mathrm{kg}\) uniform crate \(C\) rests on the elevator floor where the coefficient of static friction is \(\mu_{s} = 0.4\). Determine the largest initial angular acceleration \(\alpha\), starting from rest at \(\theta = 90^{\circ}\), without causing the crate to slip. No tipping occurs. figure — add your own
  31. 54.The two uniform \(4\,\mathrm{kg}\) bars \(DC\) and \(EF\) are fixed (welded) together at \(E\). Determine the normal force \(N_{E}\), shear force \(V_{E}\), and moment \(M_{E}\), which \(DC\) exerts on \(EF\) at \(E\) if at the instant \(\theta = 60^{\circ}\), \(BC\) has an angular velocity \(\omega = 2\,\mathrm{rad/s}\) and an angular acceleration \(\alpha = 4\,\mathrm{rad/s^2}\) as shown. figure — add your own
  32. 55.The arched pipe has a mass of \(80\,\mathrm{kg}\) and rests on the surface of the platform for which the coefficient of static friction is \(\mu_{s} = 0.3\). Determine the greatest angular acceleration \(\alpha\) of the platform, starting from rest when \(\theta = 45^{\circ}\), without causing the pipe to slip on the platform. figure — add your own
  33. 56.Determine the force developed in the links and the acceleration of the bar's mass center immediately after the cord fails. Neglect the mass of links \(AB\) and \(CD\). The uniform bar has a mass of \(20\,\mathrm{kg}\). figure — add your own
17.3 Rotation about a Fixed Axis 36 problems
Fundamental Problems
  1. F7.The \(100\,\mathrm{kg}\) wheel has a radius of gyration about its center \(O\) of \(k_{O} = 500\,\mathrm{mm}\). If the wheel starts from rest, determine its angular velocity in \(t = 3\,\mathrm{s}\). figure — add your own
  2. F8.The \(50\,\mathrm{kg}\) disk is subjected to the couple moment of \(M = (9t)\,\mathrm{N\cdot m}\), where \(t\) is in seconds. Determine the angular velocity of the disk when \(t = 4\,\mathrm{s}\) starting from rest. figure — add your own
  3. F9.At the instant shown, the uniform \(30\,\mathrm{kg}\) slender rod has a counterclockwise angular velocity of \(\omega = 6\,\mathrm{rad/s}\). Determine the tangential and normal components of reaction of pin \(O\) on the rod and the angular acceleration of the rod at this instant. figure — add your own
  4. F10.At the instant shown, the \(30\,\mathrm{kg}\) disk has a counterclockwise angular velocity of \(\omega = 10\,\mathrm{rad/s}\). Determine the tangential and normal components of reaction of the pin \(O\) on the disk and the angular acceleration of the disk at this instant. figure — add your own
  5. F11.The uniform slender rod has a mass of \(15\,\mathrm{kg}\). Determine the horizontal and vertical components of reaction at the pin \(O\), and the angular acceleration of the rod just after the cord is cut. figure — add your own
  6. F12.The uniform \(30\,\mathrm{kg}\) slender rod is being pulled by the cord that passes over the small smooth peg at \(A\). If the rod has a counterclockwise angular velocity of \(\omega = 6\,\mathrm{rad/s}\) at the instant shown, determine the tangential and normal components of reaction at the pin \(O\) and the angular acceleration of the rod. figure — add your own
Skills
  1. 57.The 10-kg wheel has a radius of gyration \(k_{A} = 200\,\mathrm{mm}\). If the wheel is subjected to a moment \(M = (5t)\,\mathrm{N\cdot m}\) where \(t\) is in seconds, determine its angular velocity when \(t = 3\,\mathrm{s}\) starting from rest. Also, compute the reactions which the fixed pin \(A\) exerts on the wheel during the motion. figure — add your own
  2. 58.The uniform 24-kg plate is released from rest at the position shown. Determine its initial angular acceleration and the horizontal and vertical reactions at the pin \(A\). figure — add your own
  3. 59.The uniform slender rod has a mass \(m\). If it is released from rest when \(\theta = 0^{\circ}\), determine the magnitude of the reactive force exerted on it by pin \(B\) when \(\theta = 90^{\circ}\). figure — add your own
  4. 60.The bent rod has a mass of \(2\,\mathrm{kg/m}\). If it is released from rest in the position shown, determine its initial angular acceleration and the horizontal and vertical components of reaction at \(A\). figure — add your own
  5. 61.If a horizontal force of \(P = 100\,\mathrm{N}\) is applied to the 300-kg reel of cable, determine its initial angular acceleration. The reel rests on rollers at \(A\) and \(B\) and has a radius of gyration of \(k_{O} = 0.6\,\mathrm{m}\). figure — add your own
  6. 62.The \(20\,\mathrm{kg}\) roll of paper has a radius of gyration \(k_{A} = 90\,\mathrm{mm}\) about an axis passing through point \(A\). It is pin supported at both ends by two brackets \(AB\). If the roll rests against a wall for which the coefficient of kinetic friction is \(\mu_{k} = 0.2\) and a vertical force \(F = 30\,\mathrm{N}\) is applied to the end of the paper, determine the angular acceleration of the roll as the paper unrolls. figure — add your own
  7. 63.The \(20\,\mathrm{kg}\) roll of paper has a radius of gyration \(k_{A} = 90\,\mathrm{mm}\) about an axis passing through point \(A\). It is pin supported at both ends by two brackets \(AB\). If the roll rests against a wall for which the coefficient of kinetic friction is \(\mu_{k} = 0.2\), determine the constant vertical force \(F\) that must be applied to the roll to pull off \(1\,\mathrm{m}\) of paper in \(t = 3\,\mathrm{s}\) starting from rest. Neglect the mass of paper that is removed. figure — add your own
  8. 64.The kinetic diagram representing the general rotational motion of a rigid body about a fixed axis passing through \(O\) is shown in the figure. Show that \(I_{G}\alpha\) may be eliminated by moving the vectors \(m(\mathbf{a}_{G})_{t}\) and \(m(\mathbf{a}_{G})_{n}\) to point \(P\), located a distance \(r_{GP} = k_{G}^{2}/r_{OG}\) from the center of mass \(G\) of the body. Here \(k_{G}\) represents the radius of gyration of the body about an axis passing through \(G\). The point \(P\) is called the center of percussion of the body. figure — add your own
  9. 65.Gears \(A\) and \(B\) have a mass of \(50\,\mathrm{kg}\) and \(15\,\mathrm{kg}\) respectively. Their radii of gyration about their respective centers of mass are \(k_{C} = 250\,\mathrm{mm}\) and \(k_{D} = 150\,\mathrm{mm}\). If a torque of \(M = 200(1 - e^{-0.2t})\,\mathrm{N\cdot m}\), where \(t\) is in seconds, is applied to gear \(A\), determine the angular velocity of both gears when \(t = 3\,\mathrm{s}\), starting from rest. figure — add your own
  10. 66.The reel of cable has a mass of \(400\,\mathrm{kg}\) and a radius of gyration of \(k_{A} = 0.75\,\mathrm{m}\). Determine its angular velocity when \(t = 2\,\mathrm{s}\), starting from rest, if the force \(\mathbf{P} = (20t^{2} + 80)\,\mathrm{N}\), when \(t\) is in seconds. Neglect the mass of the unwound cable, and assume it is always at a radius of \(0.5\,\mathrm{m}\). figure — add your own
  11. 67.The door will close automatically using torsional springs mounted on the hinges. Each spring has a stiffness \(k = 50\,\mathrm{N\cdot m/rad}\) so that the torque on each hinge is \(M = (50\theta)\,\mathrm{N\cdot m}\), where \(\theta\) is measured in radians. If the door is released from rest when it is open at \(\theta = 90^{\circ}\), determine its angular velocity at the instant \(\theta = 0^{\circ}\). For the calculation, treat the door as a thin plate having a mass of \(70\,\mathrm{kg}\). figure — add your own
  12. 68.The door will close automatically using torsional springs mounted on the hinges. If the torque on each hinge is \(M = k\theta\), where \(\theta\) is measured in radians, determine the required torsional stiffness \(k\) so that the door will close \((\theta = 0^{\circ})\) with an angular velocity \(\omega = 2\,\mathrm{rad/s}\) when it is released from rest at \(\theta = 90^{\circ}\). For the calculation, treat the door as a thin plate having a mass of \(70\,\mathrm{kg}\). figure — add your own
  13. 69.If the cord at \(B\) suddenly fails, determine the horizontal and vertical components of the initial reaction at the pin \(A\) and the angular acceleration of the 120-kg beam. Treat the beam as a uniform slender rod. figure — add your own
  14. 70.The device acts as a pop-up barrier to prevent the passage of a vehicle. It consists of a 100-kg steel plate \(AC\) and a 200-kg counterweight solid concrete block located as shown. Determine the moment of inertia of the plate and block about the hinged axis through \(A\). Neglect the mass of the supporting arms \(AB\). Also, determine the initial angular acceleration of the assembly when it is released from rest at \(\theta = 45^{\circ}\). figure — add your own
  15. 71.A cord is wrapped around the outer surface of the \(8\,\mathrm{kg}\) disk. If a force of \(F = (1/4\theta^{2})\,\mathrm{N}\), where \(\theta\) is in radians, is applied to the cord, determine the disk's angular acceleration when it has turned 5 revolutions. The disk has an initial angular velocity of \(\omega_{0} = 1\,\mathrm{rad/s}\). figure — add your own
  16. 72.Block \(A\) has a mass \(m\) and rests on a surface having a coefficient of kinetic friction \(\mu_{k}\). The cord attached to \(A\) passes over a pulley at \(C\) and is attached to a block \(B\) having a mass \(2m\). If \(B\) is released, determine the acceleration of \(A\). Assume that the cord does not slip over the pulley. The pulley can be approximated as a thin disk of radius \(r\) and mass \(\frac{1}{4}m\). Neglect the mass of the cord. figure — add your own
  17. 73.The two blocks \(A\) and \(B\) have a mass of \(5\,\mathrm{kg}\) and \(10\,\mathrm{kg}\), respectively. If the pulley can be treated as a disk of mass \(3\,\mathrm{kg}\) and radius \(0.15\,\mathrm{m}\), determine the acceleration of block \(A\). Neglect the mass of the cord and any slipping on the pulley. figure — add your own
  18. 74.The two blocks \(A\) and \(B\) have a mass \(m_{A}\) and \(m_{B}\) respectively, where \(m_{B} > m_{A}\). If the pulley can be treated as a disk of mass \(M\), determine the acceleration of block \(A\). Neglect the mass of the cord and any slipping on the pulley. figure — add your own
  19. 75.The \(30\,\mathrm{kg}\) disk is originally spinning at \(\omega = 125\,\mathrm{rad/s}\). If it is placed on the ground, for which the coefficient of kinetic friction is \(\mu_{C} = 0.5\), determine the time required for the motion to stop. What are the horizontal and vertical components of force which the member \(AB\) exerts on the pin at \(A\) during this time? Neglect the mass of \(AB\). figure — add your own
  20. 76.The wheel has a mass of \(25\,\mathrm{kg}\) and a radius of gyration \(k_{B} = 0.15\,\mathrm{m}\). It is originally spinning at \(\omega = 40\,\mathrm{rad/s}\). If it is placed on the ground, for which the coefficient of kinetic friction is \(\mu_{C} = 0.5\), determine the time required for the motion to stop. What are the horizontal and vertical components of reaction which the pin at \(A\) exerts on \(AB\) during this time? Neglect the mass of \(AB\). figure — add your own
  21. 77.The disk has a mass of \(20\,\mathrm{kg}\) and is originally spinning at the end of the strut with an angular velocity of \(\omega = 60\,\mathrm{rad/s}\). If it is then placed against the wall, for which the coefficient of kinetic friction is \(\mu_{k} = 0.3\), determine the time required for the motion to stop. What is the force in strut \(BC\) during this time? figure — add your own
  22. 78.The \(5\,\mathrm{kg}\) cylinder is initially at rest when it is placed in contact with the wall \(B\) and the rotor at \(A\). If the rotor always maintains a constant clockwise angular velocity \(\omega = 6\,\mathrm{rad/s}\), determine the initial angular acceleration of the cylinder. The coefficient of kinetic friction at the contacting surfaces \(B\) and \(C\) is \(\mu_{k} = 0.2\). figure — add your own
  23. 79.Cable is unwound from a spool supported on small rollers at \(A\) and \(B\) by exerting a force \(T = 300\,\mathrm{N}\) on the cable. Compute the time needed to unravel \(5\,\mathrm{m}\) of cable from the spool if the spool and cable have a total mass of \(600\,\mathrm{kg}\) and a radius of gyration of \(k_{O} = 1.2\,\mathrm{m}\). For the calculation, neglect the mass of the cable being unwound and the mass of the rollers at \(A\) and \(B\). The rollers turn with no friction. figure — add your own
  24. 80.The \(20\,\mathrm{kg}\) roll of paper has a radius of gyration \(k_{A} = 120\,\mathrm{mm}\) about an axis passing through point \(A\). It is pin supported at both ends by two brackets \(AB\). The roll rests on the floor, for which the coefficient of kinetic friction is \(\mu_{k} = 0.2\). If a horizontal force \(F = 60\,\mathrm{N}\) is applied to the end of the paper, determine the initial angular acceleration of the roll as the paper unrolls. figure — add your own
  25. 81.The armature (slender rod) \(AB\) has a mass of \(0.2\,\mathrm{kg}\) and can pivot about the pin at \(A\). Movement is controlled by the electromagnet \(E\), which exerts a horizontal attractive force on the armature at \(B\) of \(F_{B} = (0.2(10^{-3})I^{-2})\,\mathrm{N}\), where \(l\) in meters is the gap between the armature and the magnet at any instant. If the armature lies in the horizontal plane, and is originally at rest, determine the speed of the contact at \(B\) the instant \(l = 0.01\,\mathrm{m}\). Originally \(l = 0.02\,\mathrm{m}\). figure — add your own
  26. 82.The \(4\,\mathrm{kg}\) slender rod is initially supported horizontally by a spring at \(B\) and pin at \(A\). Determine the angular acceleration of the rod and the acceleration of the rod's mass center at the instant the \(100\,\mathrm{N}\) force is applied. figure — add your own
  27. 83.The bar has a weight per length of \(w\). If it is rotating in the vertical plane at a constant rate \(\omega\) about point \(O\), determine the internal normal force, shear force, and moment as a function of \(x\) and \(\theta\). figure — add your own
  28. 84.Determine the angular acceleration of the \(25\,\mathrm{kg}\) diving board and the horizontal and vertical components of reaction at the pin \(A\) the instant the man jumps off. Assume that the board is uniform and rigid, and that at the instant he jumps off the spring is compressed a maximum amount of \(200\,\mathrm{mm}\), \(\omega = 0\), and the board is horizontal. Take \(k = 7\,\mathrm{kN/m}\). figure — add your own
  29. 85.The lightweight turbine consists of a rotor which is powered from a torque applied at its center. At the instant the rotor is horizontal it has an angular velocity of \(15\,\mathrm{rad/s}\) and a clockwise angular acceleration of \(8\,\mathrm{rad/s^2}\). Determine the internal normal force, shear force, and moment at a section through \(A\). Assume the rotor is a \(50\,\mathrm{m}\)-long slender rod, having a mass of \(3\,\mathrm{kg/m}\). figure — add your own
  30. 86.The two-bar assembly is released from rest in the position shown. Determine the initial bending moment at the fixed joint \(B\). Each bar has a mass \(m\) and length \(l\). figure — add your own
  31. 87.The \(100\,\mathrm{kg}\) pendulum has a center of mass at \(G\) and a radius of gyration about \(G\) of \(k_{G} = 250\,\mathrm{mm}\). Determine the horizontal and vertical components of reaction on the beam by the pin \(A\) and the normal reaction of the roller \(B\) at the instant \(\theta = 90^{\circ}\) when the pendulum is rotating at \(\omega = 8\,\mathrm{rad/s}\). Neglect the weight of the beam and the support. figure — add your own
  32. 88.The \(100\,\mathrm{kg}\) pendulum has a center of mass at \(G\) and a radius of gyration about \(G\) of \(k_{G} = 250\,\mathrm{mm}\). Determine the horizontal and vertical components of reaction on the beam by the pin \(A\) and the normal reaction of the roller \(B\) at the instant \(\theta = 0^{\circ}\) when the pendulum is rotating at \(\omega = 4\,\mathrm{rad/s}\). Neglect the weight of the beam and the support. figure — add your own
  33. 89.The "Catherine wheel" is a firework that consists of a coiled tube of powder which is pinned at its center. If the powder burns at a constant rate of \(20\,\mathrm{g/s}\) such that the exhaust gases always exert a force having a constant magnitude of \(0.3\,\mathrm{N}\), directed tangent to the wheel, determine the angular velocity of the wheel when \(75\%\) of the mass is burned off. Initially, the wheel is at rest and has a mass of \(100\,\mathrm{g}\) and a radius of \(r = 75\,\mathrm{mm}\). For the calculation, consider the wheel to always be a thin disk. figure — add your own
17.4 General Plane Motion 34 problems
Fundamental Problems
  1. F13.The uniform \(60\,\mathrm{kg}\) slender bar is initially at rest on a smooth horizontal plane when the forces are applied. Determine the acceleration of the bar's mass center and the angular acceleration of the bar at this instant. figure — add your own
  2. F14.The 100-kg cylinder rolls without slipping on the horizontal plane. Determine the acceleration of its mass center and its angular acceleration. figure — add your own
  3. F15.The \(20\,\mathrm{kg}\) wheel has a radius of gyration about its center \(O\) of \(k_{O} = 300\,\mathrm{mm}\). When the wheel is subjected to the couple moment, it slips as it rolls. Determine the angular acceleration of the wheel and the acceleration of the wheel's center \(O\). The coefficient of kinetic friction between the wheel and the plane is \(\mu_{k} = 0.5\). figure — add your own
  4. F16.The \(20\,\mathrm{kg}\) sphere rolls down the inclined plane without slipping. Determine the angular acceleration of the sphere and the acceleration of its mass center. figure — add your own
  5. F17.The \(200\,\mathrm{kg}\) spool has a radius of gyration about its mass center of \(k_{G} = 300\,\mathrm{mm}\). If the couple moment is applied to the spool and the coefficient of kinetic friction between the spool and the ground is \(\mu_{k} = 0.2\), determine the angular acceleration of the spool, the acceleration of \(G\), and the tension in the cable. figure — add your own
  6. F18.The \(12\,\mathrm{kg}\) slender rod is pinned to a small roller \(A\) that slides freely along the slot. If the rod is released from rest at \(\theta = 0^{\circ}\), determine the angular acceleration of the rod and the acceleration of the roller immediately after the release. figure — add your own
Skills
  1. 90.If the disk in Fig. 17-19 rolls without slipping, show that when moments are summed about the instantaneous center of zero velocity, \(IC\), it is possible to use the moment equation \(\Sigma M_{IC} = I_{IC}\alpha\), where \(I_{IC}\) represents the moment of inertia of the disk calculated about the instantaneous axis of zero velocity. figure — add your own
  2. 91.The slender \(12\,\mathrm{kg}\) bar has a clockwise angular velocity of \(\omega = 2\,\mathrm{rad/s}\) when it is in the position shown. Determine its angular acceleration and the normal reactions of the smooth surface \(A\) and \(B\) at this instant. figure — add your own
  3. 92.The 2-kg slender bar is supported by cord \(BC\) and then released from rest at \(A\). Determine the initial angular acceleration of the bar and the tension in the cord. figure — add your own
  4. 93.The spool has a mass of \(500\,\mathrm{kg}\) and a radius of gyration \(k_{G} = 1.30\,\mathrm{m}\). It rests on the surface of a conveyor belt for which the coefficient of static friction is \(\mu_{s} = 0.5\) and the coefficient of kinetic friction is \(\mu_{k} = 0.4\). If the conveyor accelerates at \(a_{C} = 1\,\mathrm{m/s^2}\), determine the initial tension in the wire and the angular acceleration of the spool. The spool is originally at rest. figure — add your own
  5. 94.The spool has a mass of \(500\,\mathrm{kg}\) and a radius of gyration \(k_{G} = 1.30\,\mathrm{m}\). It rests on the surface of a conveyor belt for which the coefficient of static friction is \(\mu_{s} = 0.5\). Determine the greatest acceleration \(a_{C}\) of the conveyor so that the spool will not slip. Also, what are the initial tension in the wire and the angular acceleration of the spool? The spool is originally at rest. figure — add your own
  6. 95.The \(20\,\mathrm{kg}\) punching bag has a radius of gyration about its center of mass \(G\) of \(k_{G} = 0.4\,\mathrm{m}\). If it is initially at rest and is subjected to a horizontal force \(F = 30\,\mathrm{N}\), determine the initial angular acceleration of the bag and the tension in the supporting cable \(AB\). figure — add your own
  7. 96.The assembly consists of an 8-kg disk and a 10-kg bar which is pin connected to the disk. If the system is released from rest, determine the angular acceleration of the disk. The coefficients of static and kinetic friction between the disk and the inclined plane are \(\mu_{s} = 0.6\) and \(\mu_{k} = 0.4\) respectively. Neglect friction at \(B\). figure — add your own
  8. 97.Solve Prob. 17-96 if the bar is removed. The coefficients of static and kinetic friction between the disk and inclined plane are \(\mu_{s} = 0.15\) and \(\mu_{k} = 0.1\) respectively. figure — add your own
  9. 98.A force of \(F = 10\,\mathrm{N}\) is applied to the 10-kg ring as shown. If slipping does not occur, determine the ring's initial angular acceleration, and the acceleration of its mass center, \(G\). Neglect the thickness of the ring. figure — add your own
  10. 99.If the coefficient of static friction at \(C\) is \(\mu_{s} = 0.3\), determine the largest force \(\mathbf{F}\) that can be applied to the 5-kg ring, without causing it to slip. Neglect the thickness of the ring. figure — add your own
  11. 100.Wheel \(C\) has a mass of \(60\,\mathrm{kg}\) and a radius of gyration of \(0.4\,\mathrm{m}\), whereas wheel \(D\) has a mass of \(40\,\mathrm{kg}\) and a radius of gyration of \(0.35\,\mathrm{m}\). Determine the angular acceleration of each wheel at the instant shown. Neglect the mass of the link and assume that the assembly does not slip on the plane. figure — add your own
  12. 101.The spool has a mass of \(100\,\mathrm{kg}\) and a radius of gyration of \(k_{G} = 0.3\,\mathrm{m}\). If the coefficients of static and kinetic friction at \(A\) are \(\mu_{s} = 0.2\) and \(\mu_{k} = 0.15\), respectively, determine the angular acceleration of the spool if \(P = 50\,\mathrm{N}\). figure — add your own
  13. 102.Solve Prob. 17-101 if the cord and force \(P = 50\,\mathrm{N}\) are directed vertically upwards. figure — add your own
  14. 103.The spool has a mass of \(100\,\mathrm{kg}\) and a radius of gyration \(k_{G} = 0.3\,\mathrm{m}\). If the coefficients of static and kinetic friction at \(A\) are \(\mu_{s} = 0.2\) and \(\mu_{k} = 0.15\), respectively, determine the angular acceleration of the spool if \(P = 600\,\mathrm{N}\). figure — add your own
  15. 104.The uniform bar of mass \(m\) and length \(L\) is balanced in the vertical position when the horizontal force \(\mathbf{P}\) is applied to the roller at \(A\). Determine the bar's initial angular acceleration and the acceleration of its top point \(B\). figure — add your own
  16. 105.Solve Prob. 17-104 if the roller is removed and the coefficient of kinetic friction at the ground is \(\mu_{k}\). figure — add your own
  17. 106.A "lifted" truck can become a road hazard since the bumper is high enough to ride up a standard car in the event the car is rear-ended. As a model of this case consider the truck to have a mass of \(2.70\,\mathrm{Mg}\), a mass center \(G\), and a radius of gyration about \(\mathbf{G}\) of \(\mathbf{k}_{G} = 1.45\,\mathrm{m}\). Determine the horizontal and vertical components of acceleration of the mass center \(G\) and the angular acceleration of the truck, at the moment its front wheels at \(C\) have just left the ground and its smooth front bumper begins to ride up the back of the stopped car so that point \(B\) has a velocity of \(v_{B} = 8\,\mathrm{m/s}\) at \(20^{\circ}\) from the horizontal. Assume the wheels are free to roll, and neglect the size of the wheels and the deformation of the material. figure — add your own
  18. 107.The \(500\,\mathrm{kg}\) concrete culvert has a mean radius of \(0.5\,\mathrm{m}\). If the truck has an acceleration of \(3\,\mathrm{m/s^2}\), determine the culvert's angular acceleration. Assume that the culvert does not slip on the truck bed, and neglect its thickness. figure — add your own
  19. 108.The \(12\,\mathrm{kg}\) uniform bar is supported by a roller at \(A\). If a horizontal force of \(F = 80\,\mathrm{N}\) is applied to the roller, determine the acceleration of the center of the roller at the instant the force is applied. Neglect the weight and the size of the roller. figure — add your own
  20. 109.The semicircular disk having a mass of \(10\,\mathrm{kg}\) is rotating at \(\omega = 4\,\mathrm{rad/s}\) at the instant \(\theta = 60^{\circ}\). If the coefficient of static friction at \(A\) is \(\mu_{s} = 0.5\), determine if the disk slips at this instant. figure — add your own
  21. 110.The uniform disk of mass \(m\) is rotating with an angular velocity of \(\omega_{0}\) when it is placed on the floor. Determine the initial angular acceleration of the disk and the acceleration of its mass center. The coefficient of kinetic friction between the disk and the floor is \(\mu_{k}\). figure — add your own
  22. 111.The uniform disk of mass \(m\) is rotating with an angular velocity of \(\omega_{0}\) when it is placed on the floor. Determine the time before it starts to roll without slipping. What is the angular velocity of the disk at this instant? The coefficient of kinetic friction between the disk and the floor is \(\mu_{k}\). figure — add your own
  23. 112.The \(20\,\mathrm{kg}\) disk \(A\) is attached to the \(10\,\mathrm{kg}\) block \(B\) using the cable and pulley system shown. If the disk rolls without slipping, determine its angular acceleration and the acceleration of the block when they are released. Also, what is the tension in the cable? Neglect the mass of the pulleys. figure — add your own
  24. 113.The \(30\,\mathrm{kg}\) uniform slender rod \(AB\) rests in the position shown when the couple moment of \(M = 150\,\mathrm{N\cdot m}\) is applied. Determine the initial angular acceleration of the rod. Neglect the mass of the rollers. figure — add your own
  25. 114.The \(30\,\mathrm{kg}\) slender rod \(AB\) rests in the position shown when the horizontal force \(P = 50\,\mathrm{N}\) is applied. Determine the initial angular acceleration of the rod. Neglect the mass of the rollers. figure — add your own
  26. 115.The solid ball of radius \(r\) and mass \(m\) rolls without slipping down the \(60^{\circ}\) trough. Determine its angular acceleration. figure — add your own
  27. 116.A cord is wrapped around each of the two \(10\,\mathrm{kg}\) disks. If they are released from rest, determine the angular acceleration of each disk and the tension in the cord \(C\). Neglect the mass of the cord. figure — add your own
  28. 117.The disk of mass \(m\) and radius \(r\) rolls without slipping on the circular path. Determine the normal force which the path exerts on the disk and the disk's angular acceleration if at the instant shown the disk has an angular velocity of \(\omega\). figure — add your own
  29. 118.A long strip of paper is wrapped into two rolls, each having a mass of \(8\,\mathrm{kg}\). Roll \(A\) is pin supported about its center whereas roll \(B\) is not centrally supported. If \(B\) is brought into contact with \(A\) and released from rest, determine the initial tension in the paper between the rolls and the angular acceleration of each roll. For the calculation, assume the rolls to be approximated by cylinders. figure — add your own
  30. 119.The uniform beam has a weight \(W\). If it is originally at rest while being supported at \(A\) and \(B\) by cables, determine the tension in cable \(A\) if cable \(B\) suddenly fails. Assume the beam is a slender rod. figure — add your own
  31. 120.By pressing down with the finger at \(B\), a thin ring having a mass \(m\) is given an initial velocity \(\mathbf{v}_{0}\) and a backspin \(\omega_{0}\) when the finger is released. If the coefficient of kinetic friction between the table and the ring is \(\mu_{k}\), determine the distance the ring travels forward before backspinning stops. figure — add your own
R Review Problems 8 problems
Review
  1. R1.The handcart has a mass of \(200\,\mathrm{kg}\) and center of mass at \(G\). Determine the normal reactions at each of the wheels at \(A\) and \(B\) if a force \(P = 50\,\mathrm{N}\) is applied to the handle. Neglect the mass and rolling resistance of the wheels. figure — add your own
  2. R2.The two 1.5-kg rods \(EF\) and \(HI\) are fixed (welded) to the link \(AC\) at \(E\). Determine the internal axial force \(E_{x}\), shear force \(E_{y}\), and moment \(M_{E}\), which the bar \(AC\) exerts on \(FE\) at \(E\) if at the instant \(\theta = 30^{\circ}\), link \(AB\) has an angular velocity \(\omega = 5\,\mathrm{rad/s}\) and an angular acceleration \(\alpha = 8\,\mathrm{rad/s^2}\) as shown. figure — add your own
  3. R3.The car has a mass of \(1.50\,\mathrm{Mg}\) and a mass center at \(G\). Determine the maximum acceleration it can have if power is supplied only to the rear wheels. Neglect the mass of the wheels in the calculation, and assume that the wheels that do not receive power are free to roll. Also, assume that slipping of the powered wheels occurs, where the coefficient of kinetic friction is \(\mu_{k} = 0.3\). figure — add your own
  4. R4.A \(20\,\mathrm{kg}\) roll of paper, originally at rest, is pin-supported at its ends to bracket \(AB\). The roll rests against a wall for which the coefficient of kinetic friction at \(C\) is \(\mu_{C} = 0.3\). If a force of \(40\,\mathrm{N}\) is applied uniformly to the end of the sheet, determine the initial angular acceleration of the roll and the tension in the bracket as the paper unwraps. For the calculation, treat the roll as a cylinder. figure — add your own
  5. R5.At the instant shown, two forces act on the 15-kg slender rod which is pinned at \(O\). Determine the magnitude of force \(\mathbf{F}\) and the initial angular acceleration of the rod so that the horizontal reaction which the pin exerts on the rod is \(25\,\mathrm{N}\) directed to the right. figure — add your own
  6. R6.The pendulum consists of a 15-kg sphere and a 5-kg slender rod. Compute the reaction at the pin \(O\) just after the cord \(AB\) is cut. figure — add your own
  7. R7.The spool and wire wrapped around its core have a mass of \(20\,\mathrm{kg}\) and a centroidal radius of gyration \(k_{G} = 250\,\mathrm{mm}\). If the coefficient of kinetic friction at the ground is \(\mu_{B} = 0.1\), determine the angular acceleration of the spool when the \(30\,\mathrm{N\cdot m}\) couple moment is applied. figure — add your own
  8. R8.Determine the backspin \(\omega\) which should be given to the 9-kg ball so that when its center is given an initial horizontal velocity \(v_{G} = 6\,\mathrm{m/s}\) it stops spinning and translating at the same instant. The coefficient of kinetic friction is \(\mu_{A} = 0.3\). figure — add your own