Engineering Mechanics · Dynamics · Open Problem Companion

Chapter Thirteen
Kinetics of a Particle: Force and Acceleration

13.4 Equations of Motion: Rectangular Coordinates 45 problems
Fundamental Problems
  1. F13‑1The motor winds in the cable with a constant acceleration, such that the 20‑kg crate moves a distance \(s = 6\ \mathrm{m}\) in 3 s, starting from rest. Determine the tension developed in the cable. The coefficient of kinetic friction between the crate and the plane is \(\mu_k = 0.3\). figure — add your own
  2. F13‑2If motor \(M\) exerts a force of \(F = (10t^2 + 100)\ \mathrm{N}\) on the cable, where \(t\) is in seconds, determine the velocity of the 25‑kg crate when \(t = 4\ \mathrm{s}\). The coefficients of static and kinetic friction between the crate and the plane are \(\mu_s = 0.3\) and \(\mu_k = 0.25\), respectively. The crate is initially at rest. figure — add your own
  3. F13‑3A spring of stiffness \(k = 500\ \mathrm{N/m}\) is mounted against the 10‑kg block. If the block is subjected to the force of \(F = 500\ \mathrm{N}\), determine its velocity at \(s = 0.5\ \mathrm{m}\). When \(s = 0\), the block is at rest and the spring is uncompressed. The contact surface is smooth. figure — add your own
  4. F13‑4The 2‑Mg car is being towed by a winch. If the winch exerts a force of \(T = 100(s + 1)\ \mathrm{N}\) on the cable, where \(s\) is the displacement of the car in meters, determine the speed of the car when \(s = 10\ \mathrm{m}\), starting from rest. Neglect rolling resistance of the car. figure — add your own
  5. F13‑5The spring has a stiffness \(k = 200\ \mathrm{N/m}\) and is unstretched when the 25‑kg block is at \(A\). Determine the acceleration of the block when \(s = 0.4\ \mathrm{m}\). The contact surface between the block and the plane is smooth. figure — add your own
  6. F13‑6Block \(B\) rests upon a smooth surface. If the coefficients of static and kinetic friction between \(A\) and \(B\) are \(\mu_s = 0.4\) and \(\mu_k = 0.3\), respectively, determine the acceleration of each block if \(P = 30\ \mathrm{N}\). figure — add your own
Problems
  1. 13‑1The crate has a mass of \(80\ \mathrm{kg}\) and is being towed by a chain which is always directed at \(20^\circ\) from the horizontal as shown. If the magnitude of \(\mathbf{P}\) is increased until the crate begins to slide, determine the crate's initial acceleration if the coefficient of static friction is \(\mu_s = 0.5\) and the coefficient of kinetic friction is \(\mu_k = 0.3\). figure — add your own
  2. 13‑2The crate has a mass of \(80\ \mathrm{kg}\) and is being towed by a chain which is always directed at \(20^\circ\) from the horizontal as shown. Determine the crate's acceleration in \(t = 2\ \mathrm{s}\) if the coefficient of static friction is \(\mu_s = 0.4\), the coefficient of kinetic friction is \(\mu_k = 0.3\), and the towing force is \(P = (90t^2)\ \mathrm{N}\), where \(t\) is in seconds. figure — add your own
  3. 13‑3If blocks \(A\) and \(B\) of mass \(10\ \mathrm{kg}\) and \(6\ \mathrm{kg}\) respectively, are placed on the inclined plane and released, determine the force developed in the link. The coefficients of kinetic friction between the blocks and the inclined plane are \(\mu_A = 0.1\) and \(\mu_B = 0.3\). Neglect the mass of the link. figure — add your own
  4. 13‑4If \(P = 400\ \mathrm{N}\) and the coefficient of kinetic friction between the \(50\ \mathrm{kg}\) crate and the inclined plane is \(\mu_k = 0.25\), determine the velocity of the crate after it travels \(6\ \mathrm{m}\) up the plane. The crate starts from rest. figure — add your own
  5. 13‑5If the \(50\ \mathrm{kg}\) crate starts from rest and travels a distance of \(6\ \mathrm{m}\) up the plane in 4 s, determine the magnitude of force \(\mathbf{P}\) acting on the crate. The coefficient of kinetic friction between the crate and the ground is \(\mu_k = 0.25\). figure — add your own
  6. 13‑6If the coefficient of kinetic friction between the \(50\ \mathrm{kg}\) crate and the ground is \(\mu_k = 0.3\), determine the distance the crate travels and its velocity when \(t = 3\ \mathrm{s}\). The crate starts from rest, and \(P = 200\ \mathrm{N}\). figure — add your own
  7. 13‑7If the \(50\ \mathrm{kg}\) crate starts from rest and achieves a velocity of \(v = 4\ \mathrm{m/s}\) when it travels a distance of \(5\ \mathrm{m}\) to the right, determine the magnitude of force \(\mathbf{P}\) acting on the crate. The coefficient of kinetic friction between the crate and the ground is \(\mu_k = 0.3\). figure — add your own
  8. 13‑8The conveyor belt is moving at \(4\ \mathrm{m/s}\). If the coefficient of static friction between the conveyor and the \(10\ \mathrm{kg}\) package \(B\) is \(\mu_s = 0.2\), determine the shortest time the belt can stop so that the package does not slide on the belt. figure — add your own
  9. 13‑9The conveyor belt is designed to transport packages of various weights. Each \(10\ \mathrm{kg}\) package has a coefficient of kinetic friction \(\mu_k = 0.15\). If the speed of the conveyor is \(5\ \mathrm{m/s}\) and then it suddenly stops, determine the distance the package will slide on the belt before coming to rest. figure — add your own
  10. 13‑10The winding drum \(D\) is drawing in the cable at an accelerated rate of \(5\ \mathrm{m/s^2}\). Determine the cable tension if the suspended crate has a mass of \(800\ \mathrm{kg}\). figure — add your own
  11. 13‑11Cylinder \(B\) has a mass \(m\) and is hoisted using the cord and pulley system shown. Determine the magnitude of force \(\mathbf{F}\) as a function of the block's vertical position \(y\) so that when \(\mathbf{F}\) is applied, the block rises with a constant acceleration \(\mathbf{a}_B\). Neglect the mass of the cord and pulleys. figure — add your own
  12. 13‑12The elevator \(E\) has a mass of \(500\ \mathrm{kg}\) and the counterweight at \(A\) has a mass of \(150\ \mathrm{kg}\). If the elevator attains a speed of \(10\ \mathrm{m/s}\) after it rises \(40\ \mathrm{m}\), determine the constant force developed in the cable at \(B\). Neglect the mass of the pulleys and cable. figure — add your own
  13. 13‑13The \(400\ \mathrm{kg}\) mine car is hoisted up the incline using the cable and motor \(M\). For a short time, the force in the cable is \(F = (3200t^2)\ \mathrm{N}\), where \(t\) is in seconds. If the car has an initial velocity \(v_1 = 2\ \mathrm{m/s}\) when \(t = 0\), determine its velocity when \(t = 2\ \mathrm{s}\). figure — add your own
  14. 13‑14The \(400\ \mathrm{kg}\) mine car is hoisted up the incline using the cable and motor \(M\). For a short time, the force in the cable is \(F = (3200t^2)\ \mathrm{N}\) where \(t\) is in seconds. If the car has an initial velocity \(v_1 = 2\ \mathrm{m/s}\) at \(s = 0\) and \(t = 0\), determine the distance it moves up the plane when \(t = 2\ \mathrm{s}\). figure — add your own
  15. 13‑15The \(75\ \mathrm{kg}\) man pushes on the \(150\ \mathrm{kg}\) crate with a horizontal force \(\mathbf{F}\). If the coefficients of static and kinetic friction between the crate and the surface are \(\mu_s = 0.3\) and \(\mu_k = 0.2\), and the coefficient of static friction between the man's shoes and the surface is \(\mu_s = 0.8\), show that the man is able to move the crate. What is the greatest acceleration the man can give the crate? figure — add your own
  16. 13‑16The \(2\ \mathrm{Mg}\) truck is traveling at \(15\ \mathrm{m/s}\) when the brakes on all its wheels are applied, causing it to skid for a distance of \(10\ \mathrm{m}\) before coming to rest. Determine the constant horizontal force developed between the tires of the truck and the road during this time. The total mass of the boat and trailer is \(1\ \mathrm{Mg}\). figure — add your own
  17. 13‑17Determine the acceleration of the blocks when the system is released. The coefficient of kinetic friction is \(\mu_k\) and the mass of each block is \(m\). Neglect the mass of the pulleys and cord. figure — add your own
  18. 13‑18The motor lifts the \(50\ \mathrm{kg}\) crate with an acceleration of \(6\ \mathrm{m/s^2}\). Determine the components of force reaction and the couple moment at the fixed support \(A\). figure — add your own
  19. 13‑19A crate having a mass of \(60\ \mathrm{kg}\) falls horizontally off the back of a truck which is traveling at \(80\ \mathrm{km/h}\). Determine the coefficient of kinetic friction between the road and the crate if the crate slides \(45\ \mathrm{m}\) on the ground with no tumbling along the road before coming to rest. Assume that the initial speed of the crate along the road is \(80\ \mathrm{km/h}\). figure — add your own
  20. 13‑20Determine the required mass of block \(A\) so that when it is released from rest it moves the 5‑kg block \(B\) 0.75 m up along the smooth inclined plane in \(t = 2\ \mathrm{s}\). Neglect the mass of the pulleys and cords. figure — add your own
  21. 13‑21The force of the motor \(M\) on the cable is shown in the graph. Determine the velocity of the 400‑kg crate \(A\) when \(t = 2\ \mathrm{s}\). figure — add your own
  22. 13‑22The bullet of mass \(m\) is given a velocity due to gas pressure caused by the burning of powder within the chamber of the gun. Assuming this pressure creates a force of \(F = F_0 \sin(\pi t / t_0)\) on the bullet, determine the velocity of the bullet at any instant it is in the barrel. What is the bullet's maximum velocity? Also, determine the position of the bullet in the barrel as a function of time. figure — add your own
  23. 13‑23The 50‑kg block \(A\) is released from rest. Determine the velocity of the 15‑kg block \(B\) in 2 s. figure — add your own
  24. 13‑24If the supplied force \(F = 150\ \mathrm{N}\), determine the velocity of the 50‑kg block \(A\) when it has risen 3 m, starting from rest. figure — add your own
  25. 13‑25A 60‑kg suitcase slides from rest \(5\ \mathrm{m}\) down the smooth ramp. Determine the distance \(R\) where it strikes the ground at \(B\). How long does it take to go from \(A\) to \(B\)? figure — add your own
  26. 13‑26Solve Prob. 13‑25 if the suitcase has an initial velocity down the ramp of \(v_A = 2\ \mathrm{m/s}\), and the coefficient of kinetic friction along \(AC\) is \(\mu_k = 0.2\). figure — add your own
  27. 13‑27The conveyor belt delivers each 12‑kg crate to the ramp at \(A\) such that the crate's speed is \(v_A = 2.5\ \mathrm{m/s}\) directed down along the ramp. If the coefficient of kinetic friction between each crate and the ramp is \(\mu_k = 0.3\), determine the speed at which each crate slides off the ramp at \(B\). Assume that no tipping occurs. Take \(\theta = 30^\circ\). figure — add your own
  28. 13‑28The conveyor belt delivers each 12‑kg crate to the ramp at \(A\) such that the crate's speed is \(v_A = 2.5\ \mathrm{m/s}\) directed down along the ramp. If the coefficient of kinetic friction between each crate and the ramp is \(\mu_k = 0.3\), determine the smallest incline \(\theta\) of the ramp so that the crates will slide off and fall into the cart. figure — add your own
  29. 13‑29The conveyor belt is moving downward at \(4\ \mathrm{m/s}\). If the coefficient of static friction between the conveyor and the 15‑kg package \(B\) is \(\mu_s = 0.8\), determine the shortest time the belt can stop so that the package does not slide on the belt. figure — add your own
  30. 13‑30The \(1.5\ \mathrm{Mg}\) sports car has a tractive force of \(F = 4.5\ \mathrm{kN}\). If it produces the velocity described by \(v\)–\(t\) graph shown, plot the air resistance \(R\) versus \(t\) for this time period. figure — add your own
  31. 13‑31Crate \(B\) has a mass \(m\) and is released from rest when it is on top of cart \(A\), which has a mass \(3m\). Determine the tension in cord \(CD\) needed to hold the cart from moving while \(B\) is sliding down \(A\). Neglect friction. figure — add your own
  32. 13‑32The \(4\ \mathrm{kg}\) smooth cylinder is supported by the spring having a stiffness of \(k_{AB} = 120\ \mathrm{N/m}\). Determine the velocity of the cylinder when it moves downward \(s = 0.2\ \mathrm{m}\) from its equilibrium position, which is caused by the application of the force \(F = 60\ \mathrm{N}\). figure — add your own
  33. 13‑33The coefficient of static friction between the \(200\ \mathrm{kg}\) crate and the flat bed of the truck is \(\mu_s = 0.3\). Determine the shortest time for the truck to reach a speed of \(60\ \mathrm{km/h}\) starting from rest with constant acceleration, so that the crate does not slip. figure — add your own
  34. 13‑34The \(300\ \mathrm{kg}\) bar \(B\), originally at rest, is being towed over a series of small rollers. Determine the force in the cable when \(t = 5\ \mathrm{s}\), if the motor \(M\) is drawing in the cable for a short time at a rate of \(v = (0.4t^2)\ \mathrm{m/s}\), where \(t\) is in seconds \((0 \le t \le 6\ \mathrm{s})\). How far does the bar move in \(5\ \mathrm{s}\)? Neglect the mass of the cable, pulley, and the rollers. figure — add your own
  35. 13‑35An electron of mass \(m\) is discharged with an initial horizontal velocity of \(\mathbf{v}_0\). If it is subjected to two fields of force for which \(F_x = F_0\) and \(F_y = 0.3F_0\), where \(F_0\) is constant, determine the equation of the path, and the speed of the electron at any time \(t\). figure — add your own
  36. 13‑36A car of mass \(m\) is traveling at a slow velocity \(v_0\). If it is subjected to the drag resistance of the wind, which is proportional to its velocity, i.e., \(F_D = k v\), determine the distance and the time the car will travel before its velocity becomes \(0.5v_0\). Assume no other frictional forces act on the car. figure — add your own
  37. 13‑37The \(10\ \mathrm{kg}\) block \(A\) rests on the \(50\ \mathrm{kg}\) plate \(B\) in the position shown. Neglecting the mass of the rope and pulley, and using the coefficients of kinetic friction indicated, determine the time needed for block \(A\) to slide \(0.5\ \mathrm{m}\) on the plate when the system is released from rest. figure — add your own
  38. 13‑38Blocks \(A\) and \(B\) each has a mass \(m\). Determine the largest horizontal force \(\mathbf{P}\) which can be applied to \(B\) so that it will not slide on \(A\). Also, what is the corresponding acceleration? The coefficient of static friction between \(A\) and \(B\) is \(\mu_s\). Neglect any friction between \(A\) and the horizontal surface. figure — add your own
  39. 13‑39The tractor is used to lift the \(150\ \mathrm{kg}\) load \(B\) with the \(24\ \mathrm{m}\) long rope, boom, and pulley system. If the tractor travels to the right at a constant speed of \(4\ \mathrm{m/s}\), determine the tension in the rope when \(s_A = 5\ \mathrm{m}\). When \(s_A = 0\), \(s_B = 0\). figure — add your own
  40. 13‑40The tractor is used to lift the \(150\ \mathrm{kg}\) load \(B\) with the \(24\ \mathrm{m}\) long rope, boom, and pulley system. If the tractor travels to the right with an acceleration of \(3\ \mathrm{m/s}^2\) and has a velocity of \(4\ \mathrm{m/s}\) at the instant \(s_A = 5\ \mathrm{m}\), determine the tension in the rope at this instant. When \(s_A = 0\), \(s_B = 0\). figure — add your own
  41. 13‑41A freight elevator, including its load, has a mass of \(1\ \mathrm{Mg}\). It is prevented from rotating due to the track and wheels mounted along its sides. If the motor \(M\) develops a constant tension \(T = 4\ \mathrm{kN}\) in its attached cable, determine the velocity of the elevator when it has moved upward \(6\ \mathrm{m}\) starting from rest. Neglect the mass of the pulleys and cables. figure — add your own
  42. 13‑42The \(10\ \mathrm{kg}\) block is subjected to the forces shown. In each case, determine its velocity when \(t = 2\ \mathrm{s}\) if \(v = 0\) when \(t = 0\). figure — add your own
  43. 13‑43If the force exerted on cable \(AB\) by the motor is \(F = (100t^{3/2})\ \mathrm{N}\), where \(t\) is in seconds, determine the \(50\ \mathrm{kg}\) crate's velocity when \(t = 5\ \mathrm{s}\). The coefficients of static and kinetic friction between the crate and the ground are \(\mu_s = 0.4\) and \(\mu_k = 0.3\), respectively. Initially the crate is at rest. figure — add your own
  44. 13‑44A parachutist having a mass \(m\) opens his parachute from an at‑rest position at a very high altitude. If the atmospheric drag resistance is \(F_D = k v^2\), where \(k\) is a constant, determine his velocity when he has fallen for a time \(t\). What is his velocity when he lands on the ground? This velocity is referred to as the terminal velocity, which is found by letting the time of fall \(t \to \infty\). figure — add your own
  45. 13‑45Each of the three plates has a mass of \(10\ \mathrm{kg}\). If the coefficients of static and kinetic friction at each surface of contact are \(\mu_s = 0.3\) and \(\mu_k = 0.2\), respectively, determine the acceleration of each plate when the three horizontal forces are applied. figure — add your own
Applications (Rectangular)
  1. 13‑46Blocks \(A\) and \(B\) each have a mass \(m\). Determine the largest horizontal force \(\mathbf{P}\) which can be applied to \(B\) so that \(A\) will not move relative to \(B\). All surfaces are smooth. figure — add your own
  2. 13‑47Blocks \(A\) and \(B\) each have a mass \(m\). Determine the largest horizontal force \(\mathbf{P}\) which can be applied to \(B\) so that \(A\) will not slip on \(B\). The coefficient of static friction between \(A\) and \(B\) is \(\mu_s\). Neglect any friction between \(B\) and \(C\). figure — add your own
  3. 13‑48The smooth block \(B\) of negligible size has a mass \(m\) and rests on the horizontal plane. If the board \(AC\) pushes on the block at an angle \(\theta\) with a constant acceleration \(\mathbf{a}_0\), determine the velocity of the block along the board and the distance \(s\) the block moves along the board as a function of time \(t\). The block starts from rest when \(s = 0, t = 0\). figure — add your own
  4. 13‑49Block \(A\) has a mass \(m_A\) and is attached to a spring having a stiffness \(k\) and unstretched length \(l_0\). If another block \(B\), having a mass \(m_B\), is pressed against \(A\) so that the spring deforms a distance \(d\), determine the distance both blocks slide on the smooth surface before they begin to separate. What is their velocity at this instant? figure — add your own
  5. 13‑50Block \(A\) has a mass \(m_A\) and is attached to a spring having a stiffness \(k\) and unstretched length \(l_0\). If another block \(B\), having a mass \(m_B\), is pressed against \(A\) so that the spring deforms a distance \(d\), show that for separation to occur it is necessary that \(d > 2\mu_k g(m_A + m_B) / k\), where \(\mu_k\) is the coefficient of kinetic friction between the blocks and the ground. Also, what is the distance the blocks slide on the surface before they separate? figure — add your own
  6. 13‑51The block \(A\) has a mass \(m_A\) and rests on the pan \(B\) which has a mass \(m_B\). Both are supported by a spring having a stiffness \(k\) that is attached to the bottom of the pan and to the ground. Determine the distance \(d\) the pan should be pushed down from the equilibrium position and then released from rest so that separation of the block will take place from the surface of the pan at the instant the spring becomes unstretched. figure — add your own
13.5 Equations of Motion: Normal and Tangential Coordinates 35 problems
Fundamental Problems
  1. F13‑7The block rests at a distance of \(2\ \mathrm{m}\) from the center of the platform. If the coefficient of static friction between the block and the platform is \(\mu_s = 0.3\), determine the maximum speed which the block can attain before it begins to slip. Assume the angular motion of the disk is slowly increasing. figure — add your own
  2. F13‑8Determine the maximum speed that the jeep can travel over the crest of the hill and not lose contact with the road. figure — add your own
  3. F13‑9A pilot weighs \(70\ \mathrm{kg}\) and is traveling at a constant speed of \(36\ \mathrm{m/s}\). Determine the normal force he exerts on the seat of the plane when he is upside down at \(A\). The loop has a radius of curvature of \(120\ \mathrm{m}\). figure — add your own
  4. F13‑10The sports car is traveling along a \(30^\circ\) banked road having a radius of curvature of \(\rho = 150\ \mathrm{m}\). If the coefficient of static friction between the tires and the road is \(\mu_s = 0.2\), determine the maximum safe speed so no slipping occurs. Neglect the size of the car. figure — add your own
  5. F13‑11If the \(10\ \mathrm{kg}\) ball has a velocity of \(3\ \mathrm{m/s}\) when it is at the position \(A\) along the vertical path, determine the tension in the cord and the increase in the speed of the ball at this position. figure — add your own
  6. F13‑12The motorcycle has a mass of \(0.5\ \mathrm{Mg}\) and a negligible size. It passes point \(A\) traveling with a speed of \(15\ \mathrm{m/s}\), which is increasing at a constant rate of \(1.5\ \mathrm{m/s^2}\). Determine the resultant frictional force exerted by the road on the tires at this instant. figure — add your own
Problems
  1. 13‑52A girl having a mass of \(25\ \mathrm{kg}\) sits at the edge of the merry‑go‑round so her center of mass \(G\) is at a distance of \(1.5\ \mathrm{m}\) from the axis of rotation. If the angular motion of the platform is slowly increased so that the girl's tangential component of acceleration can be neglected, determine the maximum speed which she can have before she begins to slip off the merry‑go‑round. The coefficient of static friction between the girl and the merry‑go‑round is \(\mu_s = 0.3\). figure — add your own
  2. 13‑53A girl, having a mass of \(15\ \mathrm{kg}\) sits motionless relative to the surface of a horizontal platform at a distance of \(r = 5\ \mathrm{m}\) from the platform's center. If the angular motion of the platform is slowly increased so that the girl's tangential component of acceleration can be neglected, determine the maximum speed which the girl will have before she begins to slip off the platform. The coefficient of static friction between the girl and the platform is \(\mu = 0.2\). figure — add your own
  3. 13‑54The \(2\ \mathrm{kg\) block \(B\) and \(15\ \mathrm{kg\) cylinder \(A\) are connected to a light cord that passes through a hole in the center of the smooth table. If the block is given a speed of \(v = 10\ \mathrm{m/s}\), determine the radius \(r\) of the circular path along which it travels. figure — add your own
  4. 13‑55A \(5\ \mathrm{Mg}\) airplane is flying at a constant speed of \(350\ \mathrm{km/h}\) along a horizontal circular path of radius \(r = 3000\ \mathrm{m}\). Determine the uplift force \(L\) acting on the airplane and the banking angle \(\theta\). Neglect the size of the airplane. figure — add your own
  5. 13‑56A \(5\ \mathrm{Mg}\) airplane is flying at a constant speed of \(350\ \mathrm{km/h}\) along a horizontal circular path. If the banking angle \(\theta = 15^\circ\), determine the uplift force \(L\) acting on the airplane and the radius \(r\) of the circular path. Neglect the size of the airplane. figure — add your own
  6. 13‑57The \(2\ \mathrm{kg\) block \(B\) and \(15\ \mathrm{kg\) cylinder \(A\) are connected to a light cord that passes through a hole in the center of the smooth table. If the block is given a speed of \(v = 10\ \mathrm{m/s}\), determine the radius \(r\) of the circular path along which it travels. figure — add your own
  7. 13‑58The \(2\ \mathrm{kg\) block \(B\) and \(15\ \mathrm{kg\) cylinder \(A\) are connected to a light cord that passes through a hole in the center of the smooth table. If the block travels along a circular path of radius \(r = 1.5\ \mathrm{m}\), determine the speed of the block. figure — add your own
  8. 13‑59Cartons having a mass of \(5\ \mathrm{kg}\) are required to move along the assembly line at a constant speed of \(8\ \mathrm{m/s}\). Determine the smallest radius of curvature, \(\rho\), for the conveyor so the cartons do not slip. The coefficients of static and kinetic friction between a carton and the conveyor are \(\mu_s = 0.7\) and \(\mu_k = 0.5\), respectively. figure — add your own
  9. 13‑60Determine the maximum constant speed at which the pilot can travel around the vertical curve having a radius of curvature \(\rho = 800\ \mathrm{m}\) so that he experiences a maximum acceleration \(a_n = 8g = 78.5\ \mathrm{m/s^2}\). If he has a mass of \(70\ \mathrm{kg}\) determine the normal force he exerts on the seat of the airplane when the plane is traveling at this speed and is at its lowest point. figure — add your own
  10. 13‑61The \(2\ \mathrm{kg}\) spool \(S\) fits loosely on the inclined rod for which the coefficient of static friction is \(\mu_s = 0.2\). If the spool is located \(0.25\ \mathrm{m}\) from \(A\), determine the minimum constant speed the spool can have so that it does not slip down the rod. figure — add your own
  11. 13‑62The \(2\ \mathrm{kg}\) spool \(S\) fits loosely on the inclined rod for which the coefficient of static friction is \(\mu_s = 0.2\). If the spool is located \(0.25\ \mathrm{m}\) from \(A\), determine the maximum constant speed the spool can have so that it does not slip up the rod. figure — add your own
  12. 13‑63The \(1.40\ \mathrm{Mg}\) helicopter is traveling at a constant speed of \(40\ \mathrm{m/s}\) along the horizontal curved path while banking at \(\theta = 30^\circ\). Determine the force acting normal to the blade, i.e., in the \(y'\) direction, and the radius of curvature of the path. figure — add your own
  13. 13‑64The \(1.40\ \mathrm{Mg}\) helicopter is traveling at a constant speed of \(33\ \mathrm{m/s}\) along the horizontal curved path having a radius of curvature of \(\rho = 300\ \mathrm{m}\). Determine the force the blade exerts on the frame and the bank angle \(\theta\). figure — add your own
  14. 13‑65The vehicle is designed to combine the feel of a motorcycle with the comfort and safety of an automobile. If the vehicle is traveling at a constant speed of \(80\ \mathrm{km/h}\) along a circular curved road of radius \(100\ \mathrm{m}\), determine the tilt angle \(\theta\) of the vehicle so that only a normal force from the seat acts on the driver. Neglect the size of the driver. figure — add your own
  15. 13‑66Determine the constant speed of the passengers on the amusement‑park ride if it is observed that the supporting cables are directed at \(\theta = 30^\circ\) from the vertical. Each chair including its passenger has a mass of \(80\ \mathrm{kg}\). Also, what are the components of force in the \(n, t\), and \(b\) directions which the chair exerts on a \(50\ \mathrm{kg}\) passenger during the motion? figure — add your own
  16. 13‑67Bobs \(A\) and \(B\) of mass \(m_A\) and \(m_B\) (\(m_A > m_B\)) are connected to an inextensible light string of length \(l\) that passes through the smooth ring at \(C\). If bob \(B\) moves as a conical pendulum such that \(A\) is suspended a distance of \(h\) from \(C\), determine the angle \(\theta\) and the speed of bob \(B\). Neglect the size of both bobs. figure — add your own
  17. 13‑68Prove that if the block is released from rest at point \(B\) of a smooth path of arbitrary shape, the speed it attains when it reaches point \(A\) is equal to the speed it attains when it falls freely through a distance \(h\); i.e., \(v = \sqrt{2gh}\). figure — add your own
  18. 13‑69The skier starts from rest at \(A(10\ \mathrm{m},0)\) and descends the smooth slope, which may be approximated by a parabola. If she has a mass of \(52\ \mathrm{kg}\), determine the normal force the ground exerts on the skier at the instant she arrives at point \(B\). Neglect the size of the skier. figure — add your own
  19. 13‑70The \(800\ \mathrm{kg}\) motorbike travels with a constant speed of \(80\ \mathrm{km/h}\) up the hill. Determine the normal force the surface exerts on its wheels when it reaches point \(A\). Neglect its size. figure — add your own
  20. 13‑71A ball having a mass \(2\ \mathrm{kg}\) and negligible size moves within a smooth vertical circular slot. If it is released from rest when \(\theta = 10^\circ\), determine the force of the slot on the ball when the ball arrives at points \(A\) and \(B\). figure — add your own
  21. 13‑72The \(0.8\ \mathrm{Mg}\) car travels over the hill having the shape of a parabola. If the driver maintains a constant speed of \(9\ \mathrm{m/s}\), determine both the resultant normal force and the resultant frictional force that all the wheels of the car exert on the road at the instant it reaches point \(A\). Neglect the size of the car. figure — add your own
  22. 13‑73The \(0.8\ \mathrm{Mg}\) car travels over the hill having the shape of a parabola. When the car is at point \(A\), it is traveling at \(9\ \mathrm{m/s}\) and increasing its speed at \(3\ \mathrm{m/s^2}\). Determine both the resultant normal force and the resultant frictional force that all the wheels of the car exert on the road at this instant. Neglect the size of the car. figure — add your own
  23. 13‑74The block \(B\) having a mass of \(0.2\ \mathrm{kg}\) is attached to the vertex \(A\) of the right circular cone using a light cord. The cone is rotating at a constant angular rate about the \(z\) axis such that the block attains a speed of \(0.5\ \mathrm{m/s}\). At this speed, determine the tension in the cord and the reaction which the cone exerts on the block. Neglect the size of the block and the effect of friction. figure — add your own
  24. 13‑75Determine the maximum speed at which the car with mass \(m\) can pass over the top point \(A\) of the vertical curved road and still maintain contact with the road. If the car maintains this speed, what is the normal reaction the road exerts on the car when it passes the lowest point \(B\) on the road? figure — add your own
  25. 13‑76The \(35\ \mathrm{kg}\) box has a speed \(2\ \mathrm{m/s}\) when it is at \(A\) on the smooth ramp. If the surface is in the shape of a parabola, determine the normal force on the box at the instant \(x = 3\ \mathrm{m}\). Also, what is the rate of increase in its speed at this instant? figure — add your own
  26. 13‑77The box has a mass \(m\) and slides down the smooth chute having the shape of a parabola. If it has an initial velocity of \(v_0\) at the origin, determine its velocity as a function of \(x\). Also, what is the normal force on the box, and the tangential acceleration as a function of \(x\)? figure — add your own
  27. 13‑78Determine the maximum constant speed at which the \(2\ \mathrm{Mg}\) car can travel over the crest of the hill at \(A\) without leaving the surface of the road. Neglect the size of the car in the calculation. figure — add your own
  28. 13‑79A collar having a mass \(0.75\ \mathrm{kg}\) and negligible size slides over the surface of a horizontal circular rod for which the coefficient of kinetic friction is \(\mu_k = 0.3\). If the collar is given a speed of \(4\ \mathrm{m/s}\) and then released at \(\theta = 0^\circ\), determine how far, \(s\), it slides on the rod before coming to rest. figure — add your own
  29. 13‑80The \(8\ \mathrm{kg}\) sack slides down the smooth ramp. If it has a speed of \(1.5\ \mathrm{m/s}\) when \(y = 0.2\ \mathrm{m}\), determine the normal reaction the ramp exerts on the sack and the rate of increase in the speed of sack at this instant. figure — add your own
  30. 13‑81The \(2\ \mathrm{kg}\) pendulum bob moves in the vertical plane with a velocity of \(8\ \mathrm{m/s}\) when \(\theta = 0^\circ\). Determine the initial tension in the cord and also at the instant the bob reaches \(\theta = 30^\circ\). Neglect the size of the bob. figure — add your own
  31. 13‑82The \(2\ \mathrm{kg}\) pendulum bob moves in the vertical plane with a velocity of \(6\ \mathrm{m/s}\) when \(\theta = 0^\circ\). Determine the angle \(\theta\) where the tension in the cord becomes zero. figure — add your own
  32. 13‑83The airplane, traveling at a constant speed of \(50\ \mathrm{m/s}\) is executing a horizontal turn. If the plane is banked at \(\theta = 15^\circ\), when the pilot experiences only a normal force on the seat of the plane, determine the radius of curvature \(\rho\) of the turn. Also, what is the normal force of the seat on the pilot if he has a mass of \(70\ \mathrm{kg}\). figure — add your own
  33. 13‑84The ball has a mass \(m\) and is attached to the cord of length \(l\). The cord is tied at the top to a swivel and the ball is given a velocity \(\mathbf{v}_0\). Show that the angle \(\theta\) which the cord makes with the vertical as the ball travels around the circular path must satisfy the equation \(\tan \theta \sin \theta = v_0^2 / g l\). Neglect air resistance and the size of the ball. figure — add your own
13.6 Equations of Motion: Cylindrical Coordinates 31 problems
Fundamental Problems
  1. F13‑13Determine the constant angular velocity \(\dot{\theta}\) of the vertical shaft of the amusement ride if \(\phi = 45^\circ\). Neglect the mass of the cables and the size of the passengers. figure — add your own
  2. F13‑14The \(0.2\ \mathrm{kg}\) ball is blown through the smooth vertical circular tube whose shape is defined by \(r = (0.6\sin \theta)\ \mathrm{m}\) where \(\theta\) is in radians. If \(\theta = (\pi t^2)\) rad, where \(t\) is in seconds, determine the magnitude of force \(\mathbf{F}\) exerted by the blower on the ball when \(t = 0.5\ \mathrm{s}\). figure — add your own
  3. F13‑15The \(2\ \mathrm{Mg}\) car is traveling along the curved road described by \(r = (50e^{2\theta})\ \mathrm{m}\) where \(\theta\) is in radians. If a camera is located at \(A\) and it rotates with an angular velocity of \(\dot{\theta} = 0.05\ \mathrm{rad/s}\) and an angular acceleration of \(\ddot{\theta} = 0.01\ \mathrm{rad/s}^2\) at the instant \(\theta = \pi/6\) rad, determine the resultant friction force developed between the tires and the road at this instant. figure — add your own
  4. F13‑16The \(0.2\ \mathrm{kg\) ball is guided along the smooth vertical circular tube using the arm \(OA\). If the arm has an angular velocity \(\dot{\theta} = 0.4\ \mathrm{rad/s}\) and an angular acceleration \(\ddot{\theta} = 0.8\ \mathrm{rad/s}^2\) at the instant \(\theta = 30^\circ\), determine the force of the arm on the ball. Neglect friction and the size of the ball. figure — add your own
Problems
  1. 13‑85Using air pressure, the \(0.5\ \mathrm{kg}\) ball is forced to move through the tube lying in the horizontal plane and having the shape of a logarithmic spiral. If the tangential force exerted on the ball due to the air is \(6\ \mathrm{N}\), determine the rate of increase in the ball's speed at the instant \(\theta = \pi/2\). What direction does it act in? figure — add your own
  2. 13‑86Solve Prob. 13‑85 if the tube lies in a vertical plane. figure — add your own
  3. 13‑87The \(0.75\ \mathrm{kg}\) smooth can is guided along the circular path using the arm guide. If the arm has an angular velocity \(\dot{\theta} = 2\ \mathrm{rad/s}\) and an angular acceleration \(\ddot{\theta} = 0.4\ \mathrm{rad/s}^2\) at the instant \(\theta = 30^\circ\), determine the force of the guide on the can. Motion occurs in the horizontal plane. figure — add your own
  4. 13‑88Using a forked rod, a \(0.5\ \mathrm{kg}\) smooth peg \(P\) is forced to move along the vertical slotted path \(r = (0.5\theta)\ \mathrm{m}\), where \(\theta\) is in radians. If the angular position of the arm is \(\theta = (\pi/8)t^2\) rad, where \(t\) is in seconds, determine the force of the rod on the peg and the normal force of the slot on the peg at the instant \(t = 2\ \mathrm{s}\). The peg is in contact with only one edge of the rod and slot at any instant. figure — add your own
  5. 13‑89The arm is rotating at a rate of \(\dot{\theta} = 4\ \mathrm{rad/s}\) when \(\ddot{\theta} = 3\ \mathrm{rad/s}^2\) and \(\theta = 180^\circ\). Determine the force it must exert on the \(0.5\ \mathrm{kg}\) smooth cylinder if it is confined to move along the slotted path. Motion occurs in the horizontal plane. figure — add your own
  6. 13‑90The boy of mass \(40\ \mathrm{kg}\) is sliding down the spiral slide at a constant speed such that his position, measured from the top of the chute, has components \(r = 1.5\ \mathrm{m}\), \(\theta = (0.7t)\) rad, and \(z = (-0.5t)\) m, where \(t\) is in seconds. Determine the components of force \(\mathbf{F}_r, \mathbf{F}_\theta\), and \(\mathbf{F}_z\) which the slide exerts on him at the instant \(t = 2\ \mathrm{s}\). Neglect the size of the boy. figure — add your own
  7. 13‑91The \(40\ \mathrm{kg}\) boy is sliding down the smooth spiral slide such that \(z = -2\ \mathrm{m/s}\) and his speed is \(2\ \mathrm{m/s}\). Determine the \(r, \theta, z\) components of force the slide exerts on him at this instant. Neglect the size of the boy. figure — add your own
  8. 13‑92The tube rotates in the horizontal plane at a constant rate of \(\dot{\theta} = 4\ \mathrm{rad/s}\). If a \(0.2\ \mathrm{kg}\) ball \(B\) starts at the origin \(o\) with an initial radial velocity \(\dot{r} = 1.5\ \mathrm{m/s}\) and moves outward through the tube, determine the radial and transverse components of the ball's velocity at the instant it leaves the outer end at \(C\), \(r = 0.5\ \mathrm{m}\). Hint: Show that the equation of motion in the \(r\) direction is \(\ddot{r} - 16r = 0\). The solution is of the form \(r = A e^{-4t} + B e^{4t}\). Evaluate the integration constants \(A\) and \(B\) and determine the time \(t\) when \(r = 0.5\ \mathrm{m}\). Proceed to obtain \(v_r\) and \(v_\theta\). figure — add your own
  9. 13‑93The girl has a mass of \(50\ \mathrm{kg}\). She is seated on the horse of the merry‑go‑round which undergoes constant rotational motion \(\dot{\theta} = 1.5\ \mathrm{rad/s}\). If the path of the horse is defined by \(r = 4\ \mathrm{m}\), \(z = (0.5\sin \theta)\ \mathrm{m}\), determine the maximum and minimum force \(F_z\) the horse exerts on her during the motion. figure — add your own
  10. 13‑94Using a forked rod, a smooth cylinder \(P\) having a mass of \(0.4\ \mathrm{kg}\) is forced to move along the vertical slotted path \(r = (0.6\theta)\ \mathrm{m}\) where \(\theta\) is in radians. If the cylinder has a constant speed of \(v_C = 2\ \mathrm{m/s}\), determine the force of the rod and the normal force of the slot on the cylinder at the instant \(\theta = \pi\) rad. Assume the cylinder is in contact with only one edge of the rod and slot at any instant. Hint: To obtain the time derivatives necessary to compute the cylinder's acceleration components \(a_r\) and \(a_\theta\), take the first and second time derivatives of \(r = 0.6\theta\). Then, for further information, use Eq. 12‑26 to determine \(\dot{\theta}\). Also, take the time derivative of Eq. 12‑26, noting that \(\dot{v} = 0\) to determine \(\ddot{\theta}\). figure — add your own
  11. 13‑95A car of a roller coaster travels along a track which for a short distance is defined by a conical spiral, \(r = \frac{3}{4}z\), \(\theta = -1.5z\), where \(r\) and \(z\) are in meters and \(\theta\) in radians. If the angular motion \(\dot{\theta} = 1\ \mathrm{rad/s}\) is always maintained, determine the \(r, \theta, z\) components of reaction exerted on the car by the track at the instant \(z = 6\ \mathrm{m}\). The car and passengers have a total mass of \(200\ \mathrm{kg}\). figure — add your own
  12. 13‑96The particle has a mass of \(0.5\ \mathrm{kg}\) and is confined to move along the smooth vertical slot due to the rotation of the arm \(OA\). Determine the force of the rod on the particle and the normal force of the slot on the particle when \(\theta = 30^\circ\). The rod is rotating with a constant angular velocity \(\dot{\theta} = 2\ \mathrm{rad/s}\). Assume the particle contacts only one side of the slot at any instant. figure — add your own
  13. 13‑97A smooth can \(C\) having a mass of \(3\ \mathrm{kg}\) is lifted from a feed at \(A\) to a ramp at \(B\) by a rotating rod. If the rod maintains a constant angular velocity of \(\dot{\theta} = 0.5\ \mathrm{rad/s}\), determine the force which the rod exerts on the can at the instant \(\theta = 30^\circ\). Neglect the effects of friction in the calculation and the size of the can so that \(r = (1.2\cos \theta)\ \mathrm{m}\). The ramp from \(A\) to \(B\) is circular, having a radius of \(600\ \mathrm{mm}\). figure — add your own
  14. 13‑98The spring‑held follower \(AB\) has a mass of \(0.5\ \mathrm{kg}\) and moves back and forth as its end rolls on the contoured surface of the cam, where \(r = 0.15\ \mathrm{m}\) and \(z = (0.02\cos 2\theta)\ \mathrm{m}\). If the cam is rotating at a constant rate of \(30\ \mathrm{rad/s}\), determine the force component \(F_z\) at the end \(A\) of the follower when \(\theta = 30^\circ\). The spring is uncompressed when \(\theta = 90^\circ\). Neglect friction at the bearing \(C\). figure — add your own
  15. 13‑99The spring‑held follower \(AB\) has a mass of \(0.5\ \mathrm{kg}\) and moves back and forth as its end rolls on the contoured surface of the cam, where \(r = 0.15\ \mathrm{m}\) and \(z = (0.02\cos 2\theta)\ \mathrm{m}\). If the cam is rotating at a constant rate of \(30\ \mathrm{rad/s}\), determine the maximum and minimum force components \(F_z\) the follower exerts on the cam if the spring is uncompressed when \(\theta = 90^\circ\). figure — add your own
  16. 13‑100Determine the normal and frictional driving forces that the partial spiral track exerts on the \(200\ \mathrm{kg}\) motorcycle at the instant \(\theta = \frac{5}{3}\pi\) rad, \(\dot{\theta} = 0.4\ \mathrm{rad/s}\), \(\ddot{\theta} = 0.8\ \mathrm{rad/s}^2\). Neglect the size of the motorcycle. figure — add your own
  17. 13‑101The \(0.5\ \mathrm{kg}\) ball is guided along the vertical circular path \(r = 2r_c\cos \theta\) using the arm \(OA\). If the arm has an angular velocity \(\dot{\theta} = 0.4\ \mathrm{rad/s}\) and an angular acceleration \(\ddot{\theta} = 0.8\ \mathrm{rad/s}^2\) at the instant \(\theta = 30^\circ\), determine the force of the arm on the ball. Neglect friction and the size of the ball. Set \(r_c = 0.4\ \mathrm{m}\). figure — add your own
  18. 13‑102The ball of mass \(m\) is guided along the vertical circular path \(r = 2r_c\cos \theta\) using the arm \(OA\). If the arm has a constant angular velocity \(\dot{\theta}_0\), determine the angle \(\theta \le 45^\circ\) at which the ball starts to leave the surface of the semicylinder. Neglect friction and the size of the ball. figure — add your own
  19. 13‑103Rod \(OA\) rotates counterclockwise at a constant angular rate \(\dot{\theta} = 4\ \mathrm{rad/s}\). The double collar \(B\) is pin‑connected together such that one collar slides over the rotating rod and the other collar slides over the circular rod described by the equation \(r = (1.6\cos \theta)\ \mathrm{m}\). If both collars have a mass of \(0.5\ \mathrm{kg}\), determine the force which the circular rod exerts on one of the collars and the force that \(OA\) exerts on the other collar at the instant \(\theta = 45^\circ\). Motion is in the horizontal plane. figure — add your own
  20. 13‑104Solve Prob. 13‑103 if motion is in the vertical plane. figure — add your own
  21. 13‑105The smooth surface of the vertical cam is defined in part by the curve \(r = (0.2\cos \theta + 0.3)\ \mathrm{m}\). The forked rod is rotating with an angular acceleration of \(\ddot{\theta} = 2\ \mathrm{rad/s}^2\), and when \(\theta = 45^\circ\) the angular velocity is \(\dot{\theta} = 6\ \mathrm{rad/s}\). Determine the force the cam and the rod exert on the 2‑kg roller at this instant. The attached spring has a stiffness \(k = 100\ \mathrm{N/m}\) and an unstretched length of \(0.1\ \mathrm{m}\). figure — add your own
  22. 13‑106The collar has a mass of \(2\ \mathrm{kg}\) and travels along the smooth horizontal rod defined by the equiangular spiral \(r = (e^\theta)\ \mathrm{m}\), where \(\theta\) is in radians. Determine the tangential force \(F\) and the normal force \(N\) acting on the collar when \(\theta = 45^\circ\), if the force \(F\) maintains a constant angular motion \(\dot{\theta} = 2\ \mathrm{rad/s}\). figure — add your own
  23. 13‑107The pilot of the airplane executes a vertical loop which in part follows the path of a cardioid, \(r = 200(1 + \cos \theta)\ \mathrm{m}\), where \(\theta\) is in radians. If his speed at \(A\) is a constant \(v_p = 85\ \mathrm{m/s}\), determine the vertical reaction the seat of the plane exerts on the pilot when the plane is at \(A\). He has a mass of \(80\ \mathrm{kg}\). Hint: To determine the time derivatives necessary to calculate the acceleration components \(a_r\) and \(a_\theta\), take the first and second time derivatives of \(r = 200(1 + \cos \theta)\). Then, for further information, use Eq. 12‑26 to determine \(\dot{\theta}\). figure — add your own
  24. 13‑108The pilot of an airplane executes a vertical loop which in part follows the path of a “four‑leaved rose,” \(r = (-180 \cos 2\theta)\ \mathrm{m}\), where \(\theta\) is in radians. If his speed at \(A\) is a constant \(v_p = 24\ \mathrm{m/s}\), determine the vertical reaction the seat of the plane exerts on the pilot when the plane is at \(A\). He weighs \(650\ \mathrm{N}\). Hint: To determine the time derivatives necessary to compute the acceleration components \(a_r\) and \(a_\theta\), take the first and second time derivatives of \(r = -180(1 + \cos \theta)\). Then, for further information, use Eq. 12‑26 to determine \(\dot{\theta}\). Also, take the time derivative of Eq. 12‑26, noting that \(\dot{v}_C = 0\) to determine \(\ddot{\theta}\). figure — add your own
  25. 13‑109The particle has a mass of \(0.5\ \mathrm{kg}\) and is confined to move along the smooth horizontal slot due to the rotation of the arm \(OA\). Determine the force of the rod on the particle and the normal force of the slot on the particle when \(\theta = 30^\circ\). The rod is rotating with a constant angular velocity \(\dot{\theta} = 2\ \mathrm{rad/s}\). Assume the particle contacts only one side of the slot at any instant. figure — add your own
  26. 13‑110Solve Prob. 13‑109 if the arm has an angular acceleration of \(\ddot{\theta} = 3\ \mathrm{rad/s}^2\) when \(\dot{\theta} = 2\ \mathrm{rad/s}\) at \(\theta = 30^\circ\). figure — add your own
  27. 13‑111A \(0.2\ \mathrm{kg}\) spool slides down along a smooth rod. If the rod has a constant angular rate of rotation \(\dot{\theta} = 2\ \mathrm{rad/s}\) in the vertical plane, show that the equations of motion for the spool are \(\ddot{r} - 4r - 9.81 \sin \theta = 0\) and \(0.8 \dot{r} + N_s - 1.962 \cos \theta = 0\), where \(N_s\) is the magnitude of the normal force of the rod on the spool. Using the methods of differential equations, it can be shown that the solution of the first of these equations is \(r = C_1 e^{-2t} + C_2 e^{2t} - (9.81/8) \sin 2t\). If \(r, \dot{r}\), and \(\theta\) are zero when \(t = 0\), evaluate the constants \(C_1\) and \(C_2\) to determine \(r\) at the instant \(\theta = \pi/4\) rad. figure — add your own
  28. 13‑112The ball has a mass of \(2\ \mathrm{kg}\) and a negligible size. It is originally traveling around the horizontal circular path of radius \(r_0 = 0.5\ \mathrm{m}\) such that the angular rate of rotation is \(\dot{\theta}_0 = 1\ \mathrm{rad/s}\). If the attached cord \(ABC\) is drawn down through the hole at a constant speed of \(0.2\ \mathrm{m/s}\), determine the tension the cord exerts on the ball at the instant \(r = 0.25\ \mathrm{m}\). Also, compute the angular velocity of the ball at this instant. Neglect the effects of friction between the ball and horizontal plane. Hint: First show that the equation of motion in the \(\theta\) direction yields \(a_\theta = r\ddot{\theta} + 2\dot{r}\dot{\theta} = (1/r)(d(r^2\dot{\theta})/dt) = 0\). When integrated, \(r^2\dot{\theta} = c\), where the constant \(c\) is determined from the problem data. figure — add your own
13.7 Central‑Force Motion and Space Mechanics 18 problems
Problems
  1. 13‑113The earth has an orbit with eccentricity 0.0167 around the sun. Knowing that the earth's minimum distance from the sun is \(146(10^6)\) km, find the speed at which the earth travels when it is at this distance. Determine the equation in polar coordinates which describes the earth's orbit about the sun. figure — add your own
  2. 13‑114A communications satellite is in a circular orbit above the earth such that it always remains directly over a point on the earth's surface. As a result, the period of the satellite must equal the rotation of the earth, which is approximately 24 hours. Determine the satellite's altitude \(h\) above the earth's surface and its orbital speed. figure — add your own
  3. 13‑115The speed of a satellite launched into a circular orbit about the earth is given by Eq. 13‑24. Determine the speed of a satellite launched parallel to the surface of the earth so that it travels in a circular orbit \(800\ \mathrm{km}\) from the earth's surface. figure — add your own
  4. 13‑116The rocket is in circular orbit about the earth at an altitude of \(20\ \mathrm{Mm}\). Determine the minimum increment in speed it must have in order to escape the earth's gravitational field. figure — add your own
  5. 13‑117Prove Kepler's third law of motion. Hint: Use Eqs. 13‑18, 13‑27, 13‑28, and 13‑30. figure — add your own
  6. 13‑118The satellite is moving in an elliptical orbit with an eccentricity \(e = 0.25\). Determine its speed when it is at its maximum distance \(A\) and minimum distance \(B\) from the earth. figure — add your own
  7. 13‑119The rocket is traveling in free flight along the elliptical orbit. The planet has no atmosphere, and its mass is 0.60 times that of the earth. If the rocket has the orbit shown, determine the rocket's speed when it is at \(A\) and at \(B\). figure — add your own
  8. 13‑120Determine the constant speed of satellite \(S\) so that it circles the earth with an orbit of radius \(r = 15\ \mathrm{Mm}\). Hint: Use Eq. 13‑1. figure — add your own
  9. 13‑121The rocket is in free flight along an elliptical trajectory \(A'A\). The planet has no atmosphere, and its mass is 0.70 times that of the earth. If the rocket has an apoapsis and periapsis as shown in the figure, determine the speed of the rocket when it is at point \(A\). figure — add your own
  10. 13‑122The Viking Explorer approaches the planet Mars on a parabolic trajectory as shown. When it reaches point \(A\) its velocity is \(10\ \mathrm{Mm/h}\). Determine \(r_0\) and the required change in velocity at \(A\) so that it can then maintain a circular orbit as shown. The mass of Mars is 0.1074 times the mass of the earth. figure — add your own
  11. 13‑123The rocket is initially in free‑flight circular orbit around the earth. Determine the speed of the rocket at \(A\). What change in the speed at \(A\) is required so that it can move in an elliptical orbit to reach point \(A'\)? figure — add your own
  12. 13‑124The rocket is in free‑flight circular orbit around the earth. Determine the time needed for the rocket to travel from the inner orbit at \(A\) to the outer orbit at \(A'\). figure — add your own
  13. 13‑125A rocket is in a free‑flight elliptical orbit about the earth such that the eccentricity of its orbit is \(e\) and its perigee is \(r_0\). Determine the minimum increment of speed it should have in order to escape the earth's gravitational field when it is at this point along its orbit. figure — add your own
  14. 13‑126The rocket is traveling around the earth in free flight along the elliptical orbit. If the rocket has the orbit shown, determine the speed of the rocket when it is at \(A\) and at \(B\). figure — add your own
  15. 13‑127An elliptical path of a satellite has an eccentricity \(e = 0.130\). If it has a speed of \(15\ \mathrm{Mm/h}\) when it is at perigee, \(P\), determine its speed when it arrives at apogee, \(A\). Also, how far is it from the earth's surface when it is at \(A\)? figure — add your own
  16. 13‑128A rocket is in free‑flight elliptical orbit around the planet Venus. Knowing that the periapsis and apoapsis of the orbit are \(8\ \mathrm{Mm}\) and \(26\ \mathrm{Mm}\) respectively, determine (a) the speed of the rocket at point \(A'\), (b) the required speed it must attain at \(A\) just after braking so that it undergoes an 8‑Mm free‑flight circular orbit around Venus, and (c) the periods of both the circular and elliptical orbits. The mass of Venus is 0.816 times the mass of the earth. figure — add your own
  17. 13‑129The rocket is traveling in a free flight along an elliptical trajectory \(A'A\). The planet has no atmosphere, and its mass is 0.60 times that of the earth. If the rocket has the orbit shown, determine the rocket's velocity when it is at point \(A\). figure — add your own
  18. 13‑130If the rocket is to land on the surface of the planet, determine the required free‑flight speed it must have at \(A'\) so that the landing occurs at \(B\). How long does it take for the rocket to land, going from \(A'\) to \(B\)? The planet has no atmosphere, and its mass is 0.6 times that of the earth. figure — add your own
  19. 13‑131The rocket is traveling around the earth in free flight along an elliptical orbit \(AC\). If the rocket has the orbit shown, determine the rocket's velocity when it is at point \(A\). figure — add your own
  20. 13‑132The rocket is traveling around the earth in free flight along the elliptical orbit \(AC\). Determine its change in speed when it reaches \(A\) so that it travels along the elliptical orbit \(AB\). figure — add your own
Review Review Problems 8 problems
Review
  1. R13‑1The van is traveling at \(20\ \mathrm{km/h}\) when the coupling of the trailer at \(A\) fails. If the trailer has a mass of \(250\ \mathrm{kg}\) and coasts \(45\ \mathrm{m}\) before coming to rest, determine the constant horizontal force \(F\) created by rolling friction which causes the trailer to stop. figure — add your own
  2. R13‑2The motor \(M\) pulls in its attached rope with an acceleration \(a_p = 6\ \mathrm{m/s}^2\). Determine the towing force exerted by \(M\) on the rope in order to move the \(50\ \mathrm{kg}\) crate up the inclined plane. The coefficient of kinetic friction between the crate and the plane is \(\mu_k = 0.3\). Neglect the mass of the pulleys and rope. figure — add your own
  3. R13‑3Block \(B\) rests on a smooth surface. If the coefficients of friction between \(A\) and \(B\) are \(\mu_s = 0.4\) and \(\mu_k = 0.3\), determine the acceleration of each block if \(F = 250\ \mathrm{N}\). figure — add your own
  4. R13‑4If the motor draws in the cable at a rate of \(v = (0.05s^{3/2})\ \mathrm{m/s}\), where \(s\) is in meters, determine the tension developed in the cable when \(s = 10\ \mathrm{m}\). The crate has a mass of \(20\ \mathrm{kg}\), and the coefficient of kinetic friction between the crate and the ground is \(\mu_k = 0.2\). figure — add your own
  5. R13‑5The \(10\ \mathrm{kg}\) block is subjected to the forces shown. In each case, determine its velocity at \(s = 8\ \mathrm{m}\) if \(v = 3\ \mathrm{m/s}\) at \(s = 0\). Motion occurs to the right. figure — add your own
  6. R13‑6The bottle rests at a distance of \(1.5\ \mathrm{m}\) from the center of the horizontal platform. If the coefficient of static friction between the bottle and the platform is \(\mu_s = 0.3\), determine the maximum speed that the bottle can attain before slipping. Assume the angular motion of the platform is slowly increasing. figure — add your own
  7. R13‑7The 5‑kg suitcase slides down the curved ramp for which the coefficient of kinetic friction is \(\mu_k = 0.2\). If at the instant it reaches point \(A\) it has a speed of \(2\ \mathrm{m/s}\), determine the normal force on the suitcase and the rate of increase of its speed. figure — add your own
  8. R13‑8The spool, which has a mass of \(4\ \mathrm{kg}\), slides along the rotating rod. At the instant shown, the angular rate of rotation of the rod is \(\dot{\theta} = 6\ \mathrm{rad/s}\) and this rotation is increasing at \(\ddot{\theta} = 2\ \mathrm{rad/s}^2\). At this same instant, the spool has a velocity of \(3\ \mathrm{m/s}\) and an acceleration of \(1\ \mathrm{m/s}^2\), both measured relative to the rod and directed away from the center \(O\) when \(r = 0.5\ \mathrm{m}\). Determine the radial frictional force and the normal force, both exerted by the rod on the spool at this instant. figure — add your own