13-51. The block A has a mass mA and rests on the pan B.

The motor winds in the cable with a constant acceleration, such that the 20-kg crate moves a distance s = 6 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.

If motor M exerts a force of \(F=\left(10 t^{2}+100\right) \mathrm{N}\) on the cable, where t is in seconds, determine the velocity of the 25-kg crate when t = 4 s. The coefficients of static and kinetic friction between the crate and the plane are \(\mu_{s}=0.3 \text { and } \mu_{k}=0.25\), respectively.The crate is initially at rest.

F13-3. A spring of stiffness * = 500N/m is mounted against the 10-kg block. If the block is subjected to the force of F = 500 N, determine its velocity at s = 0.5 m. When .s' = 0, the block is at rest and the spring is uncompressed. The contact surface is smooth.

F13-4. The 2-Mg car is being towed by a winch. If the winch exerts a force of T = 100(j? + 1)N on the cable, where 5 is the displacement of the car in meters, determine the speed of the car when s = 10 m, starting from rest. Neglect rolling resistance of the car.

F13-5. The spring has a stiffness k = 200 N/m and is unstretched when the 25-kg block is at A. Determine the acceleration of the block when s = 0.4 m. The contact surface between the block and the plane is smooth.

Block 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 \(P=6 \mathrm{lb}\).

FI3-7. The block rests at a distance of 2 m from the center of the platform. If the coefficient of static friction between the block and the platform is = 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.

FI 3-8. Determine the maximum speed that the jeep can travel over the crest of the hill and not lose contact with the road.

A pilot weighs \(150 \mathrm{lb}\) and is traveling at a constant speed of \(120 \mathrm{ft} / \mathrm{s}\). 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 \(400 \mathrm{ft}\).

F13-10. The sports car is traveling along a 30 banked road having a radius of curvature of p = 500 ft. If the coefficient of static friction between the tires and the road is /x, = 0.2. determine the maximum safe speed so no slipping occurs. Neglect the size of the car.

FI3-11. If the 10-kg ball has a velocity of 3 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.

The motorcycle has a mass of \(0.5 \mathrm{Mg}\) and a negligible size. It passes point A traveling with a speed of \(15 \mathrm{~m} / \mathrm{s}\), which is increasing at a constant rate of \(1.5 \mathrm{~m} / \mathrm{s}^{2}\).Determine the resultant frictional force exerted by the road on the tires at this instant.

F13-13. Determine the constant angular velocity 0 of the vertical shaft of the amusement ride if <f> = 45. Neglect the mass of the cables and the size of the passengers.

F13-14. The 0.2-kg ball is blown through the smooth vertical circular tube whose shape is defined by r = (O.6sin0)m, where 0 is in radians. If 0 = (tt t2) rad, where / is in seconds, determine the magnitude of force F exerted by the blower on the ball when t = 0.5 s.

F13-15. The 2-Mg car is traveling along the curved road described by r = (50e2fl)m, where 0 is in radians. If a camera is located at A and it rotates with an angular velocity of () = 0.05 rad/s and an angular acceleration of 0 = 0.01 rad/s2 at the instant 0 = f rad, determine the resultant friction force developed between the tires and the road at this instant.

The \(0.2-\mathrm{kg}\) pin P is constrained to move in the smooth curved slot, which is defined by the lemniscate \(r=(0.6 \cos 2 \theta) \mathrm{m}\). Its motion is controlled by the rotation of the slotted arm OA, which has a constant clockwise angular velocity of \(\dot{\theta}=-3 \mathrm{rad} / \mathrm{s}\). Determine the force arm OA exerts on the pin P when \(\theta=0^{\circ}\). Motion is in the vertical plane.

The 6-lb particle is subjected to the action of its weight and forces \(\mathbf{F}_{1}=\{2 \mathbf{i}+6 \mathbf{j}-2 t \mathbf{k}\} \mathrm{lb}, \quad \mathbf{F}_{2}=\left\{t^{2} \mathbf{i}-4 t \mathbf{j}-1 \mathbf{k}\right\} \mathrm{lb} \text {, and } \mathbf{F}_{3}=\{-2 t \mathbf{i}\}\ \mathrm{lb}\), where t is in seconds. Determine the distance the ball is from the origin 2 s after being released from rest.

13-2. The 10-lb block has an initial velocity of 10 ft/s on the smooth plane. If a force F = (2.5f) lb, where t is in seconds, acts on the block for 3 s, determine the final velocity of the block and the distance the block travels during this time.

If the coefficient of kinetic friction between the 50-kg crate and the ground is \(\mu_{k}=0.3\), determine the distance the crate travels and its velocity when t = 3 s. The crate starts from rest, and P = 200 N.

*13-4. If the 50-kg crate starts from rest and achieves a velocity of v = 4 m/s when it travels a distance of 5 m to the right, determine the magnitude of force P acting on the crate. The coefficient of kinetic friction between the crate and the ground is fik = 0.3.

13-5. The water-park ride consists of an 800-lb sled which slides from rest down the incline and then into the pool. If the frictional resistance on the incline is Fr = 30 lb, and in the pool for a short distance F, = 80 lb, determine how fast the sled is traveling when s = 5 ft.

13-6. If P = 400 N and the coefficient of kinetic friction between the 50-kg crate and the inclined plane is fik = 0.25. determine the velocity of the crate after it travels 6 m up the plane.The crate starts from rest.

13-7. If the 50-kg crate starts from rest and travels a distance of 6 m up the plane in 4 s,determine the magnitude of force P acting on the crate. The coefficient of kinetic friction between the crate and the ground is fxk = 0.25.

*13-8. The speed of the 3500-lb sports car is plotted over the 30-s time period. Plot the variation of the traction force F needed to cause the motion.

13-9. The crate has a mass of 80 kg and is being towed by a chain which is always directed at 20 from the horizontal as shown. If the magnitude of P is increased until the crate begins to slide, determine the crates initial acceleration if the coefficient of static friction is /xv = 0.5 and the coefficient of kinetic friction is fik = 0.3.

13-10. The crate has a mass of 80 kg and is being towed by a chain which is always directed at 20 from the horizontal as shown. Determine the crate's acceleration in / = 2 s if the coefficient of static friction is fx, = 0.4, the coefficient of kinetic friction is fik = 0.3. and the towing force is P = (90r) N. where t is in seconds.

13-11. The safe S has a weight of 200 lb and is supported by the rope and pulley arrangement shown. If the end of the rope is given to a boy B of weight 00 lb, determine his acceleration if in the confusion he doesnt let go of the rope. Neglect the mass of the pulleys and rope.

*13-12. The boy having a weight of 80 lb hangs uniformly from the bar. Determine the force in each of his arms in / = 2 s if the bar is moving upward with (a) a constant velocity of 3 ft/s, and (b) a speed of v = (4r) ft/s, where t is in seconds.

The bullet mass m is given a velocity dues to gas pressure caused by burning of powder within the chamber of the gun. Assuming this pressure creates a force of \( F=F_{0} \sin \left(\pi l / t_{0}\right)\) on the bullet, what is the bullet's maximum velocity? also, determine the position of the bullet in the barrel as a function of time.

13-14. The 2-Mg truck is traveling at 15 m/s when the brakes on all its wheels are applied, causing it to skid for a distance of 10 m before coming to rest. Determine the constant horizontal force developed in the coupling C, and the frictional force developed between the tires of the truck and the road during this time.The total mass of the boat and trailer is 1 Mg.

13-15. A freight elevator, including its load, has a mass of 500 kg. It is prevented from rotating by the track and wheels mounted along its sides. When / = 2 s. the motor M draws in the cable with a speed of 6 m/s, measured relative to the elevator. If it starts from rest, determine the constant acceleration of the elevator and the tension in the cable. Neglect the mass of the pulleys, motor, and cables.

*13-16. The man pushes on the 60-lb crate with a force F. The force is always directed down at 30 from the horizontal as shown, and its magnitude is increased until the crate begins to slide. Determine the crates initial acceleration if the coefficient of static friction is fis = 0.6 and the coefficient of kinetic friction is /x* = 0.3.

13-17. The double inclined plane supports two blocks A and B. each having a weight of 10 lb. If the coefficient of kinetic friction between the blocks and the plane is txk = 0.1, determine the acceleration of each block.

A 40 -lb suitcase slides from rest \(20 \mathrm{ft}\) down the smooth ramp. Determine the point where it strikes the ground at C. How long does it take to go from A to C?

13-19. Solve Prob. 13-18 if the suitcase has an initial velocity down the ramp of vA = 10 ft/s and the coefficient of kinetic friction along AB is /x* = 0.2.

*13-20. The 400-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 = (3200r) N, where / is in seconds. If the car has an initial velocity V\ = 2 m/s when i = 0. determine its velocity when / = 2 s.

13-21. The 400-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 = (3200/2) N, where i is in seconds. If the car has an initial velocity V\ = 2 m/s at 5 = 0 and t = 0, determine the distance it moves up the plane when / = 2 s.

13-22. Determine 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 / = 2 s. Neglect the mass of the pulleys and cords.

13-23. The winding drum D is drawing in the cable at an accelerated rate of 5 m/s2. Determine the cable tension if the suspended crate has a mass of 800 kg.

*13-24. If the motor draws in the cable at a rate of v = (0.05s*12) m/s. where s is in meters, determine the tension developed in the cable when .s' = 10 m.Thc crate has a mass of 20 kg, and the coefficient of kinetic friction between the crate and the ground is ixk = 0.2.

If the motor draws in the cable at a rate of \(v=\left(0.05 t^2\right) \mathrm{m} / \mathrm{s}\), where t is in seconds, determine the tension developed in the cable when \(t=5 \mathrm{~s}\). The crate has a mass of \(20 \mathrm{~kg}\) and the coefficient of kinetic friction between the crate and the ground is \(\mu_k=0.2\).

13-26. The 2-kg shaft CA passes through a smooth journal bearing at B. Initially, the springs, which are coiled loosely around the shaft, are unstretched when no force is applied to the shaft. In this position s = s = 250 mm and the shaft is at rest. If a horizontal force of F = 5 kN is applied, determine the speed of the shaft at the instant s 50 mm. s' = 450 mm. The ends of the springs are attached to the bearing at B and the caps at Cand A.

13-27. The 30-lb crate is being hoisted upward with a constant acceleration of 6 ft/s2. If the uniform beam AB has a weight of 200 lb. determine the components of reaction at A. Neglect the size and mass of the pulley at B. Hint: First find the tension in the cable, then analyze the forces in the beam using statics.

*13-28. The driver attempts to tow the crate using a rope that has a tensile strength of 200 lb. If the crate is originally at rest and has a weight of 500 lb, determine the greatest acceleration it can have if the coefficient of static friction between the crate and the road is jx, = 0.4, and the coefficient of kinetic friction is /a* = 0.3.

The force exerted by the motor on the cable is shown in the graph. Determine the velocity of the 200-lb crate when t = 2.5 s.

13-30. The 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 s. F( N)

13-31. The tractor is used to lift the 150-kg load B with the 24-m-long rope, boom, and pulley system. If the tractor travels to the right at a constant speed of 4 m/s, determine the tension in the rope when sA = 5 m. When Sa = 0, sB = 0.

*13-32. The tractor is used to lift the 150-kg load B with the 24-m-long rope, boom, and pulley system. If the tractor travels to the right with an acceleration of 3 m/s2 and has a velocity of 4 m/s at the instant sA = 5 in. determine the tension in the rope at this instant. When sA = 0, sB = 0.

Each of the three plates has a mass of 10 kg. If the coefficients of static and kinetic friction at each surface of contact are \(\mu_{s}=0.3 \text { and } \mu_{k}=0.2\). respectively, determine the acceleration of each plate when the three horizontal forces are applied.

13-34. Each of the two blocks has a mass m.The coefficient of kinetic friction at all surfaces of contact is /x. If a horizontal force P moves the bottom block, determine the acceleration of the bottom block in each case.

13-35. The conveyor belt is moving at 4 m/s. If the coefficient of static friction between the conveyor and the 10-kg package B is /x = 0.2, determine the shortest time the belt can stop so that the package does not slide on the belt.

*13-36. The 2-lb collar C fits loosely on the smooth shaft. If the spring is unstretched when 5 = 0 and the collar is given a velocity of 15 ft/s, determine the velocity of the collar when a- = 1 ft.

13-37. Cylinder B has a mass m and is hoisted using the cord and pulley system shown. Determine the magnitude of force F as a function of the cylinders vertical position y so that when F is applied the cylinder rises with a constant acceleration aa. Neglect the mass of the cord, pulleys, hook and chain.

The conveyor belt delivers each 12-kg crate to the ramp at A such that the crates speed is \(v_{A}=2.5 \mathrm{\ m} / \mathrm{s}\), 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 \(\theta=30^{\circ}\).

13-39. An electron of mass m is discharged with an initial horizontal velocity of v(l. If it is subjected to two fields of force for which Fx = F() and Fy = 0.3Fo. where Fu is constant, determine the equation of the path, and the speed of the electron at any time t.

The engine of the van produces a constant driving traction force F at the wheels as it ascends the slope at a constant velocity v. Determine the acceleration of the van when it passes point A and begins to travel on a level road, provided that it maintains the same traction force.

13-41. The 2-kg collar C is free to slide along the smooth shaft AB. Determine the acceleration of collar C if (a) the shaft is fixed from moving, (b) collar A. which is fixed to shaft AB,moves downward at constant velocity along the vertical rod. and (c) collar A is subjected to a downward acceleration of 2m/s2. In all cases, the collar moves in the plane.

13-42. The 2-kg collar C is free to slide along the smooth shaft AB. Determine the acceleration of collar C if collar A

13-43. The coefficient of static friction between the 200-kg crate and the flat bed of the truck is jjlx = 0.3. Determine the shortest time for the truck to reach a speed of 60 km/h. starting from rest with constant acceleration.so that the crate does not slip.

When the blocks are released, determine their acceleration and the tension of the cable. Neglect the mass of the pulley.

13-45. If the force exerted on cable AB by the motor is F (100/3/2) N. where t is in Seconds, determine the 50-kg crates velocity when t = 5 s.The coefficients of static and kinetic friction between the crate and the ground are = 0.4 and /xk = 0.3, respectively. Initially the crate is at rest.

13-46. Blocks A and B each have a mass m. Determine the largest horizontal force P which can be applied to R so that A will not move relative to B. All surfaces are smooth.

13-47. Blocks A and B each have a mass m. Determine the largest horizontal force 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 /is. Neglect any friction between B and C

*13-48. A parachutist having a mass m opens his parachute from an at-rest position at a very high altitude. If the atmospheric drag resistance is Ff) = kv2. 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

13-49. The 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 0 with a constant acceleration a0, determine the velocity of the block along the board and the distance 5 the block moves along the board as a function of time /.The block starts from rest when s = 0. / = 0.

13-50. A projectile of mass m is fired into a liquid at an angle 00 with an initial velocity v0 as shown. If the liquid develops a frictional or drag resistance on the projectile which is proportional to its velocity, i.e.. F = kv. where k is a constant, determine the x and y components of its position at any instant. Also, what is the maximum distance that it travels?

13-51. The block A has a mass mA and rests on the pan B. which has a mass mH. 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.

A girl, having a mass of 15 kg, sits motionless relative to the surface of a horizontal platform at a distance of r = 5 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 \(\mu=0.2\).

The 2-kg block B and 15-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 \(v=10 \mathrm{\ m} / \mathrm{s} \text {, }\) determine the radius r of the circular path along which it travels.

The 2-kg block B and 15-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.

13-55. The 5-kg collar A is sliding around a smooth vertical guide rod. At the instant shown, the speed of the collar is v=4 m/s. which is increasing at 3 m/s2. Determine the normal reaction of the guide rod on the collar, and force P at this instant.

*13-56. Cartons having a mass of 5 kg arc required to move along the assembly line at a constant speed of 8 m/s. Determine the smallest radius of curvature, p, for the conveyor so the cartons do not slip.The coefficients of static and kinetic friction between a carton and the conveyor are = 0.7 and p* = 0.5, respectively.

13-57. The block 13. having a mass of 0.2 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 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.

The 2-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 m from A. determine the minimum constant speed the spool can have so that it does not slip down the rod.

13-59. The 2-kg spool S fits loosely on the inclined rod for which the coefficient of static friction is juv = 0.2. If the spool is located 0.25 m from A, determine the maximum constant speed the spool can have so that it does not slip up the rod.

At the instant \(\theta=60^{\circ}\), the boy's center of mass G has a downward speed \(v_{G}=15 \mathrm{\ ft} / \mathrm{s}\). Determine the rate of increase in his speed and the tension in each of the two supporting cords of the swing at this instant. The boy has a weight of 60 lb. Neglect his size and the mass of the seat and cords.

At the instant \(\theta=60^{\circ}\), the boy's center of mass G is momentarily at rest. Determine his speed and the tension in each of the two supporting cords of the swing when \(\theta=90^{\circ}\). The boy has a weight of 60 lb. Neglect his size and the mass of the seat and cords.

The 10-lb 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 5 ft/s, determine the normal force on the suitcase and the rate of increase of its speed.

13-63. The 150-lb man lies against the cushion for which the coefficient of static friction is /xA = 0.5. Determine the resultant normal and frictional forces the cushion exerts on him if, due to rotation about the z axis, he has a constant speed v = 20 ft/s. Neglect the size of the man. Take 0 = 60.

The 150-lb man lies against the cushion for which the coefficient of static friction is \(\mu_{s}=0.5\). If he rotates about the z axis with a constant speed v = 30 ft/s, determine the smallest angle \(\theta\) of the cushion at which he will begin to slip off.

13-65. Determine the constant speed of the passengers on the amusement-park ride if it is observed that the supporting cables are directed at 0 = 30 from the vertical. Each chair including its passenger has a mass of 80 kg. Also, what are the components of force in the n, /, and b directions which the chair exerts on a 50-kg passenger during the motion?

13-66. The man has a mass of 80 kg and sits 3 m from the center of the rotating platform. Due to the rotation his speed is increased from rest by v =0.4m/s2. If the coefficient of static friction between his clothes and the platform is fxv = 0.3, determine the time required to cause him to slip.

13-67. The 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 km/h along a circular curved road of radius 100 m. determine the tilt angle 0 of the vehicle so that only a normal force from the seat acts on the driver. Neglect the size of the driver.

The 0.8-Mg car travels over the hill having the shape of a parabola. If the driver maintains a constant speed of 9 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.

13-69. The 0.8-Mg car travels over the hill having the shape of a parabola. When the car is at point A, it is traveling at 9 m/s and increasing its speed at 3 m/s2. 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.

13-70. The package has a weight of 5 lb and slides down the chute. When it reaches the curved portion AB, it is traveling at 8 ft/s (0 = 0). If the chute is smooth, determine the speed of the package when it reaches the intermediate point C (0 = 30) and when it reaches the horizontal plane (0 = 45). Also, find the normal force on the package at C.

13-71. If the ball has a mass of 30 kg and a speed t) = 4m/s at the instant it is at its lowest point. 0 = 0, determine the tension in the cord at this instant. Also, determine the angle 0 to which the ball swings at the instant it momentarily stops. Neglect the size of the ball.

*13-72. The ball has a mass of 30 kg and a speed v = 4 m/s at the instant it is at its lowest point. 0 = 0. Determine the tension in the cord and the rate at which the balls speed is decreasing at the instant 0 = 20. Neglect the size of the ball.

Determine 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?

13-74. If the crest of the hill has a radius of curvature p = 200 ft, determine the maximum constant speed at which the car can travel over it without leaving the surface of the road. Neglect the size of the car in the calculation. The car has a weight of 3500 lb.

13-75. Bobs A and B of mass mA and mB (mA > mB) are connected to an inextensible light string of length / that passes through the smooth ring at C. If bob B moves as a conical pendulum such that A is suspended a distance of U from C. determine the angle 0 and the speed of bob B. Neglect the size of both bobs.

Prove 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., \(v=\sqrt{2 g h}\).

The 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. Hint: Use the result of Prob. 13-76.

13-78. A spring, having an unstretched length of 2 ft. has one end attached to the 10-lb ball. Determine the angle 0 of the spring if the ball has a speed of 6 fl/s tangent to the horizontal circular path.

13-79. The airplane, traveling at a constant speed of 50 m/s, is executing a horizontal turn. If the plane is banked at 0 = 15, when the pilot experiences only a normal force on the seat of the plane, determine the radius of curvature p of the turn. Also, what is the normal force of the seat on the pilot if he has a mass of 70 kg.

*13-80. A 5-Mg airplane is flying at a constant speed of 350 km/h along a horizontal circular path of radius r = 3000m. Determine the uplift force L acting on the airplane and the banking angle 6. Neglect the size of the airplane.

13-81. A 5-Mg airplane is flying at a constant speed of 350 km/h along a horizontal circular path. If the banking angle 6 = 15. determine the uplift force L acting on the airplane and the radius r of the circular path. Neglect the size of the airplane.

13-82. The 800-kg motorbike travels with a constant speed of 80 km/h up the hill. Determine the normal force the surface exerts on its wheels when it reaches point A. Neglect its size.

13-83. The ball has a mass m and is attached to the cord of length /. The cord is tied at the top to a swivel and the ball is given a velocity v. Show that the angle 0 which the cord makes with the vertical as the ball travels around the circular path must satisfy the equation tan 0 sin 0 = vl/gl. Neglect air resistance and the size of the ball.

The 5-lb collar slides on the smooth rod, so that when it is at A it has a speed of \(10 \mathrm{ft} / \mathrm{s}\). If the spring to which it is attached has an unstretched length of \(3 \mathrm{ft}\) and a stiffness of \(k=10 \mathrm{lb} / \mathrm{ft}\),determine the normal force on the collar and the magnitude of the acceleration of the collar at this instant.

The spring-held follower AB has a weight of 0.75 lb and moves back and forth as its end rolls on the contoured surface of the cam, where r = 0.2 ft and \(z=(0.1\sin2\theta)\mathrm{\ ft}\). The cam is rotating at a constant rate of 6 rad/s. determine the force at the end A of the follower when \(\theta=45^{\circ}\). In this position the spring is compressed 0.4 ft. Neglect friction at the bearing C.

Determine the magnitude of the resultant force acting on a 5-kg particle at the instant \(t=2 \mathrm{~s}\), if the particle is moving along a horizontal path defined by the equations \(r=(2 t+10) \mathrm{m}\) and \(\theta=\left(1.5 t^2-6 t\right)\) rad, where t is in seconds.

13-87. The path of motion of a 5-lb particle in the horizontal plane is described in terms of polar coordinates as r = (2/ + 1) ft and 0 = (0.5/2 - /) rad, where / is in seconds. Determine the magnitude of the unbalanced force acting on the particle when / = 2 s.

A particle, having a mass of 1.5 kg. moves along a path defined by the equations r = (4 + 3t) m, \(\theta=\left(t^{2}+2\right) \mathrm{rad}, \text { and } z=\left(6-t^{3}\right) \mathrm{m}\), where t is in seconds. Determine the \(r, \theta, \text { and } z\) components of force which the path exerts on the particle when t = 2 s.

13-89. Rod OA rotates counterclockwise with a constant angular velocity of 0 =5 rad/s. The double collar B is pin-connected together such that one collar slides over the rotating rod and the other slides over the horizontal curved rod. of which the shape is described by the equation r = 1.5(2 - cos 0) ft. If both collars weigh 0.75 lb. determine the normal force which the curved rod exerts on one collar at the instant 0 = 120. Neglect friction.

The 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 \(r=1.5 \mathrm{~m}\), \(\theta=(0.7 t) \mathrm{rad}\), and \(z=(-0.5 t) \mathrm{m}\), 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.

13-91. The 0.5-lb particle is guided along the circular path using the slotted arm guide. If the arm has an angular velocity 0=4 rad/s. and an angular acceleration 0=8 rad/s. at the instant 0 = 30. determine the force of the guide on the particle. Motion occurs in the horizontal plane.

*13-92. Using a forked rod. a smooth cylinder C having a mass of 0.5 kg is forced to move along the vertical slotted path r = (0.5 0) m, where 0 is in radians. If the angular position of the arm is 0 = (0.5/2) rad, where / is in seconds, determine the force of the rod on the cylinder and the normal force of the slot on the cylinder at the instant / = 2 s. The cylinder is in contact with only one edge of the rod and slot at any instant.

13-93. If arm OA rotates with a constant clockwise angular velocity of 0 = 1.5 rad/s. determine the force arm OA exerts on the smooth 4-lb cylinder B when 0 = 45.

The collar has a mass of \(2 \mathrm{~kg}\) and travels along the smooth horizontal rod defined by the equiangular spiral \(r=\left(e^\theta\right) \mathrm{m}\), where \(\theta\) is in radians. Determine the tangential force F and the normal force N acting on the collar when \(\theta=90^{\circ}\), if the force F maintains a constant angular motion \(\dot{\theta}=2 \mathrm{rad} / \mathrm{s}\).

13-95. The ball has a mass of 2 kg and a negligible size. It is originally traveling around the horizontal circular path of radius r0 = 0.5 m such that the angular rate of rotation is tf0 = I rad/s. If the attached cord ABC is drawn down through the hole at a constant speed of 0.2 m/s, determine the tension the cord exerts on the ball at the instant r = 0.25 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 0 direction yields aft = rO 4- 2'rd = (1/r) (d(rf)/dt) = 0. When integrated. rd = Cs where the constant c is determined from the problem data.

*13-96. The particle has a mass of 0.5 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 0 = 30. The rod is rotating with a constant angular velocity 0 = 2 rad/s. Assume the particle contacts only one side of the slot at any instant.

13-97. Solve Problem 13-96 if the arm has an angular acceleration of 0 = 3 rad/s2 and 0=2 rad/s at this instant. Assume the particle contacts only one side of the slot at any instant.

13-98. The collar has a mass of 2 kg and travels along the smooth horizontal rod defined by the equiangular spiral r = (e6) m, where 0 is in radians. Determine the tangential force F and the normal force N acting on the collar when 0 = 45, if the force F maintains a constant angular motion 0 = 2 rad/s.

13-99. For a short time, the 250-kg roller coaster car is traveling along the spiral track such that its position measured from the top of the track has components r = 8 m. 0 = (0.1/ + 0.5) rad, and z (0.20 m. where / is in seconds. Determine the magnitudes of the components of force which the track exerts on the car in the r. 0, and z directions at the instant / = 2 s. Neglect the size of the car.

*13-100. The 0.5-lb ball is guided along the vertical circular path r = 2rc cos 0 using the arm OA. If the arm has an angular velocity 0 = 0.4 rad/s and an angular acceleration 6 = 0.8 rad/s2 at the instant 0 = 30. determine the force of the arm on the ball. Neglect friction and the size of the ball. Set r(. = 0.4ft.

The ball of mass m is guided along the vertical circular path \(r=2 r_{c} \cos \theta\) using the arm OA. If the arm has a constant angular velocity \(\dot{\theta}_{0}\), determine the angle \(\theta \leq 45^{\circ}\) at which the ball starts to leave the surface of the semicylinder. Neglect friction and the size of the ball.

Using 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 \(v_C=2 \mathrm{~m} / \mathrm{s}\), 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 \(\vec{v}=0\) to determine \(\ddot{\theta}\).

13-103. A ride in an amusement park consists of a cart which is supported by small wheels. Initially the cart is traveling in a circular path of radius rQ = 16 ft such that the angular rate of rotation is 0O = 0.2 rad/s. If the attached cable OC is drawn inward at a constant speed of r = -0.5 ft/s, determine the tension it exerts on the cart at the instant r = 4 ft. The cart and its passengers have a total weight of 400 lb. Neglect the effects of friction. Hint: First show that the equation of motion in the 0 direction yields a6 = r0 + 2/0 = (\/r)d(r 0 )/dt = 0. When integrated, r 0 = c, where the constant c is determined from the problem data.

*13-104. The arm is rotating at a rate of 0 = 5 rad/s when 0 = 2 rad/s2 and 0 = 90. Determine the normal force it must exert on the 0.5-kg particle if the particle is confined to move along the slotted path defined by the horizontal hyperbolic spiral rO = 0.2 in.

The forked rod is used to move the smooth 2-lb particle around the horizontal path in the shape of a limacon, \(r=(2+\cos \theta)\) ft. If at all times \(\dot{\theta}=0.5 \mathrm{\ rad} / \mathrm{s}\), determine the force which the rod exerts on the particle at the instant \(\theta=90^{\circ}\). The fork and path contact the particle on only one side.

13-106. Solve Prob. 13-105 at the instant 0 = 60.

13-107. The forked rod is used to move the smooth 2-lb particle around the horizontal path in the shape of a lima^on, r = (2 + cos 0) ft. If 0 = (0.5r) rad. where t is in seconds, determine the force which the rod exerts on the particle at the instant / = I s.Thc fork and path contact the particle on only one side.

The collar, which has a weight of 3 lb. slides along the smooth rod lying in the horizontal plane and having the shape of a parabola \(r=4 /(1-\cos \theta)\), where \(\theta\) is in radians and r is in feet. If the collar's angular rale is constant and equals \(\dot{\theta}=4 \mathrm{rad} / \mathrm{s} \text {, }\) determine the tangential retarding force P needed to cause the motion and the normal force that the collar exerts on the rod at the instant \(\theta=90^{\circ}\).

13-109. The smooth particle has a mass of 80 g. It is attached to an elastic cord extending from O to P and due to the slotted arm guide moves along the horizontal circular path r = (0.8 sin 0) m. If the cord has a stiffness k = 30 N/m and an unstretched length of 0.25 m. determine the force of the guide on the particle when 0 = 60. The guide has a constant angular velocity 0 = 5 rad/s.

13-110. Solve Prob. 13-109 if 0 = 2 rad/s2 when () = 5 rad/s and 0 = 60.

A 0.2-kg spool slides down along a smooth rod. If the rod has a constant angular rate of rotation \(\dot{\theta}=2 \mathrm{\ rad} / \mathrm{s}\) in the vertical plane, show that the equations of motion for the spool arc \(\ddot{r}-4 r-9.81 \sin \theta=0 \text { and } 0.8 \dot{r}+N_{s}-1.962 \cos \theta=0 \text {, where } N_{s}\) 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 \(r=C_1e^{-2t}+C_2e^{2t}-(9.81/8)\sin2t\text{. If }r,\ r\text{, and }\theta\) are zero when t = 0, evaluate the constants \(C_{1} \text { and } C_{2}\) determine r at the instant \(\theta=\pi / 4 \mathrm{\ rad}\).

*13-112. The pilot of an airplane executes a vertical loop which in part follows the path of a cardioid. r = 600(1 + cos 0) ft. If his speed at A (0 = 0) is a constant vP = 80 ft/s, determine the vertical force the seat belt must exert on him to hold him to his seat when the plane is upside down at A. He weighs 150 lb.

The earth has an orbit with eccentricity \(e=0.0821\) around the sun. Knowing that the earth's minimum distance from the sun is \(151.3\left(10^{6}\right)\) 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.

A 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.

13-115. The speed of a satellite launched into a circular orbit about the earth is given by Eq. 13-25. Determine the speed of a satellite launched parallel to the surface of the earth so that it travels in a circular orbit 800 km from the earth's surface.

*13-116. The rocket is in circular orbit about the earth at an altitude of h = 4 Mm. Determine the minimum increment in speed it must have in order to escape the earth's gravitational field.

Prove Keplers third law of motion. Hint: Use Eqs. 13-19.13-28,13-29. and 13-31.

13-118. The 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.

13-119. The elliptical orbit of a satellite orbiting the earth has an eccentricity of e = 0.45. If the satellite has an altitude of 6 Mm at perigee, determine the velocity of the satellite at apogee and the period. that it circles the earth with an orbit of radius / = 15 Mm. Hint: Use Eq. 13-1.

13-120 Determine the constant speed of Satellite S that it circles the earth with an orbit of radius / = 15 Mm. Hint: Use Eq. 13-1.

13-121. The rocket is in free flight along an elliptical trajectory /tM.The planet has no atmosphere, and its mass is 0.70 times that of the earth. If the rocket has an apoapsis and pcriapsis as shown in the figure, determine the speed of the rocket when it is at point A.

13-122. A satellite S travels in a circular orbit around the earth. A rocket is located at the apogee of its elliptical orbit for which e = 0.58. Determine the sudden change in speed that must occur at A so that the rocket can enter the satellites orbit while in free flight along the blue elliptical trajectory. When it arrives at B. determine the sudden adjustment in speed that must be given to the rocket in order to maintain the circular orbit.

13-123. An asteroid is in an elliptical orbit about the sun such that its perihelion distance is 9.30(109)km. If the eccentricity of the orbit is e = 0.073. determine the aphelion distance of the orbit.

*13-124. An elliptical path of a satellite has an eccentricity e = 0.130. If it has a speed of 15 Mm/h when it is at perigee, P. determine its speed when it arrives at apogee, A. Also, how far is it from the earths surface when it is at A?

13-125. A satellite is launched with an initial velocity Vo = 2500 mi/h parallel to the surface of the earth. Determine the required altitude (or range of altitudes) above the earths surface for launching if the free-flight trajectory is to be (a) circular, (b) parabolic, (c) elliptical, and (d) hyperbolic. Take G = 34.4(10 9)(lb ft2)/slug-, Me = 409(I021) slug, the earth's radius re = 3960 mi. and 1 mi = 5280 ft.

13-126. A probe has a circular orbit around a planet of radius R and mass M. If the radius of the orbit is nR and the explorer is traveling with a constant speed u0. determine the angle 0 at which it lands on the surface of the planet B when its speed is reduced to kv(h where k < 1 at point A.

Upon completion of the moon exploration mission, the command module, which was originally in a circular orbit as shown, is given a boost so that it escapes from the moon's gravitational field. Determine the necessary increase in velocity so that the command module follows a parabolic trajectory. The mass of the moon is \(0.01230 M_e\)

*13-128. The rocket is traveling in a frce-flight elliptical orbit about the earth such that e = 0.76 and its perigee is 9 Mm as shown. Determine its speed when it is at point B. Also determine the sudden decrease in speed the rocket must experience at A in order to travel in a circular orbit about the earth.

13-129. A rocket is in circular orbit about the earth at an altitude above the earth's surface of h = 4 Mm. Determine the minimum increment in speed it must have in order to escape the earth's gravitational field.

The satellite is an elliptical orbit having an eccentricity of \(e=0.15\). If its velocity at perigee is \(v_P=15\mathrm{\ Mm}/\mathrm{h}\), determine its velocity at apogee A and the period of the satellite.

13-131. A rocket is in a free-flight elliptical orbit about the earth such that the eccentricity of its orbit is e and its perigee is r0. 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.

*13-132. The rocket shown is originally in a circular orbit 6 Mm above the surface of the earth. It is required that it travel in another circular orbit having an altitude of 14 Mm. To do this, the rocket is given a short pulse of power at A so that it travels in free flight along the gray elliptical path from the first orbit to the second orbit. Determine the necessary speed it must have at A just after the power pulse, and the time required to get to the outer orbit along the path AA\What adjustment in speed must be made at A' to maintain the second circular orbit?

P13-1. If the box is released from rest at A, use numerical values to show how you would estimate the time for it to arrive at B. Also, list the assumptions for your analysis.

P13-2. The tugboat has a known mass and its propeller provides a known maximum thrust. When the tug is fully powered you observe the time it takes for the tug to reach a speed of known value starting from rest. Show how you could determine the mass of the barge. Neglect the drag force of the water on the tug. Use numerical values to explain your answer.

P13-3. Determine the smallest speed of each car A and B so that the passengers do not lose contact with the seat while the arms turn at a constant rate. What is the largest normal force of the seat on each passenger? Use numerical values to explain your answer.

P13-4. Each car is pin connected at its ends to the rim of the wheel which turns at a constant speed. Using numerical values, show how to determine the resultant force the seat exerts on the passenger located in the top car A. The passengers are seated towards the center of the wheel. Also, list the assumptions for your analysis.