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Examples and Problems

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Problem 12-11 The acceleration of a particle as it moves along a straight line is given by a = (2t-1) m/s2, where t is in seconds. If s = 1 m and v = 2 m/s when t = 0, determine the particle’s velocity and position when t = 6 s. Also, determine the total distance the particle travels during this time period.

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Problem 12-26 Ball A is released from rest at a height of 40 ft at the same time that a second ball B is thrown upward 5 ft from the ground. If the balls pass one another at a height of 20 ft, determine the speed at which ball B was thrown upward.

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Problem 12-75 The path of a particle is defined by y2 = 4kx, and the component of velocity along the y axis is vy = ct, where both k and c are constants. Determine the x and y components of acceleration.

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Problem 12-78 The particle travels along the path defined by the parabola y = 0.5x2. If the component of velocity along the x axis is vx = (5t) ft/s, where t is in seconds, determine the particle’s distance from the origin O and the magnitude of its acceleration when t = 1 s. When t = 0, x = 0, and y = 0. (See diagram in the textbook.)

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Problem 12-85 From a videotape, it was observed that a pro football player kicked a football 126 ft during a measured time of 3.6 seconds. Determine the initial speed of the ball and the angle θ at which it was kicked. (See diagram in the textbook.)

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Problem 12-95 Determine the horizontal velocity vA of a tennis ball at A so that it just clears the net at B. Also, find the distance s where the ball strikes the ground. (See diagram in the textbook.)

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Problem At a given instant the jet plane has a speed of 400 ft/s and an acceleration of 70 ft/s2 acting in the direction shown. Determine the rate of increase in the plane’s speed and the radius of curvature ρ of the path.

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Problem When the motorcyclist is at A, he increases his speed along the vertical circular path at the rate of ft/s2, where t is in seconds. If he starts from rest at A, determine the magnitudes of his velocity and acceleration when he reaches B.

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Problem A block moves outward along the slot in the platform with a speed of m/s, where t is in seconds. The platform rotates at a constant rate of 6 rad/s. If the block starts from rest at the center, determine the magnitudes of its velocity and acceleration when t = 1s.

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Problem If the end of the cable at A is pulled down with a speed of 2 m/s, determine the speed at which block B rises.

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Problem Determine the displacement of the block at B if A is pulled down 4 ft.

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Problem A block moves outward along the slot in the platform with a speed of m/s, where t is in seconds. The platform rotates at a constant rate of 6 rad/s. If the block starts from rest at the center, determine the magnitudes of its velocity and acceleration when t = 1 s.

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Problem 13-2 The 10-Ib block has an initial velocity of 10 ft/s on the smooth plane. If a force F = (2.5t) Ib, 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.

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Problem 13-14 Each of the two blocks has a mass m. The coefficient of kinetic friction at all surfaces of contact is µ. If a horizontal force P moves the bottom block, determine the acceleration of the bottom block in each case.

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Problem 13-70 The package has a weight of 5Ib and slides down the chute. When it reaches the curved portion AB, it is traveling at 8 ft/s (θ = 0o). If the chute is smooth, determine the speed of the package when it reaches the intermediate point C (θ = 300) and when it reaches the horizontal plane (θ = 450). Also, find the normal force on the package at C.

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Problem 13-90 The 0.5-Ib particle is guided along the circular path using the slotted arm guide. If the arm has an angular velocity and an angular acceleration at the instant , determine the force of the guide on the particle. Motion occurs in the horizontal plane.

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Problem 14-2 The car having a mass of 2 Mg is originally traveling at 2 m/s. Determine the distance it must be towed by a force F = 4 kN in order to attain a sped of 5 m/s. Neglect friction and the mass of the wheels.

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Problem 14-23 Packages having a weight of 50 Ib are delivered to the chute at vA = 3 ft/s using a conveyor belt. Determine their speeds when they reach points B, C, and D. Also calculate the normal force of the chute on the packages at B and C. Neglect friction and the size of the packages.

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Problem 14-56 The 50-kg crate is hoisted up the 30o incline by the pulley system and motor M. If the crate starts from rest and by constant acceleration attains a speed of 4 m/s after traveling 8 m along the plane, determine the power that must be supplied to the motor at this instant. Neglect friction along the plane. The motor has an efficiency of e = 0.74.

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Problem 14-72 The girl has a mass of 40 kg and center of mass at G. If she is swinging to a maximum height defined by θ = 60o, determine the force developed along each of the four supporting posts such as AB at the instant θ = 0o. The swing is centrally located between the posts.

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Problem 14-89 The 2-kg ball of negligible size is fired from point A with an initial velocity of 10 m/s up the smooth inclined plane. Determine the distance from the point C to where it hits the horizontal surface at D. Also, what is its velocity when it strikes the surface?

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Problem 15-2 A 2-Ib ball is thrown in the direction shown with an initial speed vA = 18 ft/s. Determine the time needed for it to reach its highest point B and the speed at which it is traveling at B. Use the principle of impulse and momentum for the solution. (See diagram in the textbook.)

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Problem 15-33 The car A has a weight of 4500 Ib and is traveling to the right at 3 ft/s. Meanwhile a 3000-Ib car B is traveling at 6 ft/s to the left. If the cars crash head-on and become entangled, determine their common velocity just after collision. Assume that the brakes are not applied during collision.

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Problem 15-50 The 20-Ib cart B is supported on rollers of negligible size. If a 10-Ib suitcase A is thrown horizontally on it at 10 ft/s, determine the time t and the distance B moves before A stops relative to B. The coefficient of kinetic friction between A and B is mk = 0.4.

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Problem 15-61 The man A has a weight of 175 Ib and jumps from rest h = 8 ft onto a platform P that has a weight of 60 Ib. The platform is mounted on a spring, which has a stiffness k = 200 Ib/ft. Determine (a) the velocities of A and P just after impact and (b) the maximum compression imparted to the spring by the impact. Assume the coefficient of restitution between the man and the platform is e = 0.6, and the man holds himself rigid during the motion.

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Problem 15-84 Two coins A and B have the initial velocities shown just before they collide at point O. If they have weights of WA = 13.2(10-3) Ib and WB = 6.60(10-3) Ib and the surface upon which they slide is smooth, determine their speeds just after impact. The coefficient of restitution is e = 0.65.

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Problem 15-96 Determine the total angular momentum Ho for the system of three particles about point O. All the particles are moving in the x-y plane.

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Problem An amusement park ride consists of a car which is attached to the cable OA. The car rotates in a horizontal circular path and is brought to a speed v1 = 4 ft/s when r = 12 ft. The cable is then pulled in at the constant rate of 0.5 ft/s. Determine the speed of the car in 3 s.

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Help Session

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Problem 15-28 Block A weighs 10 Ib and block B weighs 3 Ib. If B is moving downward with a velocity (vB)1 = 3 ft/s at t = 0, determine the velocity of A when t = 1 s. The coefficient of kinetic friction between the horizontal plane and block A is mA = 0.15.

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Problem 15-47 The 10-kg block is held at rest on the smooth inclined plane by the stop block at A. If the 10-g bullet is traveling at 300 m/s when it becomes embedded in the 10-kg block, determine the distance the block will slide up along the plane before momentarily stopping.

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Problem 15-64 If the girl throws the ball with a horizontal velocity of 8 ft/s, determine the distance d so that the ball bounces once on the smooth surface and then lands in the cup at C. Take e = 0.8.

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Problem 16-5 Due to an increase in power, the motor M rotates the shaft A with an angular acceleration of a = (0.06q 2) rad/s2, where q is in radians. If the shaft is initially turning at wo = 50 rad/s, determine the angular velocity of gear B after the shaft undergoes an angular displacement Dq = 10 rev.

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Problem 16-19 Starting from rest when s = 0, pulley A is given a constant angular acceleration ac = 6 rad/s2. Determine the speed of block B when it has risen s = 6 m. The pulley has an inner hub D which is fixed to C and turns with it.

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Problem 16-55 Determine the velocity of the slider block at C at the instant q = 45o, if link AB is rotating at 4 rad/s.

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Problem 16-87 The shaper mechanism is designed to give a slow cutting stroke and a quick return to a blade attached to the slider at C. Determine the angular velocity of the link CB at the instant shown, if the link AB is rotating at 4 rad/s.

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Problem The ends of the bar AB are confined to move along the paths shown. At a given instant, A has a velocity of 8 ft/s and an acceleration of 3 ft/s2. Determine the angular velocity and angular acceleration of AB at this instant.

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Problem The man stands on the platform at O and runs out toward the edge such that when he is at A, y = 5 ft, his mass center has a velocity of 2 ft/s and an acceleration of 3 ft/s2, both measured with respect to the platform and directed along the y axis. If the platform has the angular motions shown, determine the velocity and acceleration of his mass center at this instant.

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Problem Block B of the mechanism is confined to move within the slot member CD. If AB is rotating at a constant rate of wAB = 3 rad/s, determine the angular velocity and angular acceleration of member CD at the instant shown.

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Problem At a given instant the wheel is rotating with the angular motions shown. Determine the acceleration of the collar at A at this instant.

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Problem 17-20 The pendulum consists of two slender rods AB and OC which have a mass of 3 kg/m. The thin plate has a mass of 12 kg/m2. Determine the location y of the center of mass G of the pendulum, then calculate the moment of inertia of the pendulum about an axis perpendicular to the page and passing through G.

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Problem 17-28 The jet aircraft has a total mass of 22 Mg and a center of mass at G. Initially at take-off the engines provide a thrust 2T = 4 kN and T’ = 1.5 kN. Determine the acceleration of the plane and the normal reactions on the nose wheel and each of the two wing wheels located at B. Neglect the mass of the wheels and, due to low velocity, neglect any lift caused by the wings.

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Problem (modified) The van has a weight of 4500 Ib and center of gravity at Gv. It carries a fixed 800-Ib load which has a center of gravity at Gl. If the van is travelling at 40 ft/s, determine the distance it skids before stopping. The brakes cause all the wheels to lock or skid. The coefficient of kinetic friction between the wheels and the pavement is mk = 0.3. Also, determine the reactions on the wheels. Neglect the mass of the wheels.

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Problem 17-57 The spool is supported on small rollers at A and B. Determine the constant force P that must be applied to the cable in order to unwind 8 m of cable in 4 s starting from rest. Also calculate the normal forces at A and B during this time. The spool has a mass of 60 kg and a radius of gyration ko = 0.65 m. For the calculation neglect the mass of the cable and the mass of the rollers at A and B.

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Problem 17-71 If the support at B is suddenly removed, determine the initial downward acceleration of point C. Segments AC and CB each have a weight of 10 Ib.

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Problem 17-97 The spool has a mass of 100 kg and a radius of gyration of kG = 0.3 m. If the coefficients of static and kinetic friction at A are ms = 0.2 and mk = 0.15, respectively, determine the angular acceleration of the spool if P = 600 N.

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Problem 18-14 A motor supplies a constant torque or twist M = 120 Ib.ft to the drum. If the drum has a weight of 30 Ib and a radius of gyration of KO = 0.8 ft, determine the speed of the 15-Ib crate A after it rises s = 4 ft starting from rest. Neglect the mass of the cord.

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Problem 18-43 The 50-Ib wheel has a radius of gyration about its center of gravity G of kG = 0.7 ft. If it rolls without slipping, determine its angular velocity when it has rotated clockwise 90o from the position shown. The spring AB has a stiffness k = 120 Ib/ft and an unstreched length of 0.5 ft. The wheel is released from rest.

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Problem 18-50 The assembly consists of a 3-kg pulley A and 10-kg pulley B. If a 2-kg block is suspended from the cord, determine the block’s speed after it descends 0.5 m starting from rest. Neglect the mass of the cord and treat the pulleys as thin disks. No slipping occurs.

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Problem 19-6 Solve Prob using the principle of impulse and momentum. Problem 17-54 The 10-kg wheel has a radius of gyration kA = 200 mm. If the wheel is subjected to a moment M = (5t) N.m, where t is in seconds, determine its angular velocity when t = 3 s starting from rest. Also, compute the reactions which the fixed pin A exerts on the wheel during the motion.

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Problem 19-9 Solve Prob using the principle of impulse and momentum. Problem The disk has a mass of 20 kg and is originally spinning at the end of the strut with an angular velocity of w = 60 rad/s. If it is then placed against the wall, for which the coefficient of kinetic friction is mk = 0.3 determine the time required for the motion to stop. What is the force in the strut BC during this time?

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Problem 19-51 The 4-Ib rod AB is hanging in the vertical position. A 2-Ib block, sliding on a smooth horizontal surface with a velocity of 12 ft/s, strikes the rod at its end B. Determine the velocity of the block immediately after collision. The coefficient of restitution between the block and the rod at B is e = 0.8.

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