Engineering Mechanics: Dynamics · Kinetics of a Particle · Impulse and Momentum

Chapter Fifteen
Impulse and Momentum

15.1 Impulse and Momentum 34 problems
Fundamental Problems
  1. F15-1The \(0.5\mathrm{-kg}\) ball strikes the rough ground and rebounds with the velocities shown. Determine the magnitude of the impulse the ground exerts on the ball. Assume that the ball does not slip when it strikes the ground, and neglect the size of the ball and the impulse produced by the weight of the ball. figure — add your own
  2. F15-2If the coefficient of kinetic friction between the \(75\mathrm{-kg}\) crate and the ground is \(\mu_{k} = 0.2\), determine the speed of the crate when \(t = 4\mathrm{~s}\). The crate starts from rest and is towed by the \(500\mathrm{-N}\) force. figure — add your own
  3. F15-3The motor exerts a force of \(F = (20t^2)\mathrm{N}\) on the cable, where \(t\) is in seconds. Determine the speed of the \(25\mathrm{-kg}\) 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. figure — add your own
  4. F15-4The wheels of the \(1.5\mathrm{-Mg}\) car generate the traction force \(\mathbf{F}\) described by the graph. If the car starts from rest, determine its speed when \(t = 6\mathrm{~s}\). figure — add your own
  5. F15-5The \(2.5\mathrm{-Mg}\) four-wheel-drive SUV tows the \(1.5\mathrm{-Mg}\) trailer. The traction force developed at the wheels is \(F_{D} = 9\mathrm{kN}\). Determine the speed of the truck in \(20\mathrm{~s}\), starting from rest. Also, determine the tension developed in the mass of the wheels. figure — add your own
  6. F15-6The \(10\mathrm{-kg}\) block \(A\) attains a velocity of \(1\mathrm{~m/s}\) in 5 seconds, starting from rest. Determine the tension in the cord and the coefficient of kinetic friction between block \(A\) and the horizontal plane. Neglect the weight of the pulley. Block \(B\) has a mass of \(8\mathrm{~kg}\). figure — add your own
  7. F15-7The freight cars \(A\) and \(B\) have a mass of \(20~\mathrm{Mg}\) and \(15~\mathrm{Mg}\) respectively. Determine the velocity of \(A\) after collision if the cars collide and rebound, such that \(B\) moves to the right with a speed of \(2\mathrm{m/s}\). If \(A\) and \(B\) are in contact for \(0.5\mathrm{s}\) find the average impulsive force which acts between them. figure — add your own
  8. F15-8The spring is fixed to block \(A\) and block \(B\) is pressed against the spring. If the spring is compressed \(s = 200~\mathrm{mm}\) and then the blocks are released, determine their velocity at the instant block \(B\) loses contact with the spring. The masses of blocks \(A\) and \(B\) are \(10\mathrm{kg}\) and \(15\mathrm{kg}\) respectively. figure — add your own
  9. F15-9The \(5\mathrm{-kg}\) block \(A\) has an initial speed of \(5\mathrm{m/s}\) as it slides down the smooth ramp, after which it collides with the stationary block \(B\) of mass \(8\mathrm{kg}\). If the two blocks couple together after collision, determine their common velocity immediately after collision. figure — add your own
  10. F15-10Blocks \(A\) and \(B\) have a mass of \(15\mathrm{kg}\) and \(10\mathrm{kg}\) respectively. If \(A\) is stationary and \(B\) has a velocity of \(15~\mathrm{m/s}\) just before collision, and the blocks couple together after impact, determine the maximum compression of the spring. figure — add your own
  11. F15-11The cannon and support without a projectile have a mass of \(250~\mathrm{kg}\). If a \(20\mathrm{-kg}\) projectile is fired from the cannon with a velocity of \(400~\mathrm{m/s}\) measured relative to the cannon, determine the speed of the projectile as it leaves the barrel of the cannon. Neglect rolling resistance. figure — add your own
  12. F15-12The \(0.5\mathrm{-kg}\) ball is fired from the tube at \(A\) with a velocity of \(v = 6 \mathrm{m/s}\). If the coefficient of restitution between the ball and the surface is \(e = 0.8\), determine the height \(h\) after it bounces off the surface. figure — add your own
Skills
  1. 1.Determine the impulse of the force for \(t = 2\) s. figure — add your own
  2. 2.Determine the linear momentum of the 10-kg block. figure — add your own
  3. 3.A man kicks the \(150\mathrm{-g}\) ball such that it leaves the ground at an angle of \(60^{\circ}\) and strikes the ground at the same elevation a distance of \(12\mathrm{m}\) away. Determine the impulse of his foot on the ball at \(A\). Neglect the impulse caused by the ball's weight while it's being kicked. figure — add your own
  4. 4.A \(2.5\mathrm{-kg}\) block is given an initial velocity of \(3\mathrm{m/s}\) up a \(45^{\circ}\) smooth slope. Determine the time for it to travel up the slope before it stops. figure — add your own
  5. 5.A hockey puck is traveling to the left with a velocity of \(v_{1} = 10\mathrm{m/s}\) when it is struck by a hockey stick and given a velocity of \(v_{2} = 20\mathrm{m/s}\) as shown. Determine the magnitude of the net impulse exerted by the hockey stick on the puck. The puck has a mass of \(0.2\mathrm{kg}\). figure — add your own
  6. 6.A train consists of a \(50\mathrm{-Mg}\) engine and three cars, each having a mass of \(30\mathrm{Mg}\). If it takes \(80\mathrm{s}\) for the train to increase its speed uniformly to \(40\mathrm{km/h}\), starting from rest, determine the force \(T\) developed at the coupling between the engine \(E\) and the first car \(A\). The wheels of the engine provide a resultant frictional tractive force \(\mathbf{F}\) which gives the train forward motion, whereas the car wheels roll freely. Also, determine \(F\) acting on the engine wheels. figure — add your own
  7. 7.The winch delivers a horizontal towing force \(\mathbf{F}\) to its cable at \(A\) which varies as shown in the graph. Determine the speed of the \(70\mathrm{-kg}\) bucket when \(t = 18\mathrm{s}\). Originally the bucket is moving upward at \(v_{1} = 3\mathrm{m/s}\). figure — add your own
  8. 8.The winch delivers a horizontal towing force \(\mathbf{F}\) to its cable at \(A\) which varies as shown in the graph. Determine the speed of the \(80\mathrm{-kg}\) bucket when \(t = 24\mathrm{s}\). Originally the bucket is released from rest. figure — add your own
  9. 9.The 50-kg crate is pulled by the constant force P. If the crate starts from rest and achieves a speed of \(10\mathrm{m/s}\) in \(5\mathrm{s}\), determine the magnitude of P. The coefficient of kinetic friction between the crate and the ground is \(\mu_{k} = 0.2\). figure — add your own
  10. 10.If the jets exert a vertical thrust of \(T = (500t^{3/2})\mathrm{N}\) where \(t\) is in seconds, determine the man's speed when \(t = 3\) s. The total mass of the man and the jet suit is \(100\mathrm{kg}\). Neglect the loss of mass due to the fuel consumed during the lift which begins from rest on the ground. figure — add your own
  11. 11.The train consists of a \(30\mathrm{-Mg}\) engine \(E\) and cars \(A\), \(B\) and \(C\) which have a mass of \(15\mathrm{Mg}\), \(10\mathrm{Mg}\) and \(8\mathrm{Mg}\) respectively. If the tracks provide a traction force of \(F = 30\mathrm{kN}\) on the engine wheels, determine the speed of the train when \(t = 30\) s, starting from rest. Also, find the horizontal coupling force at \(D\) between the engine \(E\) and car \(A\). Neglect rolling resistance. figure — add your own
  12. 12.The \(200\mathrm{-kg}\) crate rests on the ground for which the coefficients of static and kinetic friction are \(\mu_{s} = 0.5\) and \(\mu_{k} = 0.4\) respectively. The winch delivers a horizontal towing force T to its cable at \(A\) which varies as shown in the graph. Determine the speed of the crate when \(t = 4\) s. Originally the tension in the cable is zero. Hint: First determine the force needed to begin moving the crate. figure — add your own
  13. 13.The \(2.5\mathrm{-Mg}\) van is traveling with a speed of \(100\mathrm{km/h}\) when the brakes are applied and all four wheels lock. If the speed decreases to \(40\mathrm{km/h}\) in \(5\mathrm{s}\), determine the coefficient of kinetic friction between the tires and the road. figure — add your own
  14. 14.During operation the jack hammer strikes the concrete surface with a force which is indicated in the graph. To achieve this the \(2\mathrm{-kg}\) spike \(S\) is fired into the surface at \(90\mathrm{m/s}\). Determine the speed of the spike just after rebounding. figure — add your own
  15. 15.For a short period of time, the frictional driving force acting on the wheels of the \(2.5\mathrm{-Mg}\) van is \(F_{D} = (600t^{2})\) N, where \(t\) is in seconds. If the van has a speed of \(20\mathrm{km/h}\) when \(t = 0\), determine its speed when \(t = 5\) s. figure — add your own
  16. 16.Under a constant thrust of \(T = 40 \mathrm{kN}\), the 1.5-Mg dragster reaches its maximum speed of \(125 \mathrm{m/s}\) in \(8 \mathrm{s}\) starting from rest. Determine the average drag resistance \(\mathbf{F}_{D}\) during this period of time. figure — add your own
  17. 17.The motor, \(M\) pulls on the cable with a force \(F = (10t^{2} + 300)\mathrm{N}\), where \(t\) is in seconds. If the \(100\mathrm{kg}\) crate is originally at rest at \(t = 0\), determine its speed when \(t = 4\) s. Neglect the mass of the cable and pulleys. Hint: First find the time needed to begin lifting the crate. figure — add your own
  18. 18.A tankcar has a mass of \(20\mathrm{Mg}\) and is freely rolling to the right with a speed of \(0.75\mathrm{m/s}\). If it strikes the barrier, determine the horizontal impulse needed to stop the car if the spring in the bumper \(B\) has a stiffness (a) \(k\rightarrow \infty\) (bumper is rigid), and (b) \(k = 15\mathrm{kN/m}\). figure — add your own
  19. 19.The thrust on the 4-Mg rocket sled is shown in the graph. Determine the sleds maximum velocity and the distance the sled travels when \(t = 35\) s. Neglect friction. figure — add your own
  20. 20.A 50-kg crate rests against a stop block \(s\) which prevents the crate from moving down the plane. If the coefficients of static and kinetic friction between the plane and the crate are \(\mu_{s} = 0.3\) and \(\mu_{k} = 0.2\) respectively, determine the time needed for the force \(\mathbf{F}\) to give the crate a speed of \(2\mathrm{m/s}\) up the plane. The force always acts parallel to the plane and has a magnitude of \(F = (300t)\) N, where \(t\) is in seconds. Hint: First determine the time needed to overcome static friction and start the crate moving. figure — add your own
  21. 21.The choice of a seating material for moving vehicles depends upon its ability to resist shock and vibration. From the data shown in the graphs, determine the impulses created by a falling weight onto a sample of urethane foam and CONFOR foam. figure — add your own
  22. 22.The towing force acting on the 400-kg safe varies as shown on the graph. Determine its speed, starting from rest, when \(t = 8\) s. How far has it traveled during this time? figure — add your own
  23. 23.If it takes \(35\mathrm{s}\) for the \(50\mathrm{-Mg}\) tugboat to increase its speed uniformly to \(25\mathrm{km/h}\) starting from rest, determine the force of the rope on the tugboat. The propeller provides motion, whereas the barge moves freely. Also, determine \(F\) acting on the tugboat. The barge has a mass of \(75\mathrm{Mg}\). figure — add your own
  24. 24.The crate \(B\) and cylinder \(A\) have a mass of \(200\mathrm{kg}\) and \(75\mathrm{kg}\), respectively. If the system is released from rest, determine the speed of the crate and cylinder when \(t = 3\) s. Neglect the mass of the pulleys. figure — add your own
  25. 25.The motor exerts a force \(F\) on the 40-kg crate as shown in the graph. Determine the speed of the crate when \(t = 3\) s and when \(t = 6\) s. When \(t = 0\) the crate is moving downward at \(10\mathrm{m/s}\). figure — add your own
  26. 26.The 30-kg slider block is moving to the left with a speed of \(5\mathrm{m/s}\) when it is acted upon by the forces \(\mathbf{F}_{1}\) and \(\mathbf{F}_{2}\). If these loadings vary in the manner shown on the graph, determine the speed of the block at \(t = 6\) s. Neglect friction and the mass of the pulleys and cords. figure — add your own
  27. 27.The balloon has a total mass of \(400\mathrm{kg}\) including the passengers and ballast. The balloon is rising at a constant velocity of \(18\mathrm{km/h}\) when \(h = 10 \mathrm{m}\). If the man drops the 40-kg sand bag, determine the velocity of the balloon when the bag strikes the ground. Neglect air resistance. figure — add your own
  28. 28.As indicated by the derivation, the principle of impulse and momentum is valid for observers in any inertial reference frame. Show that this is so, by considering the 10-kg block which slides along the smooth surface and is subjected to a horizontal force of \(6\mathrm{N}\). If observer \(A\) is in a fixed frame \(x\) determine the final speed of the block in 4 s if it has an initial speed of \(5\mathrm{m/s}\) measured from the fixed frame. Compare the result with that obtained by an observer \(B\) attached to the \(x^{\prime}\) axis that moves at a constant velocity of \(2\mathrm{m/s}\) relative to \(A\). figure — add your own
  29. 29.In case of emergency, the gas actuator is used to move a 75-kg block \(B\) by exploding a charge \(C\) near a pressurized cylinder of negligible mass. As a result of the explosion, the cylinder fractures and the released gas forces the front part of the cylinder, \(A\) to move \(B\) forward, giving it a speed of \(200\mathrm{mm/s}\) in 0.4 s. If the coefficient of kinetic friction between \(B\) and the floor is \(\mu_{k} = 0.5\), determine the impulse that the actuator imparts to \(B\). figure — add your own
  30. 30.The 20-kg crate is lifted by a force of \(F = (100 + 5t^2)\mathrm{N}\), where \(t\) is in seconds. Determine the speed of the crate when \(t = 3\) s, starting from rest. figure — add your own
  31. 31.The 20-kg crate is lifted by a force of \(F = (100 + 5t^2)\mathrm{N}\), where \(t\) is in seconds. Determine how high the crate has moved upward when \(t = 3\) s, starting from rest. figure — add your own
  32. 32.A jet plane having a mass of \(7\mathrm{Mg}\) takes off from an aircraft carrier such that the engine thrust varies as shown by the graph. If the carrier is traveling forward with a speed of \(40\mathrm{km/h}\), determine the plane's airspeed after 5 s. figure — add your own
  33. 33.The 6-kg block is moving downward at \(v_{1} = 3 \mathrm{m/s}\) when it is \(8 \mathrm{m}\) from the sandy surface. Determine the impulse of the sand on the block necessary to stop its motion. Neglect the distance the block dents into the sand and assume the block does not rebound. Neglect the weight of the block during the impact with the sand. figure — add your own
  34. 34.The 6-kg block is falling downward at \(v_{1} = 3 \mathrm{m/s}\) when it is \(8 \mathrm{m}\) from the sandy surface. Determine the average impulsive force acting on the block by the sand if the motion of the block is stopped in time 1.2 s once the block strikes the sand. Neglect the distance the block dents into the sand and assume the block does not rebound. Neglect the weight of the block during the impact with the sand. figure — add your own
15.2 Conservation of Linear Momentum for a System of Particles 23 problems
Skills
  1. 35.The 5-Mg bus \(B\) is traveling to the right at \(20~\mathrm{m/s}\). Meanwhile a 2-Mg car \(A\) is traveling at \(15~\mathrm{m/s}\) to the right. If the vehicles crash and become entangled, determine their common velocity just after the collision. Assume that the vehicles are free to roll during collision. figure — add your own
  2. 36.The 50-kg boy jumps on the 5-kg skateboard with a horizontal velocity of \(5\mathrm{m/s}\). Determine the distance \(s\) the boy reaches up the inclined plane before momentarily coming to rest. Neglect the skateboard's rolling resistance. figure — add your own
  3. 37.The 2.5-Mg pickup truck is towing the 1.5-Mg car using a cable as shown. If the car is initially at rest and the truck is coasting with a velocity of \(30~\mathrm{km/h}\) when the cable is slack, determine the common velocity of the truck and the car just after the cable becomes taut. Also, find the loss of energy. figure — add your own
  4. 38.A railroad car having a mass of \(15\mathrm{Mg}\) is coasting at \(1.5\mathrm{m/s}\) on a horizontal track. At the same time another car having a mass of \(12\mathrm{Mg}\) is coasting at \(0.75\mathrm{m/s}\) in the opposite direction. If the cars meet and couple together, determine the speed of both cars just after the coupling. Find the difference between the total kinetic energy before and after coupling has occurred, and explain qualitatively what happened to this energy. figure — add your own
  5. 39.A ballistic pendulum consists of a 4-kg wooden block originally at rest, \(\theta = 0^{\circ}\). When a 2-g bullet strikes and becomes embedded in it, it is observed that the block swings upward to a maximum angle of \(\theta = 6^{\circ}\). Estimate the initial speed of the bullet. figure — add your own
  6. 40.The boy \(B\) jumps off the canoe at \(A\) with a velocity \(5\mathrm{m/s}\) relative to the canoe as shown. If he lands in the second canoe \(C\), determine the final speed of both canoes after the motion. Each canoe has a mass of \(40~\mathrm{kg}\). The boy's mass is \(30~\mathrm{kg}\), and the girl \(D\) has a mass of \(25~\mathrm{kg}\). Both canoes are originally at rest. figure — add your own
  7. 41.The block of mass \(m\) travels at \(v_{1}\) in the direction \(\theta_{1}\) shown at the top of the smooth slope. Determine its speed \(v_{2}\) and its direction \(\theta_{2}\) when it reaches the bottom. figure — add your own
  8. 42.A toboggan having a mass of \(10\mathrm{kg}\) starts from rest at \(A\) and carries a girl and boy having a mass of \(40\mathrm{kg}\) and \(45\mathrm{kg}\) respectively. When the toboggan reaches the bottom of the slope at \(B\) the boy is pushed off from the back with a horizontal velocity of \(\mathrm{v}_{b/t} = 2 \mathrm{m/s}\) measured relative to the toboggan. Determine the velocity of the toboggan afterwards. Neglect friction in the calculation. figure — add your own
  9. 43.The \(20\mathrm{-g}\) bullet is traveling at \(400 \mathrm{m/s}\) when it becomes embedded in the \(2\mathrm{-kg}\) stationary block. Determine the distance the block will slide before it stops. The coefficient of kinetic friction between the block and the plane is \(\mu_{k} = 0.2\). figure — add your own
  10. 44.A \(4\mathrm{-kg}\) projectile travels with a horizontal velocity of \(600 \mathrm{m/s}\) before it explodes and breaks into two fragments \(A\) and \(B\) of mass \(1.5\mathrm{kg}\) and \(2.5\mathrm{kg}\) respectively. If the fragments travel along the parabolic trajectories shown, determine the magnitude of velocity of each fragment just after the explosion and the horizontal distance \(d_{A}\) where segment \(A\) strikes the ground at \(C\). figure — add your own
  11. 45.A \(4\mathrm{-kg}\) projectile travels with a horizontal velocity of \(600 \mathrm{m/s}\) before it explodes and breaks into two fragments \(A\) and \(B\) of mass \(1.5\mathrm{kg}\) and \(2.5\mathrm{kg}\) respectively. If the fragments travel along the parabolic trajectories shown, determine the magnitude of velocity of each fragment just after the explosion and the horizontal distance \(d_{B}\) where segment \(B\) strikes the ground at \(D\). figure — add your own
  12. 46.The \(10\mathrm{-Mg}\) barge \(B\) supports a \(2\mathrm{-Mg}\) automobile \(A\). If someone drives the automobile to the other side of the barge, determine how far the barge moves. Neglect the resistance of the water. figure — add your own
  13. 47.Block \(A\) has a mass of \(2\mathrm{kg}\) and slides into an open ended box \(B\) with a velocity of \(2\mathrm{m/s}\). If the box \(B\) has a mass of \(3\mathrm{kg}\) and rests on top of a plate \(P\) that has a mass of \(3\mathrm{kg}\), determine the distance the plate moves after it stops sliding on the floor. Also, how long is it after impact before all motion ceases? The coefficient of kinetic friction between the box and the plate is \(\mu_{k} = 0.2\) and between the plate and the floor \(\mu_{k}^{\prime} = 0.4\). Also, the coefficient of static friction between the plate and the floor is \(\mu_{s}^{\prime} = 0.5\). figure — add your own
  14. 48.Block \(A\) has a mass of \(2\mathrm{kg}\) and slides into an open ended box \(B\) with a velocity of \(2\mathrm{m/s}\). If the box \(B\) has a mass of \(3\mathrm{kg}\) and rests on top of a plate \(P\) that has a mass of \(3\mathrm{kg}\), determine the distance the plate moves after it stops sliding on the floor. Also, how long is it after impact before all motion ceases? The coefficient of kinetic friction between the box and the plate is \(\mu_{k} = 0.2\) and between the plate and the floor \(\mu_{k}^{\prime} = 0.1\). Also, the coefficient of static friction between the plate and the floor is \(\mu_{s}^{\prime} = 0.12\). figure — add your own
  15. 49.The \(10\mathrm{-kg}\) block is held at rest on the smooth inclined plane by the stop block at \(A\). If the \(10\mathrm{-g}\) bullet is traveling at \(300\mathrm{m/s}\) when it becomes embedded in the \(10\mathrm{-kg}\) block, determine the distance the block will slide up along the plane before momentarily stopping. figure — add your own
  16. 50.The cart has a mass of \(3\mathrm{kg}\) and rolls freely from \(A\) down the slope. When it reaches the bottom, a spring loaded gun fires a \(0.5\mathrm{-kg}\) ball out the back with a horizontal velocity of \(v_{b/c} = 0.6\mathrm{m/s}\) measured relative to the cart. Determine the final velocity of the cart. figure — add your own
  17. 51.The \(30\mathrm{-Mg}\) freight car \(A\) and \(15\mathrm{-Mg}\) freight car \(B\) are moving towards each other with the velocities shown. Determine the maximum compression of the spring mounted on car \(A\). Neglect rolling resistance. figure — add your own
  18. 52.The two blocks \(A\) and \(B\) each have a mass of \(5\mathrm{kg}\) and are suspended from parallel cords. A spring, having a stiffness of \(k = 60~\mathrm{N/m}\), is attached to \(B\) and is compressed \(0.3\mathrm{m}\) against \(A\) and \(B\) as shown. Determine the maximum angles \(\theta\) and \(\phi\) of the cords when the blocks are released from rest and the spring becomes unstretched. figure — add your own
  19. 53.Blocks \(A\) and \(B\) have masses of \(40\mathrm{kg}\) and \(60\mathrm{kg}\) respectively. They are placed on a smooth surface and the spring connected between them is stretched \(2\mathrm{m}\). If they are released from rest, determine the speeds of both blocks the instant the spring becomes unstretched. figure — add your own
  20. 54.Two boxes \(A\) and \(B\), each having a mass of \(80\mathrm{kg}\), sit on the \(250\mathrm{-kg}\) conveyor which is free to roll on the ground. If the belt starts from rest and begins to run with a speed of \(1\mathrm{m/s}\), determine the final speed of the conveyor if (a) the boxes are not stacked and \(A\) falls off then \(B\) falls off, and (b) \(A\) is stacked on top of \(B\) and both fall off together. figure — add your own
  21. 55.Block \(A\) has a mass of \(5\mathrm{kg}\) and is placed on the smooth triangular block \(B\) having a mass of \(30\mathrm{kg}\). If the system is released from rest, determine the distance \(B\) moves from point \(o\) when \(A\) reaches the bottom. Neglect the size of block \(A\). figure — add your own
  22. 56.Solve Prob. 15-55 if the coefficient of kinetic friction between \(A\) and \(B\) is \(\mu_{k} = 0.3\). Neglect friction between block \(B\) and the horizontal plane. figure — add your own
  23. 57.The free-rolling ramp has a mass of \(40\mathrm{kg}\). A \(10\mathrm{-kg}\) crate is released from rest at \(A\) and slides down \(3.5\mathrm{m}\) to point \(B\). If the surface of the ramp is smooth, determine the ramp's speed when the crate reaches \(B\). Also, what is the velocity of the crate? figure — add your own
  24. 58.The log has a mass of \(500 \mathrm{kg}\) and rests on the ground for which the coefficients of static and kinetic friction are \(\mu_{s} = 0.5\) and \(\mu_{k} = 0.4\) respectively. The winch delivers a horizontal towing force \(T\) to its cable at \(A\) which varies as shown in the graph. Determine the speed of the log when \(t = 5 \mathrm{s}\). Originally the tension in the cable is zero. Hint: First determine the force needed to begin moving the log. figure — add your own
  25. 59.The 0.15-kg baseball has a speed of \(v = 30 \mathrm{m/s}\) just before it is struck by the bat. It then travels along the trajectory shown before the outfielder catches it. Determine the magnitude of the average impulsive force imparted to the ball if it is in contact with the bat for 0.75 ms. figure — add your own
15.3 Impact 36 problems
Fundamental Problems
  1. F15-13Determine the coefficient of restitution \(e\) between ball \(A\) and ball \(B\). The velocities of \(A\) and \(B\) before and after the collision are shown. figure — add your own
  2. F15-14The \(15\mathrm{-Mg}\) tank car \(A\) and \(25\mathrm{-Mg}\) freight car \(B\) travel toward each other with the velocities shown. If the coefficient of restitution between the bumpers is \(e = 0.6\) determine the velocity of each car just after the collision. figure — add your own
  3. F15-15The \(15\mathrm{-kg}\) package \(A\) has a speed of \(1.5\mathrm{m/s}\) when it enters the smooth ramp. As it slides down the ramp, it strikes the \(40\mathrm{-kg}\) package \(B\) which is initially at rest. If the coefficient of restitution between \(A\) and \(B\) is \(e = 0.6\) determine the velocity of \(B\) just after the impact. figure — add your own
  4. F15-16The ball strikes the smooth wall with a velocity of \((v_{b})_{1} = 20\mathrm{m/s}\). If the coefficient of restitution between the ball and the wall is \(e = 0.75\), determine the velocity of the ball just after the impact. figure — add your own
  5. F15-17Disk \(A\) has a mass of \(2\mathrm{kg}\) and slides on the smooth horizontal plane with a velocity of \(3\mathrm{m/s}\). Disk \(B\) has a mass of \(11\mathrm{kg}\) and is initially at rest. If after impact \(A\) has a velocity of \(1\mathrm{m/s}\), parallel to the positive \(x\) axis, determine the speed of disk \(B\) after impact. figure — add your own
  6. F15-18Two disks \(A\) and \(B\) each have a mass of \(1\mathrm{kg}\) and the initial velocities shown just before they collide. If the coefficient of restitution is \(e = 0.5\), determine their speeds just after impact. figure — add your own
Skills
  1. 60.Disk \(A\) has a mass of \(250\mathrm{g}\) and is sliding on a smooth horizontal surface with an initial velocity \((v_{A})_{1} = 2\mathrm{m/s}\). It makes a direct collision with disk \(B\) which has a mass of \(175\mathrm{g}\) and is originally at rest. If both disks are of the same size and the collision is perfectly elastic \((e = 1)\) determine the velocity of each disk just after collision. Show that the kinetic energy of the disks before and after collision is the same. figure — add your own
  2. 61.The \(5\mathrm{-Mg}\) truck and \(2\mathrm{-Mg}\) car are traveling with the free-rolling velocities shown just before they collide. After the collision, the car moves with a velocity of \(15\mathrm{km/h}\) to the right relative to the truck. Determine the coefficient of restitution between the truck and car and the loss of energy due to the collision. figure — add your own
  3. 62.Disk \(A\) has a mass of \(2\mathrm{kg}\) and is sliding forward on the smooth surface with a velocity \((v_{A})_{1} = 5\mathrm{m/s}\) when it strikes the \(4\mathrm{-kg}\) disk \(B\) which is sliding towards \(A\) at \((v_{B})_{1} = 2\mathrm{m/s}\), with direct central impact. If the coefficient of restitution between the disks is \(e = 0.4\) compute the velocities of \(A\) and \(B\) just after collision. figure — add your own
  4. 63.Ball \(A\) has a mass of \(3\mathrm{kg}\) and is moving with a velocity of \(8\mathrm{m/s}\) when it makes a direct collision with ball \(B\) which has a mass of \(2\mathrm{kg}\) and is moving with a velocity of \(4\mathrm{m/s}\). If \(e = 0.7\), determine the velocity of each ball just after the collision. Neglect the size of the balls. figure — add your own
  5. 64.The \(15\mathrm{-kg}\) block \(A\) slides on the surface for which \(\mu_{k} = 0.3\). The block has a velocity \(v = 10\mathrm{m/s}\) when it is \(s = 4\mathrm{m}\) from the \(10\mathrm{-kg}\) block \(B\). If the unstretched spring has a stiffness \(k = 1000\mathrm{N/m}\), determine the maximum compression of the spring due to the collision. Take \(e = 0.6\). figure — add your own
  6. 65.The four smooth balls each have the same mass \(m\). If \(A\) and \(B\) are rolling forward with velocity \(\mathbf{v}\) and strike \(C\) explain why after collision \(C\) and \(D\) each move off with velocity \(\mathbf{v}\). Why doesn't \(D\) move off with velocity \(2\mathrm{v}\)? The collision is elastic, \(e = 1\). Neglect the size of each ball. figure — add your own
  7. 66.The four balls each have the same mass \(m\). If \(A\) and \(B\) are rolling forward with velocity \(\mathbf{v}\) and strike \(C\) determine the velocity of each ball after the first three collisions. Take \(e = 0.5\) between each ball. figure — add your own
  8. 67.Two smooth spheres \(A\) and \(B\) each have a mass \(m\). If \(A\) is given a velocity of \(v_{0}\) while sphere \(B\) is at rest, determine the velocity of \(B\) just after it strikes the wall. The coefficient of restitution for any collision is \(e\). figure — add your own
  9. 68.A pitching machine throws the \(0.5\mathrm{-kg}\) ball toward the wall with an initial velocity \(v_{A} = 10 \mathrm{m/s}\) as shown. Determine (a) the velocity at which it strikes the wall at \(B\), (b) the velocity at which it rebounds from the wall if \(e = 0.5\), and (c) the distance \(s\) from the wall to where it strikes the ground at \(C\). figure — add your own
  10. 69.A \(300\mathrm{-g}\) ball is kicked with a velocity of \(v_{A} = 25 \mathrm{m/s}\) at point \(A\) as shown. If the coefficient of restitution between the ball and the field is \(e = 0.4\), determine the magnitude and direction \(\theta\) of the velocity of the rebounding ball at \(B\). figure — add your own
  11. 70.A 1-kg ball \(A\) is traveling horizontally at \(20 \mathrm{m/s}\) when it strikes a \(10\mathrm{-kg}\) block \(B\) that is at rest. If the coefficient of restitution between \(A\) and \(B\) is \(e = 0.6\) and the coefficient of kinetic friction between the plane and the block is \(\mu_{k} = 0.4\), determine the time for the block \(B\) to stop sliding. figure — add your own
  12. 71.The three balls each have a mass \(m\). If \(A\) has a speed \(v\) just before a direct collision with \(B\), determine the speed of \(C\) after collision. The coefficient of restitution between each pair of balls is \(e\). Neglect the size of each ball. figure — add your own
  13. 72.Block \(A\), having a mass \(m\), is released from rest, falls a distance \(h\) and strikes the plate \(B\) having a mass \(2m\). If the coefficient of restitution between \(A\) and \(B\) is \(e\), determine the velocity of the plate just after collision. The spring has a stiffness \(k\). figure — add your own
  14. 73.The cue ball \(A\) is given an initial velocity \((v_{A})_{1} = 5 \mathrm{m/s}\). If it makes a direct collision with ball \(B\) \((e = 0.8)\), determine the velocity of \(B\) and the angle \(\theta\) just after it rebounds from the cushion at \(C(e^{\prime} = 0.6)\). Each ball has a mass of \(0.4 \mathrm{kg}\). Neglect their size. figure — add your own
  15. 74.The two disks \(A\) and \(B\) have a mass of \(3 \mathrm{kg}\) and \(5 \mathrm{kg}\), respectively. If they collide with the initial velocities shown, determine their velocities just after impact. The coefficient of restitution is \(e = 0.65\). figure — add your own
  16. 75.The pile \(P\) has a mass of \(800 \mathrm{kg}\) and is being driven into loose sand using the \(300 \mathrm{-kg}\) hammer \(C\) which is dropped a distance of \(0.5 \mathrm{m}\) from the top of the pile. Determine the initial speed of the pile just after it is struck by the hammer. The coefficient of restitution between the hammer and the pile is \(e = 0.1\). Neglect the impulses due to the weights of the pile and hammer and the impulse due to the sand during the impact. figure — add your own
  17. 76.The pile \(P\) has a mass of \(800 \mathrm{kg}\) and is being driven into loose sand using the \(300 \mathrm{-kg}\) hammer \(C\) which is dropped a distance of \(0.5 \mathrm{m}\) from the top of the pile. Determine the distance the pile is driven into the sand after one blow if the sand offers a frictional resistance against the pile of \(18 \mathrm{kN}\). The coefficient of restitution between the hammer and the pile is \(e = 0.1\). Neglect the impulses due to the weights of the pile and hammer and the impulse due to the sand during the impact. figure — add your own
  18. 77.A ball of mass \(m\) is dropped vertically from a height \(h_0\) above the ground. If it rebounds to a height of \(h_1\), determine the coefficient of restitution between the ball and the ground. figure — add your own
  19. 78.Two smooth disks \(A\) and \(B\) each have a mass of \(0.5\mathrm{kg}\). If both disks are moving with the velocities shown when they collide, determine their final velocities just after collision. The coefficient of restitution is \(e = 0.75\). figure — add your own
  20. 79.Two smooth disks \(A\) and \(B\) each have a mass of \(0.5\mathrm{kg}\). If both disks are moving with the velocities shown when they collide, determine the coefficient of restitution between the disks if after collision \(B\) travels along a line, \(30^{\circ}\) counterclockwise from the \(y\) axis. figure — add your own
  21. 80.A ball of negligible size and mass \(m\) is given a velocity of \(\mathbf{v}_{0}\) on the center of the cart which has a mass \(M\) and is originally at rest. If the coefficient of restitution between the ball and walls \(A\) and \(B\) is \(e\) determine the velocity of the ball and the cart just after the ball strikes \(A\). Also, determine the total time needed for the ball to strike \(A\), rebound, then strike \(B\) and rebound and then return to the center of the cart. Neglect friction. figure — add your own
  22. 81.The 2-kg ball is thrown at the suspended 20-kg block with a velocity of \(4\mathrm{m/s}\). If the coefficient of restitution between the ball and the block is \(e = 0.8\) determine the maximum height \(h\) to which the block will swing before it momentarily stops. figure — add your own
  23. 82.The 2-kg ball is thrown at the suspended 20-kg block with a velocity of \(4\mathrm{m/s}\). If the time of impact between the ball and the block is \(0.005\mathrm{s}\), determine the average normal force exerted on the block during this time. Take \(e = 0.8\). figure — add your own
  24. 83.The three balls each have the same mass \(m\). If \(A\) is released from rest at \(\theta\) determine the angle \(\phi\) to which \(C\) rises after collision. The coefficient of restitution between each ball is \(e\). figure — add your own
  25. 84.The girl throws the \(0.5\mathrm{-kg}\) ball toward the wall with an initial velocity \(v_{A} = 10\mathrm{m/s}\). Determine (a) the velocity at which it strikes the wall at \(B\), (b) the velocity at which it rebounds from the wall if the coefficient of restitution \(e = 0.5\), and (c) the distance \(s\) from the wall to where it strikes the ground at \(C\). figure — add your own
  26. 85.The 1-kg ball is dropped from rest at point \(A\), \(2\mathrm{m}\) above the smooth plane. If the coefficient of restitution between the ball and the plane is \(e = 0.6\) determine the distance \(d\) where the ball again strikes the plane. figure — add your own
  27. 86.Disks \(A\) and \(B\) have a mass of \(15\mathrm{kg}\) and \(10\mathrm{kg}\) respectively. If they are sliding on a smooth horizontal plane with the velocities shown, determine their speeds just after impact. The coefficient of restitution between them is \(e = 0.8\). figure — add your own
  28. 87.Two smooth billiard balls \(A\) and \(B\) each have a mass of \(200\mathrm{g}\). If \(A\) strikes \(B\) with a velocity \((v_{A})_{1} = 1.5\mathrm{m/s}\) as shown, determine their final velocities just after collision. Ball \(B\) is originally at rest and the coefficient of restitution is \(e = 0.85\). Neglect the size of each ball. figure — add your own
  29. 88.A ball is thrown onto a rough floor at an angle \(\theta\). If it rebounds at an angle \(\phi\) and the coefficient of kinetic friction is \(\mu\) determine the coefficient of restitution \(e\). Neglect the size of the ball. Hint: Show that during impact, the average impulses in the \(x\) and \(y\) directions are related by \(I_{x} = \mu I_{y}\). Since the time of impact is the same, \(F_{x}\Delta t = \mu F_{y}\Delta t\) or \(F_{x} = \mu F_{y}\). figure — add your own
  30. 89.A ball is thrown onto a rough floor at an angle of \(\theta = 45^{\circ}\). If it rebounds at the same angle \(\phi = 45^{\circ}\), determine the coefficient of kinetic friction between the floor and the ball. The coefficient of restitution is \(e = 0.6\). Hint: Show that during impact, the average impulses in the \(x\) and \(y\) directions are related by \(I_{x} = \mu I_{y}\). Since the time of impact is the same, \(F_{x}\Delta t = \mu F_{y}\Delta t\) or \(F_{x} = \mu F_{y}\). figure — add your own
  31. 90.The two billiard balls \(A\) and \(B\) are originally in contact with one another when a third ball \(C\) strikes each of them at the same time as shown. If ball \(C\) remains at rest after the collision, determine the coefficient of restitution. All the balls have the same mass. Neglect the size of each ball. figure — add your own
  32. 91.Disks \(A\) and \(B\) have masses of \(2kg\) and \(4kg,\) respectively. If they have the velocities shown, and \(e = 0.4,\) determine their velocities just after direct central impact. figure — add your own
  33. 92.If disk \(A\) is sliding along the tangent to disk \(B\) and strikes \(B\) with a velocity \(\mathbf{v},\) determine the velocity of \(B\) after the collision and compute the loss of kinetic energy during the collision. Neglect friction. Disk \(B\) is originally at rest. The coefficient of restitution is \(e,\) and each disk has the same size and mass \(m\). figure — add your own
  34. 93.Two smooth disks \(A\) and \(B\) have the initial velocities shown just before they collide. If they have masses \(m_{A} = 4 \mathrm{kg}\) and \(m_{B} = 2 \mathrm{kg},\) determine their speeds just after impact. The coefficient of restitution is \(e = 0.8\). figure — add your own
  35. 94.The 200-g billiard ball is moving with a speed of \(2.5m/s\) when it strikes the side of the pool table at \(A.\) If the coefficient of restitution between the ball and the side of the table is \(e = 0.6,\) determine the speed of the ball just after striking the table twice, i.e., at \(A,\) then at \(B.\) Neglect the size of the ball. figure — add your own
  36. 95.The 0.5-kg ball is fired from the tube at \(A\) with a velocity of \(v = 6 \mathrm{m/s}\). If the coefficient of restitution between the ball and the surface is \(e = 0.8\), determine the height \(h\) after it bounces off the surface. figure — add your own
15.4 Angular Momentum 27 problems
Fundamental Problems
  1. F15-19The 2-kg particle \(A\) has the velocity shown. Determine its angular momentum \(\mathbf{H}_O\) about point \(O\). figure — add your own
  2. F15-20The 2-kg particle \(A\) has the velocity shown. Determine its angular momentum \(\mathbf{H}_P\) about point \(P\). figure — add your own
  3. F15-21Initially the \(5\mathrm{-kg}\) block is moving with a constant speed of \(2\mathrm{m/s}\) around the circular path centered at \(O\) on the smooth horizontal plane. If a constant tangential force \(F = 5\mathrm{N}\) is applied to the block, determine its speed when \(t = 3\mathrm{s}\). Neglect the size of the block. figure — add your own
  4. F15-22The 5-kg block is moving around the circular path centered at \(O\) on the smooth horizontal plane when it is subjected to the force \(F = (10t)\mathrm{N}\), where \(t\) is in seconds. If the block starts from rest, determine its speed when \(t = 4\mathrm{s}\). Neglect the size of the block. The force maintains the same constant angle tangent to the path. figure — add your own
  5. F15-23The 2-kg sphere is attached to the light rigid rod, which rotates in the horizontal plane centered at \(O\). If the system is subjected to a couple moment \(M = (0.9t^2)\mathrm{N}\cdot \mathrm{m}\) where \(t\) is in seconds, determine the speed of the sphere at the instant \(t = 5\) s starting from rest. figure — add your own
  6. F15-24Two identical 10-kg spheres are attached to the light rigid rod, which rotates in the horizontal plane centered at pin \(O\). If the spheres are subjected to tangential forces of \(P = 10\mathrm{N}\), and the rod is subjected to a couple moment \(M = (8t)\mathrm{N}\cdot \mathrm{m}\), where \(t\) is in seconds, determine the speed of the spheres at the instant \(t = 4\mathrm{s}\). The system starts from rest. Neglect the size of the spheres. figure — add your own
Skills
  1. 96.Determine the angular momentum \(\mathbf{H}_{O}\) of each of the particles about point \(O\). figure — add your own
  2. 97.Determine the angular momentum \(\mathbf{H}_{p}\) of each of the particles about point \(P\). figure — add your own
  3. 98.Determine the angular momentum \(\mathbf{H}_{o}\) of each of the two particles about point \(O\). figure — add your own
  4. 99.Determine the angular momentum \(\mathbf{H}_{p}\) of each of the two particles about point \(P\). figure — add your own
  5. 100.Determine the angular momentum \(\mathbf{H}_{O}\) of the 3-kg particle about point \(O\). figure — add your own
  6. 101.Determine the angular momentum \(\mathbf{H}_{p}\) of the 3-kg particle about point \(P\). figure — add your own
  7. 102.Each ball has a negligible size and a mass of \(10\mathrm{kg}\) and is attached to the end of a rod whose mass may be neglected. If the rod is subjected to a torque \(M = (t^{2} + 2)\mathrm{N}\cdot \mathrm{m}\) where \(t\) is in seconds, determine the speed of each ball when \(t = 3\) s. Each ball has a speed \(v = 2\) m/s when \(t = 0\). figure — add your own
  8. 103.The two spheres each have a mass of \(3\mathrm{kg}\) and are attached to the rod of negligible mass. If a torque \(M = (6e^{0.2t})\mathrm{N}\cdot \mathrm{m}\), where \(t\) is in seconds, is applied to the rod as shown, determine the speed of each of the spheres in 2 s, starting from rest. figure — add your own
  9. 104.The two spheres each have a mass of \(3\mathrm{kg}\) and are attached to the rod of negligible mass. Determine the time the torque \(M = (8t)\mathrm{N}\cdot \mathrm{m}\), where \(t\) is in seconds, must be applied to the rod so that each sphere attains a speed of \(3\mathrm{m/s}\) starting from rest. figure — add your own
  10. 105.If the rod of negligible mass is subjected to a couple moment of \(M = (30t^2)\mathrm{N}\cdot \mathrm{m}\), and the engine of the car supplies a traction force of \(F = (15t)\mathrm{N}\) to the wheels, where \(t\) is in seconds, determine the speed of the car at the instant \(t = 5\) s. The car starts from rest. The total mass of the car and rider is \(150\mathrm{kg}\). Neglect the size of the car. figure — add your own
  11. 106.A child having a mass of \(50\mathrm{kg}\) holds her legs up as shown as she swings downward from rest at \(\theta_{1} = 30^{\circ}\). Her center of mass is located at point \(G_{1}\). When she is at the bottom position \(\theta = 0^{\circ}\), she suddenly lets her legs come down, shifting her center of mass to position \(G_{2}\). Determine the speed in the upswing due to this sudden movement and the angle \(\theta_{2}\) to which she swings before momentarily coming to rest. Treat the child's body as a particle. figure — add your own
  12. 107.When the \(2\mathrm{-kg}\) bob is given a horizontal speed of \(1.5\mathrm{m/s}\) it begins to move around the horizontal circular path \(A\). If the force \(\mathbf{F}\) on the cord is increased, the bob rises and then moves around the horizontal circular path \(B\). Determine the speed of the bob around path \(B\). Also, find the work done by force \(\mathbf{F}\). figure — add your own
  13. 108.A small particle having a mass \(m\) is placed inside the semicircular tube. The particle is placed at the position shown and released. Apply the principle of angular momentum about point \(O\) (\(\Sigma M_{O} = H_{O}\)), and show that the motion of the particle is governed by the differential equation \(\ddot{\theta} +(g / R)\) sin \(\theta = 0\). figure — add your own
  14. 109.At the instant \(r = 1.5\mathrm{m}\) the \(5\mathrm{-kg}\) disk is given a speed of \(v = 5\mathrm{m/s}\) perpendicular to the elastic cord. Determine the speed of the disk and the rate of shortening of the elastic cord at the instant \(r = 1.2\mathrm{m}\). The disk slides on the smooth horizontal plane. Neglect its size. The cord has an unstretched length of \(0.5\mathrm{m}\). figure — add your own
  15. 110.The two blocks \(A\) and \(B\) each have a mass of \(400\mathrm{g}\). The blocks are fixed to the horizontal rods, and their initial velocity along the circular path is \(2\mathrm{m/s}\). If a couple moment of \(M = 0.6\mathrm{N}\cdot \mathrm{m}\) is applied about \(CD\) of the frame, determine the speed of the blocks when \(t = 3\mathrm{s}\). The mass of the frame is negligible, and it is free to rotate about \(CD\). Neglect the size of the blocks. figure — add your own
  16. 111.The ball \(B\) has mass of \(10\mathrm{kg}\) and is attached to the end of a rod whose mass may be neglected. If the rod is subjected to a torque \(M = (3t^{2} + 5t + 2)\mathrm{N}\cdot \mathrm{m}\), where \(t\) is in seconds, determine the speed of the ball when \(t = 2\mathrm{s}\). The ball has a speed \(v = 2\mathrm{m/s}\) when \(t = 0\). figure — add your own
  17. 112.The amusement park ride consists of a 200-kg car and passenger that are traveling at \(3\mathrm{m/s}\) along a circular path having a radius of \(8\mathrm{m}\). If at \(t = 0\) the cable \(OA\) is pulled in toward \(O\) at \(0.5\mathrm{m/s}\), determine the speed of the car when \(t = 4\) s. Also, determine the work done to pull in the cable. figure — add your own
  18. 113.A small block having a mass of \(0.1\mathrm{kg}\) is given a horizontal velocity \(v_{1} = 0.4\mathrm{m/s}\) of when \(r_1 = 500 \mathrm{mm}\). It slides along the smooth conical surface. Determine the distance \(h\) it must descend for it to reach a speed of \(v_{2} = 2 \mathrm{m/s}\). Also, what is the angle of descent \(\theta\), that is, the angle measured from the horizontal to the tangent of the path? figure — add your own
  19. 114.A toboggan and rider, having a total mass of \(150\mathrm{kg}\), enter horizontally tangent to a \(90^{\circ}\) circular curve with a velocity of \(v_{A} = 70 \mathrm{km/h}\). If the track is flat and banked at an angle of \(60^{\circ}\), determine the speed \(v_{B}\) and the angle \(\theta\) of "descent," measured from the horizontal in a vertical \(x - z\) plane, at which the toboggan exists at \(B\). Neglect friction in the calculation. figure — add your own
  20. 115.An earth satellite of mass \(700 \mathrm{kg}\) is launched into a free-flight trajectory about the earth with an initial speed of \(v_{A} = 10 \mathrm{km/s}\) when the distance from the center of the earth is \(r_{A} = 15 \mathrm{Mm}\). If the launch angle at this position is \(\phi_{A} = 70^{\circ}\), determine the speed \(v_{B}\) of the satellite and its closest distance \(r_{B}\) from the center of the earth. The earth has a mass \(M_{e} = 5.976(10^{24}) \mathrm{kg}\). Hint: Under these conditions, the satellite is subjected only to the earth's gravitational force, \(F = GM_{e}m_{s} / r^{2}\), Eq. 13-1. For part of the solution, use the conservation of energy. figure — add your own
  21. 116.The fire boat discharges two streams of seawater, each at a flow of \(0.25\mathrm{m}^3 /\mathrm{s}\) and with a nozzle velocity of \(50~\mathrm{m/s}\). Determine the tension developed in the anchor chain, needed to secure the boat. The density of seawater is \(\rho_{sw} = 1020\mathrm{kg/m}^3\). figure — add your own
  22. 117.The chute is used to divert the flow of water, \(Q = 0.6\mathrm{m}^3/\mathrm{s}\). If the water has a cross-sectional area of \(0.05\mathrm{m}^2\), determine the force components at the pin \(D\) and roller \(C\) necessary for equilibrium. Neglect the weight of the chute and weight of the water on the chute. \(\rho_{w} = 1\mathrm{Mg/m}^3\). figure — add your own
  23. 118.The \(200\mathrm{-kg}\) boat is powered by the fan which develops a slipstream having a diameter of \(0.75\mathrm{m}\). If the fan ejects air with a speed of \(14\mathrm{m/s}\), measured relative to the boat, determine the initial acceleration of the boat if it is initially at rest. Assume that air has a constant density of \(\rho_{w} = 1.22\mathrm{kg/m}^3\) and that the entering air is essentially at rest. Neglect the drag resistance of the water. figure — add your own
15.5 Steady Flow of a Fluid Stream 18 problems
Skills
  1. 119.The bend is connected to the pipe at flanges \(A\) and \(B\) as shown. If the diameter of the pipe is \(0.3\mathrm{m}\) and it carries a discharge of \(1.35\mathrm{m}^3 /\mathrm{s}\), determine the horizontal and vertical components of force reaction and the moment reaction exerted at the fixed base \(D\) of the support. The total weight of the bend and the water within it is \(2500\mathrm{N}\), with a mass center at point \(G\). The gauge pressure of the water at the flanges at \(A\) and \(B\) are \(120\mathrm{kN/m}^2\) and \(96\mathrm{kN/m}^2\), respectively. Assume that no force is transferred to the flanges at \(A\) and \(B\). The specific weight of water is \(\gamma_{w} = 10\mathrm{kN/m}^3\). figure — add your own
  2. 120.Water is discharged at \(16\mathrm{m/s}\) against the fixed cone diffuser. If the opening diameter of the nozzle is \(40\mathrm{mm}\), determine the horizontal force exerted by the water on the diffuser, \(\rho_{w} = 1\mathrm{Mg/m}^3\). figure — add your own
  3. 121.The hemispherical bowl of mass \(m\) is held in equilibrium by the vertical jet of water discharged through a nozzle of diameter \(d\). If the discharge of the water through the nozzle is \(Q\), determine the height \(h\) at which the bowl is suspended. The water density is \(\rho_{w}\). Neglect the weight of the water jet. figure — add your own
  4. 122.A snowblower having a scoop \(S\) with a cross-sectional area of \(A_{s} = 0.12\mathrm{m}^3\) is pushed into snow with a speed of \(v_{s} = 0.5\mathrm{m/s}\). The machine discharges the snow through a tube \(T\) that has a cross-sectional area of \(A_{T} = 0.03\mathrm{m}^2\) and is directed \(60^{\circ}\) from the horizontal. If the density of snow is \(\rho_{s} = 104\mathrm{kg/m}^3\), determine the horizontal force \(P\) required to push the blower forward, and the resultant frictional force \(F\) of the wheels on the ground, necessary to prevent the blower from moving sideways. The wheels roll freely. figure — add your own
  5. 123.The gauge pressure of water at \(A\) is \(150.5\mathrm{kPa}\). Water flows through the pipe at \(A\) with a velocity of \(18~\mathrm{m/s}\) and out the pipe at \(B\) and \(C\) with the same velocity \(v\). Determine the horizontal and vertical components of force exerted on the elbow necessary to hold the pipe assembly in equilibrium. Neglect the weight of water within the pipe and the weight of the pipe. The pipe has a diameter of \(50~\mathrm{mm}\) at \(A\) and at \(B\) and \(C\) the diameter is \(30~\mathrm{mm}\). \(\rho_{w} = 1000\mathrm{kg/m}^3\). figure — add your own
  6. 124.A scoop in front of the tractor collects snow at a rate of \(200~\mathrm{kg/s}\). Determine the resultant traction force \(\mathbf{T}\) that must be developed on all the wheels as it moves forward on level ground at a constant speed of \(5\mathrm{km/h}\). The tractor has a mass of \(5\mathrm{Mg}\). figure — add your own
  7. 125.The boat has a mass of \(180~\mathrm{kg}\) and is traveling forward on a river with a constant velocity of \(70~\mathrm{km/h}\) measured relative to the river. The river is flowing in the opposite direction at \(5\mathrm{km/h}\). If a tube is placed in the water, as shown, and it collects \(40~\mathrm{kg}\) of water in the boat in \(80~\mathrm{s}\), determine the horizontal thrust \(T\) on the tube that is required to overcome the resistance due to the water collection and yet maintain the constant speed of the boat. \(\rho_{w} = 1\mathrm{Mg/m}^3\). figure — add your own
  8. 126.Water is flowing from the \(150\mathrm{-mm}\)-diameter fire hydrant with a velocity \(v_{B} = 15\mathrm{m/s}\). Determine the horizontal and vertical components of force and the moment developed at the base joint \(A\), if the static (gauge) pressure at \(A\) is \(50~\mathrm{kPa}\). The diameter of the fire hydrant at \(A\) is \(200~\mathrm{mm}\). \(\rho_{w} = 1\mathrm{Mg/m}^3\). figure — add your own
  9. 127.Water is discharged from a nozzle with a velocity of \(12~\mathrm{m/s}\) and strikes the blade mounted on the \(20\mathrm{-kg}\) cart. Determine the tension developed in the cord, needed to hold the cart stationary, and the normal reaction of the wheels on the cart. The nozzle has a diameter of \(50~\mathrm{mm}\) and the density of water is \(\rho_{w} = 1000\mathrm{kg/m}^3\). figure — add your own
  10. 128.When operating, the air-jet fan discharges air with a speed of \(v_{B} = 20 \mathrm{m/s}\) into a slipstream having a diameter of \(0.5 \mathrm{m}\). If air has a density of \(1.22 \mathrm{kg/m^3}\), determine the horizontal and vertical components of reaction at \(C\) and the vertical reaction at each of the two wheels, \(D\), when the fan is in operation. The fan and motor have a mass of \(20 \mathrm{kg}\) and a center of mass at \(G\). Neglect the weight of the frame. Due to symmetry, both of the wheels support an equal load. Assume the air entering the fan at \(A\) is essentially at rest. figure — add your own
  11. 129.Sand is discharged from the silo at \(A\) at a rate of \(50 \mathrm{kg/s}\) with a vertical velocity of \(10 \mathrm{m/s}\) onto the conveyor belt, which is moving with a constant velocity of \(1.5 \mathrm{m/s}\). If the conveyor system and the sand on it have a total mass of \(750 \mathrm{kg}\) and center of mass at point \(G\), determine the horizontal and vertical components of reaction at the pin support \(B\) and roller support \(A\). Neglect the thickness of the conveyor. figure — add your own
  12. 130.Each of the two stages \(A\) and \(B\) of the rocket has a mass of \(2 \mathrm{Mg}\) when their fuel tanks are empty. They each carry \(500 \mathrm{kg}\) of fuel and are capable of consuming it at a rate of \(50 \mathrm{kg/s}\) and eject it with a constant velocity of \(2500 \mathrm{m/s}\), measured with respect to the rocket. The rocket is launched vertically from rest by first igniting stage \(B\). Then stage \(A\) is ignited immediately after all the fuel in \(B\) is consumed and \(A\) has separated from \(B\). Determine the maximum velocity of stage \(A\). Neglect drag resistance and the variation of the rocket's weight with altitude. figure — add your own
  13. 131.Sand is deposited from a chute onto a conveyor belt which is moving at \(0.5 \mathrm{m/s}\). If the sand is assumed to fall vertically onto the belt at \(A\) at the rate of \(4 \mathrm{kg/s}\), determine the belt tension \(F_{B}\) to the right of \(A\). The belt is free to move over the conveyor rollers and its tension to the left of \(A\) is \(F_{C} = 400 \mathrm{N}\). figure — add your own
  14. 132.The water flow enters below the hydrant at \(C\) at the rate of \(0.75\mathrm{m}^{3}/\mathrm{s}\). It is then divided equally between the two outlets at \(A\) and \(B\). If the gauge pressure at \(C\) is \(300\mathrm{kPa}\), determine the horizontal and vertical force reactions and the moment reaction on the fixed support at \(C\). The diameter of the two outlets at \(A\) and \(B\) is \(75\mathrm{mm}\) and the diameter of the inlet pipe at \(C\) is \(150~\mathrm{mm}\). The density of water is \(\rho_{w} = 1000\mathrm{kg/m}^{3}\). Neglect the mass of the contained water and the hydrant. figure — add your own
  15. 133.The nozzle has a diameter of \(40\mathrm{mm}\). If it discharges water uniformly with a downward velocity of \(20m/s\) against the fixed blade, determine the vertical force exerted by the water on the blade. \(\rho_{w} = 1\mathrm{Mg/m}^{3}\). figure — add your own
  16. 134.Sand drops onto the \(2\mathrm{-Mg}\) empty rail car at \(50\mathrm{kg/s}\) from a conveyor belt. If the car is initially coasting at \(4\mathrm{m/s}\), determine the speed of the car as a function of time. figure — add your own
  17. 135.The tractor together with the empty tank has a total mass of \(4\mathrm{Mg}\). The tank is filled with \(2\mathrm{Mg}\) of water. The water is discharged at a constant rate of \(50\mathrm{kg/s}\) with a constant velocity of \(5\mathrm{m/s}\), measured relative to the tractor. If the tractor starts from rest, and the rear wheels provide a resultant traction force of \(250\mathrm{N}\), determine the velocity and acceleration of the tractor at the instant the tank becomes empty. figure — add your own
  18. 136.The earthmover initially carries \(10\mathrm{m}^{3}\) of sand having a density of \(1520\mathrm{kg/m}^{3}\). The sand is unloaded horizontally through a \(2.5\mathrm{m}^{3}\) dumping port \(P\) at a rate of \(900\mathrm{kg/s}\) measured relative to the port. Determine the resultant tractive force \(\mathbf{F}\) at its front wheels if the acceleration of the earthmover is \(0.1\mathrm{m/s}^{2}\) when half the sand is dumped. When empty, the earthmover has a mass of \(30\mathrm{Mg}\). Neglect any resistance to forward motion and the mass of the wheels. The rear wheels are free to roll. figure — add your own
  19. 137.A power lawn mower hovers very close over the ground. This is done by drawing air in at a speed of \(6\mathrm{m/s}\) through an intake unit \(A\), which has a cross-sectional area of \(A_{A} = 0.25\mathrm{m}^{2}\) and then discharging it at the ground, \(B\) where the cross-sectional area is \(A_{B} = 0.35\mathrm{m}^{2}\). If air at \(A\) is subjected only to atmospheric pressure, determine the air pressure which the lawn mower exerts on the ground when the weight of the mower is freely supported and no load is placed on the handle. The mower has a mass of \(15\mathrm{kg}\) with center of mass at \(G\). Assume that air has a constant density of \(\rho_{a} = 1.22\mathrm{kg/m}^{3}\). figure — add your own
15.6 Propulsion with Variable Mass 22 problems
Skills
  1. 138.The rocket has an initial mass \(m_{0}\), including the fuel. For practical reasons desired for the crew, it is required that it maintain a constant upward acceleration \(a_{0}\). If the fuel is expelled from the rocket at a relative speed \(v_{e/r}\), determine the rate at which the fuel should be consumed to maintain the motion. Neglect air resistance, and assume that the gravitational acceleration is constant. figure — add your own
  2. 139.The \(12\mathrm{-Mg}\) jet airplane has a constant speed of \(950~\mathrm{km/h}\) when it is flying along a horizontal straight line. Air enters the intake scoops \(S\) at the rate of \(50~\mathrm{m^3/s}\). If the engine burns fuel at the rate of \(0.4\mathrm{kg/s}\) and the gas (air and fuel) is exhausted relative to the plane with a speed of \(450~\mathrm{m/s}\), determine the resultant drag force exerted on the plane by air resistance. Assume that air has a constant density of \(1.22\mathrm{kg/m^3}\). Hint: Since mass both enters and exits the plane, Eqs. 15-28 and 15-29 must be combined to yield \(\Sigma F_{s} = m\frac{dv}{dt} -v_{D/e}\frac{dm_{e}}{dt} +v_{D/i}\frac{dm_{i}}{dt}\). figure — add your own
  3. 140.The jet is traveling at a speed of \(720~\mathrm{km/h}\). If the fuel is being spent at \(0.8\mathrm{kg/s}\), and the engine takes in air at \(200~\mathrm{kg/s}\), whereas the exhaust gas (air and fuel) has a relative speed of \(12000\mathrm{m/s}\), determine the acceleration of the plane at this instant. The drag resistance of the air is \(F_{D} = (55v^{2})\), where the speed is measured in \(\mathrm{m/s}\). The jet has a mass of \(7\mathrm{Mg}\). figure — add your own
  4. 141.The rope has a mass \(m^{\prime}\) per unit length. If the end length \(y = h\) is draped off the edge of the table, and released, determine the velocity of its end \(A\) for any position \(y\) as the rope uncoils and begins to fall. figure — add your own
  5. 142.A coil of heavy open chain is used to reduce the stopping distance of a sled that has a mass \(M\) and travels at a speed of \(v_{0}\). Determine the required mass per unit length of the chain needed to slow down the sled to \((1/2)v_{0}\) within a distance \(x = s\) if the sled is hooked to the chain at \(x = 0\). Neglect friction between the chain and the ground. figure — add your own
  6. 143.A four-engine commercial jumbo jet is cruising at a constant speed of \(800~\mathrm{km/h}\) in level flight when all four engines are in operation. Each of the engines is capable of discharging combustion gases with a velocity of \(775~\mathrm{m/s}\) relative to the plane. If during a test two of the engines, one on each side of the plane, are shut off, determine the new cruising speed of the jet. Assume that air resistance (drag) is proportional to the square of the speed, that is, \(F_{D} = c v^{2}\), where \(c\) is a constant to be determined. Neglect the loss of mass due to fuel consumption. figure — add your own
  7. 144.A commercial jet aircraft has a mass of \(150~\mathrm{Mg}\) and is cruising at a constant speed of \(850~\mathrm{km/h}\) in level flight \((\theta = 0^{\circ})\). If each of the two engines draws in air at a rate of \(1000\mathrm{kg/s}\) and ejects it with a velocity of \(900~\mathrm{m/s}\), relative to the aircraft, determine the maximum angle of inclination \(\theta\) at which the aircraft can fly with a constant speed of \(750~\mathrm{km/h}\). Assume that air resistance (drag) is proportional to the square of the speed, that is, \(F_{D} = c v^{2}\), where \(c\) is a constant to be determined. The engines are operating with the same power in both cases. Neglect the amount of fuel consumed. figure — add your own
  8. 145.A coil of heavy open chain is used to reduce the stopping distance of a sled that has a mass \(M\) and travels at a speed of \(v_{0}\). Determine the required mass per unit length of the chain needed to slow down the sled to \((1/4)v_{0}\) within a distance \(x = s\) if the sled is hooked to the chain at \(x = 0\). Neglect friction between the chain and the ground. figure — add your own
  9. 146.Determine the magnitude of force \(\mathbf{F}\) as a function of time, which must be applied to the end of the cord at \(A\) to raise the hook \(H\) with a constant speed \(v = 0.4\mathrm{m/s}\). Initially the chain is at rest on the ground. Neglect the mass of the cord and the hook. The chain has a mass of \(2\mathrm{kg/m}\). figure — add your own
  10. 147.The \(10\mathrm{-Mg}\) helicopter carries a bucket containing \(500~\mathrm{kg}\) of water, which is used to fight fires. If it hovers over the land in a fixed position and then releases \(50~\mathrm{kg/s}\) of water at \(10\mathrm{m/s}\) measured relative to the helicopter, determine the initial upward acceleration the helicopter experiences as the water is being released. figure — add your own
  11. 148.The truck has a mass of \(50~\mathrm{Mg}\) when empty. When it is unloading \(5\mathrm{m}^3\) of sand at a constant rate of \(0.8\mathrm{m}^3/\mathrm{s}\) the sand flows out the back at a speed of \(7\mathrm{m/s}\) measured relative to the truck, in the direction shown. If the truck is free to roll, determine its initial acceleration just as the load begins to empty. Neglect the mass of the wheels and any frictional resistance to motion. The density of sand is \(\rho_{s} = 1520\mathrm{kg/m^{3}}\). figure — add your own
  12. 149.The car has a mass \(m_{0}\) and is used to tow the smooth chain having a total length \(l\) and a mass per unit of length \(m^{\prime}\). If the chain is originally piled up, determine the tractive force \(F\) that must be supplied by the rear wheels of the car, necessary to maintain a constant speed \(v\) while the chain is being drawn out. figure — add your own
  13. 150.The rope has a mass \(m^{\prime}\) per unit length. If the end length \(y = h\) is draped off the edge of the table, and released, determine the velocity of its end \(A\) for any position \(y\) as the rope uncoils and begins to fall. figure — add your own
  14. 151.A coil of heavy open chain is used to reduce the stopping distance of a sled that has a mass \(M\) and travels at a speed of \(v_{0}\). Determine the required mass per unit length of the chain needed to slow down the sled to \((1/2)v_{0}\) within a distance \(x = s\) if the sled is hooked to the chain at \(x = 0\). Neglect friction between the chain and the ground. figure — add your own
  15. 152.A four-engine commercial jumbo jet is cruising at a constant speed of \(800~\mathrm{km/h}\) in level flight when all four engines are in operation. Each of the engines is capable of discharging combustion gases with a velocity of \(775~\mathrm{m/s}\) relative to the plane. If during a test two of the engines, one on each side of the plane, are shut off, determine the new cruising speed of the jet. Assume that air resistance (drag) is proportional to the square of the speed, that is, \(F_{D} = c v^{2}\), where \(c\) is a constant to be determined. Neglect the loss of mass due to fuel consumption. figure — add your own
  16. 153.A commercial jet aircraft has a mass of \(150~\mathrm{Mg}\) and is cruising at a constant speed of \(850~\mathrm{km/h}\) in level flight \((\theta = 0^{\circ})\). If each of the two engines draws in air at a rate of \(1000\mathrm{kg/s}\) and ejects it with a velocity of \(900~\mathrm{m/s}\), relative to the aircraft, determine the maximum angle of inclination \(\theta\) at which the aircraft can fly with a constant speed of \(750~\mathrm{km/h}\). Assume that air resistance (drag) is proportional to the square of the speed, that is, \(F_{D} = c v^{2}\), where \(c\) is a constant to be determined. The engines are operating with the same power in both cases. Neglect the amount of fuel consumed. figure — add your own
  17. 154.A coil of heavy open chain is used to reduce the stopping distance of a sled that has a mass \(M\) and travels at a speed of \(v_{0}\). Determine the required mass per unit length of the chain needed to slow down the sled to \((1/4)v_{0}\) within a distance \(x = s\) if the sled is hooked to the chain at \(x = 0\). Neglect friction between the chain and the ground. figure — add your own
  18. 155.Determine the magnitude of force \(\mathbf{F}\) as a function of time, which must be applied to the end of the cord at \(A\) to raise the hook \(H\) with a constant speed \(v = 0.4\mathrm{m/s}\). Initially the chain is at rest on the ground. Neglect the mass of the cord and the hook. The chain has a mass of \(2\mathrm{kg/m}\). figure — add your own
  19. 156.The \(10\mathrm{-Mg}\) helicopter carries a bucket containing \(500~\mathrm{kg}\) of water, which is used to fight fires. If it hovers over the land in a fixed position and then releases \(50~\mathrm{kg/s}\) of water at \(10\mathrm{m/s}\) measured relative to the helicopter, determine the initial upward acceleration the helicopter experiences as the water is being released. figure — add your own
  20. 157.The truck has a mass of \(50~\mathrm{Mg}\) when empty. When it is unloading \(5\mathrm{m}^3\) of sand at a constant rate of \(0.8\mathrm{m}^3/\mathrm{s}\) the sand flows out the back at a speed of \(7\mathrm{m/s}\) measured relative to the truck, in the direction shown. If the truck is free to roll, determine its initial acceleration just as the load begins to empty. Neglect the mass of the wheels and any frictional resistance to motion. The density of sand is \(\rho_{s} = 1520\mathrm{kg/m^{3}}\). figure — add your own
  21. 158.The car has a mass \(m_{0}\) and is used to tow the smooth chain having a total length \(l\) and a mass per unit of length \(m^{\prime}\). If the chain is originally piled up, determine the tractive force \(F\) that must be supplied by the rear wheels of the car, necessary to maintain a constant speed \(v\) while the chain is being drawn out. figure — add your own
  22. 159.The rope has a mass \(m^{\prime}\) per unit length. If the end length \(y = h\) is draped off the edge of the table, and released, determine the velocity of its end \(A\) for any position \(y\) as the rope uncoils and begins to fall. figure — add your own
Review Fundamental Review Problems 8 problems
Fundamental Review
  1. R15-1Packages having a mass of \(6\mathrm{kg}\) slide down a smooth chute and land horizontally with a speed of \(3\mathrm{m/s}\) on the surface of a conveyor belt. If the coefficient of kinetic friction between the belt and a package is \(\mu_{k} = 0.2\), determine the time needed to bring the package to rest on the belt if the belt is moving in the same direction as the package with a speed \(v = 1\mathrm{m/s}\). figure — add your own
  2. R15-2The \(50\mathrm{-kg}\) block is hoisted up the incline using the cable and motor arrangement shown. The coefficient of kinetic friction between the block and the surface is \(\mu_{k} = 0.4\). If the block is initially moving up the plane at \(v_{0} = 2\mathrm{m/s}\), and at this instant \((t = 0)\) the motor develops a tension in the cord of \(T = (300 + 120\sqrt{t})\) N, where \(t\) is in seconds, determine the velocity of the block when \(t = 2\) s. figure — add your own
  3. R15-3A \(20\mathrm{-kg}\) block is originally at rest on a horizontal surface for which the coefficient of static friction is \(\mu_{\mathrm{s}} = 0.6\) and the coefficient of kinetic friction is \(\mu_{\mathrm{k}} = 0.5\). If a horizontal force \(F\) is applied such that it varies with time as shown, determine the speed of the block in \(10\mathrm{s}\). Hint: First determine the time needed to overcome friction and start the block moving. figure — add your own
  4. R15-4The three freight cars \(A, B\) and \(C\) have masses of \(10\mathrm{Mg}\), \(5\mathrm{Mg}\) and \(20\mathrm{Mg}\) respectively. They are traveling along the track with the velocities shown. Car \(A\) collides with car \(B\) first, followed by car \(C\). If the three cars couple together after collision, determine the common velocity of the cars after the two collisions have taken place. figure — add your own
  5. R15-5The \(200 - \mathrm{g}\) projectile is fired with a velocity of \(900~\mathrm{m/s}\) towards the center of the \(15 - \mathrm{kg}\) wooden block, which rests on a rough surface. If the projectile penetrates and emerges from the block with a velocity of \(300~\mathrm{m/s}\), determine the velocity of the block just after the projectile emerges. How long does the block slide on the rough surface, after the projectile emerges, before it comes to rest again? The coefficient of kinetic friction between the surface and the block is \(\mu_{k} = 0.2\). figure — add your own
  6. R15-6The \(200 - \mathrm{g}\) projectile is fired with a velocity of \(900~\mathrm{m/s}\) towards the center of the \(15 - \mathrm{kg}\) wooden block, which rests on a rough surface. If the projectile penetrates and emerges from the block with a velocity of \(300~\mathrm{m/s}\), determine the velocity of the block just after the projectile emerges. How long does the block slide on the rough surface, after the projectile emerges, before it comes to rest again? The coefficient of kinetic friction between the surface and the block is \(\mu_{k} = 0.2\). figure — add your own
  7. R15-7Two smooth billiard balls \(A\) and \(B\) have an equal mass of \(m = 200 \mathrm{g}\). If \(A\) strikes \(B\) with a velocity of \((v_{A})_{1} = 2 \mathrm{m/s}\) as shown, determine their final velocities just after collision. Ball \(B\) is originally at rest and the coefficient of restitution is \(e = 0.75\). figure — add your own
  8. R15-8The small cylinder \(C\) has a mass of \(10 \mathrm{kg}\) and is attached to the end of a rod whose mass may be neglected. If the frame is subjected to a couple \(M = (8t^{2} + 5) \mathrm{N} \cdot \mathrm{m}\) where \(t\) is in seconds, and the cylinder is subjected to a force of \(60 \mathrm{N}\), which is always directed as shown, determine the speed of the cylinder when \(t = 2 \mathrm{s}\). The cylinder has a speed \(v_{0} = 2 \mathrm{m/s}\) when \(t = 0\). figure — add your own