Calculate the vertical acceleration of the systemmass center and the individual vertical

The system of three particles has the indicated particle masses, velocities, and external forces. Determine \(\overline{\mathbf{r}}, \dot{\overline{\mathbf{r}}}, \ddot{\overline{\mathbf{r}}}\), T, \(\mathbf{H}_{O}\), and \(\dot{\mathbf{H}}_{O}\) for this two-dimensional system.

For the particle system of Prob. 4 /1, determine \(\mathbf{H}_{G}\) and \(\dot{\mathbf{H}}_{G}\).

The system of three particles has the indicated particle masses, velocities, and external forces. Determine \(\overline{\mathbf{r}}, \dot{\overline{\mathbf{r}}}, \ddot{\overline{\mathbf{r}}}\), T, \(\mathbf{H}_{O}\), and \(\dot{\mathbf{H}}_{O}\) for this three-dimensional system.

For the particle system of Prob. 4 /3, determine \(\mathbf{H}_{G}\) and \(\dot{\mathbf{H}}_{G}\).

The system consists of the two smooth spheres, each weighing 3 lb and connected by a light spring, and the two bars of negligible weight hinged freely at their ends and hanging in the vertical plane. The spheres are confined to slide in the smooth horizontal guide. If a horizontal force F = 10 lb is applied to the one bar at the position shown, what is the acceleration of the center C of the spring? Why does the result not depend on the dimension b?

The two small spheres, each of mass m, are rigidly connected by a rod of negligible mass and are released from rest in the position shown and slide down the smooth circular guide in the vertical plane. Determine their common velocity v as they reach the horizontal dashed position. Also find the force N between sphere 1 and the supporting surface an instant before the sphere reaches the bottom position A.

The total linear momentum of a system of five particles at time t = 2.2 s is given by \(\mathbf{G}_{2.2}=3.4 \mathbf{i}-2.6 \mathbf{j}+4.6 \mathbf{k} \mathrm{\ kg} \cdot \mathrm{m} / \mathrm{s}\). At time t = 2.4 s, the linear momentum has changed to \(\mathbf{G}_{2.4}=3.7 \mathbf{i}-2.2 \mathbf{j}+4.9 \mathbf{k}\). Calculate the magnitude F of the time average of the resultant of the external forces acting on the system during the interval.

The angular momentum of a system of six particles about a fixed point O at time t = 4s is \(\mathbf{H}_{4}=3.65 \mathbf{i}+4.27 \mathbf{j}-5.36 \mathbf{k} \mathrm{\ kg} \cdot \mathrm{m}^{2} / \mathrm{s}\). At time t = 4.1 s, the angular momentum is \(\mathbf{H}_{4.1}=3.67 \mathbf{i}+4.30 \mathbf{j}-5.20 \mathbf{k} \mathrm{\ kg} \cdot \mathrm{m}^{2} / \mathrm{s}\). Determine the average value of the resultant moment about point O of all forces acting on all particles during the 0.1-s interval.

Three monkeys A, B, and C weighing 20, 25, and 15 lb, respectively, are climbing up and down the rope suspended from D. At the instant represented, A is descending the rope with an acceleration of \(5 \mathrm{\ ft} / \mathrm{sec}^{2}\), and C is pulling himself up with an acceleration of \(3 \mathrm{\ ft} / \mathrm{sec}^{2}\). Monkey B is climbing up with a constant speed of 2 ft/sec. Treat the rope and monkeys as a complete system and calculate the tension T in the rope at D.

The monkeys of Prob. 4 /9 are now climbing along the heavy rope wall suspended from the uniform beam. If monkeys A, B, and C have velocities of 5, 3, and 2 ft/sec, and accelerations of 1.5, 0.5, and \(2 \mathrm{\ ft} / \mathrm{sec}^{2}\), respectively, determine the changes in the reactions at D and E caused by the motion and weight of the monkeys. The support at E makes contact with only one side of the beam at a time. Assume for this analysis that the rope wall remains rigid.

The two spheres, each of mass m, are connected by the spring and hinged bars of negligible mass. The spheres are free to slide in the smooth guide up the incline \(\theta\). Determine the acceleration \(a_{C}\) of the center C of the spring.

Each of the five connected particles has a mass of 0.5 kg, and G is the mass center of the system. At a certain instant the angular velocity of the body is \(\omega=2\) rad /s and the linear velocity of G is \(v_{G}=4\) m /s in the direction shown. Determine the linear momentum of the body and its angular momentum about G and about O.

Calculate the acceleration of the center of mass of the system of the four 10-kg cylinders. Neglect friction and the mass of the pulleys and cables.

The four systems slide on a smooth horizontal surface and have the same mass m. The configurations of mass in the two pairs are identical. What can be said about the acceleration of the mass center for each system? Explain any difference in the accelerations of the members.

Calculate the vertical acceleration of the system mass center and the individual vertical accelerations of spheres 1 and 2 for the cases (a) \(\alpha=\beta\) and (b) \(\alpha \neq \beta\). Each sphere has a mass of 2 kg, and the 50-N force is applied vertically to the junction O of the two light wires. The spheres are initially at rest. State any assumptions.

The two small spheres, each of mass m, and their connecting rod of negligible mass are rotating about their mass center G with an angular velocity \(\omega\). At the same instant the mass center has a velocity v in the x-direction. Determine the angular momentum \(\mathbf{H}_{O}\) of the assembly at the instant when G has coordinates x and y.

A department-store escalator makes an angle of \(30^{\circ}\) with the horizontal and takes 40 seconds to transport a person from the first to the second floor with a vertical rise of 20 ft. At a certain instant, there are 10 people on the escalator averaging 150 lb per person and standing at rest relative to the moving steps. Additionally, three boys averaging 120 lb each are running down the escalator at a speed of 2 ft /sec relative to the moving steps. Calculate the power output P of the driving motor to maintain the constant speed of the escalator. The no-load power without passengers is 2.2 hp to overcome friction in the mechanism.

A centrifuge consists of four cylindrical containers, each of mass m, at a radial distance r from the rotation axis. Determine the time t required to bring the centrifuge to an angular velocity \(\omega\) from rest under a constant torque M applied to the shaft. The diameter of each container is small compared with r, and the mass of the shaft and supporting arms is small compared with m.

The three small spheres are welded to the light rigid frame which is rotating in a horizontal plane about a vertical axis through O with an angular velocity \(\dot{\theta}=20\) rad /s. If a couple \(M_{O}=30 \mathrm{\ N} \cdot \mathrm{m}\) is applied to the frame for 5 seconds, compute the new angular velocity \(\dot{\theta}^{\prime}\).

Billiard ball A is moving in the y-direction with a velocity of 2 m /s when it strikes ball B of identical size and mass initially at rest. Following the impact, the balls are observed to move in the directions shown. Calculate the velocities \(v_{A}\) and \(v_{B}\) which the balls have immediately after the impact. Treat the balls as particles and neglect any friction forces acting on the balls compared with the force of impact.

The 300-kg and 400-kg mine cars are rolling in opposite directions along the horizontal track with the respective speeds of 0.6 m /s and 0.3 m /s. Upon impact the cars become coupled together. Just prior to impact, a 100-kg boulder leaves the delivery chute with a velocity of 1.2 m /s in the direction shown and lands in the 300-kg car. Calculate the velocity v of the system after the boulder has come to rest relative to the car. Would the final velocity be the same if the cars were coupled before the boulder dropped?

The three freight cars are rolling along the horizontal track with the velocities shown. After the impacts occur, the three cars become coupled together and move with a common velocity v. The weights of the loaded cars A, B, and C are 130,000, 100,000, and 150,000 lb, respectively. Determine v and calculate the percentage loss n of energy of the system due to coupling.

The man of mass \(m_{1}\) and the woman of mass \(m_{2}\) are standing on opposite ends of the platform of mass \(m_{0}\) which moves with negligible friction and is initially at rest with s = 0. The man and woman begin to approach each other. Derive an expression for the displacement s of the platform when the two meet in terms of the displacement \(x_{1}\) of the man relative to the platform.

The woman A, the captain B, and the sailor C weigh 120, 180, and 160 lb, respectively, and are sitting in the 300-lb skiff, which is gliding through the water with a speed of 1 knot. If the three people change their positions as shown in the second figure, find the distance x from the skiff to the position where it would have been if the people had not moved. Neglect any resistance to motion afforded by the water. Does the sequence or timing of the change in positions affect the final result?

Each of the bars A and B has a mass of 10 kg and slides in its horizontal guideway with negligible friction. Motion is controlled by the lever of negligible mass connected to the bars as shown. Calculate the acceleration of point C on the lever when the 200-N force is applied as indicated. To verify your result, analyze the kinetics of each member separately and determine \(a_{C}\) by kinematic considerations from the calculated accelerations of the two bars.

The three small steel balls, each of mass 2.75 kg, are connected by the hinged links of negligible mass and equal length. They are released from rest in the positions shown and slide down the quarter-circular guide in the vertical plane. When the upper sphere reaches the bottom position, the spheres have a horizontal velocity of 1.560 m /s. Calculate the energy loss \(\Delta Q\) due to friction and the total impulse \(I_{x}\) on the system of three spheres during this interval.

Two steel balls, each of mass m, are welded to a light rod of length L and negligible mass and are initially at rest on a smooth horizontal surface. A horizontal force of magnitude F is suddenly applied to the rod as shown. Determine (a) the instantaneous acceleration \(\bar{a}\) of the mass center G and (b) the corresponding rate \(\ddot{\theta}\) at which the angular velocity of the assembly about G is changing with time.

The small car, which has a mass of 20 kg, rolls freely on the horizontal track and carries the 5-kg sphere mounted on the light rotating rod with r = 0.4 m. A geared motor drive maintains a constant angular speed \(\dot{\theta}=4\) rad /s of the rod. If the car has a velocity v = 0.6 m /s when \(\theta=0\), calculate v when \(\theta=60^{\circ}\). Neglect the mass of the wheels and any friction.

The cars of a roller-coaster ride have a speed of 30 km / h as they pass over the top of the circular track. Neglect any friction and calculate their speed v when they reach the horizontal bottom position. At the top position, the radius of the circular path of their mass centers is 18 m, and all six cars have the same mass.

The carriage of mass 2m is free to roll along the horizontal rails and carries the two spheres, each of mass m, mounted on rods of length l and negligible mass. The shaft to which the rods are secured is mounted in the carriage and is free to rotate. If the system is released from rest with the rods in the vertical position where \(\theta=0\), determine the velocity \(v_{x}\) of the carriage and the angular velocity \(\dot{\theta}\) of the rods for the instant when \(\theta=180^{\circ}\). Treat the carriage and the spheres as particles and neglect any friction.

The 50,000-lb flatcar supports a 15,000-lb vehicle on a \(5^{\circ}\) ramp built on the flatcar. If the vehicle is released from rest with the flatcar also at rest, determine the velocity v of the flatcar when the vehicle has rolled s = 40 ft down the ramp just before hitting the stop at B. Neglect all friction and treat the vehicle and the flatcar as particles.

A 60-kg rocket is fired from O with an initial velocity \(v_{0}=125\) m /s along the indicated trajectory. The rocket explodes 7 seconds after launch and breaks into three pieces A, B, and C having masses of 10, 30, and 20 kg, respectively. Pieces B and C are recovered at the impact coordinates shown. Instrumentation records reveal that piece B reached a maximum altitude of 1500 m after the explosion and that piece C struck the ground 6 seconds after the explosion. What are the impact coordinates for piece A? Neglect air resistance.

A horizontal bar of mass \(m_{1}\) and small diameter is suspended by two wires of length l from a carriage of mass \(m_{2}\) which is free to roll along the horizontal rails. If the bar and carriage are released from rest with the wires making an angle \(\theta\) with the vertical, determine the velocity \(v_{b / c}\) of the bar relative to the carriage and the velocity \(v_{c}\) of the carriage at the instant when \(\theta=0\). Neglect all friction and treat the carriage and the bar as particles in the vertical plane of motion.

In the unstretched position the coils of the 3-lb spring are just touching one another, as shown in part a of the figure. In the stretched position the force P, proportional to x, equals 200 lb when x = 20 in. If end A of the spring is suddenly released, determine the velocity \(v_{A}\) of the coil end A, measured positive to the left, as it approaches its unstretched position at x = 0. What happens to the kinetic energy of the spring?

The experimental race car is propelled by a rocket motor and is designed to reach a maximum speed v = 300 mi / hr under the thrust T of its motor. Prior wind-tunnel tests disclose that the wind resistance at this speed is 225 lb. If the rocket motor is burning fuel at the rate of 3.5 lb /sec, determine the velocity u of the exhaust gases relative to the car

The jet aircraft has a mass of 4.6 Mg and a drag (air resistance) of 32 kN at a speed of 1000 km / h at a particular altitude. The aircraft consumes air at the rate of 106 kg /s through its intake scoop and uses fuel at the rate of 4 kg /s. If the exhaust has a rearward velocity of 680 m /s relative to the exhaust nozzle, determine the maximum angle of elevation \(\alpha\) at which the jet can fly with a constant speed of 1000 km / h at the particular altitude in question.

Fresh water issues from the nozzle with a velocity of 30 m /s at the rate of \(0.05 \mathrm{\ m}^{3} / \mathrm{s}\) and is split into two equal streams by the fixed vane and deflected through \(60^{\circ}\) as shown. Calculate the force F required to hold the vane in place. The density of water is \(1000 \mathrm{\ kg} / \mathrm{m}^{3}\). .

In an unwise effort to remove debris, a homeowner directs the nozzle of his backpack blower directly toward the garage door. The nozzle velocity is 130 mi / hr and the flow rate is \(410 \mathrm{\ ft}^{3} / \min\). Estimate the force F exerted by the airflow on the door. The specific weight of air is \(0.0753 \mathrm{\ lb} / \mathrm{ft}^{3}\). .

The jet water ski has reached its maximum velocity of 70 km /h when operating in salt water. The water intake is in the horizontal tunnel in the bottom of the hull, so the water enters the intake at the velocity of 70 km /h relative to the ski. The motorized pump discharges water from the horizontal exhaust nozzle of 50-mm diameter at the rate of \(0.082 \mathrm{\ m}^{3} / \mathrm{s}\). Calculate the resistance R of the water to the hull at the operating speed.

The 25-mm steel slab 1.2 m wide enters the rolls at the speed of 0.4 m /s and is reduced in thickness to 19 mm. Calculate the small horizontal thrust T on the bearings of each of the two rolls.

The fire tug discharges a stream of salt water (density \(1030 \mathrm{\ kg} / \mathrm{m}^{3}\)) with a nozzle velocity of 40 m /s at the rate of \(0.080 \mathrm{\ m}^{3} / \mathrm{s}\). Calculate the propeller thrust T which must be developed by the tug to maintain a fixed position while pumping.

The pump shown draws air with a density through the fixed duct A of diameter d with a velocity u and discharges it at high velocity v through the two outlets B. The pressure in the airstreams at A and B is atmospheric. Determine the expression for the tension T exerted on the pump unit through the flange at C.

A jet-engine noise suppressor consists of a movable duct which is secured directly behind the jet exhaust by cable A and deflects the blast directly upward. During a ground test, the engine sucks in air at the rate of 43 kg /s and burns fuel at the rate of 0.8 kg /s. The exhaust velocity is 720 m /s. Determine the tension T in the cable.

The \(90^{\circ}\) vane moves to the left with a constant velocity of 10 m /s against a stream of fresh water issuing with a velocity of 20 m /s from the 25-mm diameter nozzle. Calculate the forces \(F_{x}\) and \(F_{y}\) on the vane required to support the motion.

The pipe bend shown has a cross-sectional area A and is supported in its plane by the tension T applied to its flanges by the adjacent connecting pipes (not shown). If the velocity of the liquid is v, its density \(\rho\), and its static pressure p, determine T and show that it is independent of the angle \(\theta\).

A jet of fluid with cross-sectional area A and mass density \(\rho\) issues from the nozzle with a velocity v and impinges on the inclined trough shown in section. Some of the fluid is diverted in each of the two directions. If the trough is smooth, the velocity of both diverted streams remains v, and the only force which can be exerted on the trough is normal to the bottom surface. Hence, the trough will be held in position by forces whose resultant is F normal to the trough. By writing impulse-momentum equations for the directions along and normal to the trough, determine the force F required to support the trough. Also find the volume rates of flow \(Q_{1}\) and \(Q_{2}\) for the two streams.

The 8-oz ball is supported by the vertical stream of fresh water which issues from the 1 /2-in.-diameter nozzle with a velocity of 35 ft /sec. Calculate the height h of the ball above the nozzle. Assume that the stream remains intact and there is no energy lost in the jet stream.

A jet-engine thrust reverser to reduce an aircraft speed of 200 km / h after landing employs folding vanes which deflect the exhaust gases in the direction indicated. If the engine is consuming 50 kg of air and 0.65 kg of fuel per second, calculate the braking thrust as a fraction n of the engine thrust without the deflector vanes. The exhaust gases have a velocity of 650 m /s relative to the nozzle.

Water issues from a nozzle with an initial velocity v and supports a thin plate of mass m at a height h above the nozzle exit. A hole in the center of the plate allows some of the water to travel upward to a maximum altitude 2h above the plate. Determine the mass m of the plate. Neglect any effects of water falling onto the plate after reaching maximum altitude. Let \(\rho\) be the density of water.

The axial-flow fan C pumps air through the duct of circular cross section and exhausts it with a velocity v at B. The air densities at A and B are \(\rho_{A}\) and \(\rho_{B}\), respectively, and the corresponding pressures are \(p_{A}\) and \(p_{B}\). The fixed deflecting blades at D restore axial flow to the air after it passes through the propeller blades C. Write an expression for the resultant horizontal force R exerted on the fan unit by the flange and bolts at A.

Air is pumped through the stationary duct A with a velocity of 50 ft /sec and exhausted through an experimental nozzle section BC. The average static pressure across section B is \(150 \mathrm{\ lb} / \mathrm{in}^{2}\) gage, and the specific weight of air at this pressure and at the temperature prevailing is \(0.840 \mathrm{\ lb} / \mathrm{ft}^{3}\). The average static pressure across the exit section C is measured to be \(2 \mathrm{\ lb} / \mathrm{in} .^{2}\) gage, and the corresponding specific weight of air is \(0.0760 \mathrm{\ lb} / \mathrm{ft}^{3}\). Calculate the force T exerted on the nozzle flange at B by the bolts and the gasket to hold the nozzle in place.

Air enters the pipe at A at the rate of 6 kg /s under a pressure of 1400 kPa gage and leaves the whistle at atmospheric pressure through the opening at B. The entering velocity of the air at A is 45 m /s, and the exhaust velocity at B is 360 m /s. Calculate the tension T, shear V, and bending moment M in the pipe at A. The net flow area at A is \(7500 \mathrm{\ mm}^{2}\).

The sump pump has a net mass of 310 kg and pumps fresh water against a 6-m head at the rate of \(0.125 \mathrm{\ m}^{3} / \mathrm{s}\). Determine the vertical force R between the supporting base and the pump flange at A during operation. The mass of water in the pump may be taken as the equivalent of a 200-mm-diameter column 6 m in height.

The experimental ground-effect machine has a total weight of 4200 lb. It hovers 1 or 2 ft off the ground by pumping air at atmospheric pressure through the circular intake duct at B and discharging it horizontally under the periphery of the skirt C. For an intake velocity v of 150 ft /sec, calculate the average air pressure p under the 18-ft-diameter machine at ground level. The specific weight of the air is \(0.076 \mathrm{\ lb} / \mathrm{ft}^{3}\).

The leaf blower draws in air at a rate of \(400 \mathrm{ft}^{3} / \mathrm{min}\) and discharges it at a speed v = 240 mi / hr. If the specific weight of the air being drawn into the blower is \(7.53\left(10^{-2}\right) \mathrm{lb} / \mathrm{ft}^{3}\), determine the added torque which the man must exert on the handle of the blower when it is running, compared with that when it is off, to maintain a steady orientation.

The ducted fan unit of mass m is supported in the vertical position on its flange at A. The unit draws in air with a density \(\rho\) and a velocity u through section A and discharges it through section B with a velocity v. Both inlet and outlet pressures are atmospheric. Write an expression for the force R applied to the flange of the fan unit by the supporting slab.

The fire hydrant is tested under a high standpipe pressure. The total flow of \(10 \mathrm{ft}^{3} / \mathrm{sec}\) is divided equally between the two outlets, each of which has a cross-sectional area of \(0.040 \mathrm{ft}^{2}\). The inlet cross sectional area at the base is \(0.75 \mathrm{ft}^{2}\). Neglect the weight of the hydrant and water within it and compute the tension T, the shear V, and the bending moment M in the base of the standpipe at B. The specific weight of water is \(62.4 \mathrm{lb} / \mathrm{ft}^{3}\). The static pressure of the water as it enters the base at B is \(120 \text { lb/in. }{ }^{2}\).

A rotary snow plow mounted on a large truck eats its way through a snow drift on a level road at a constant speed of 20 km / h. The plow discharges 60 Mg of snow per minute from its \(45^{\circ}\) chute with a velocity of 12 m /s relative to the plow. Calculate the tractive force P on the tires in the direction of motion necessary to move the plow and find the corresponding lateral force R between the tires and the road.

Salt water flows through the fixed 12-in.-insidediameter pipe at a speed \(v_{0}=4 \mathrm{ft} / \mathrm{sec}\) and enters the \(150^{\circ}\) bend with inside radius of 24 in. The water exits to the atmosphere through the 6-in.-diameter nozzle C. Determine the shear force V, axial force P, and bending moment M at flanges A and B which result from the flow of the salt water. The gage pressure in the pipe at flange A is \(250 \mathrm{lb} / \mathrm{in}^{2}\) and the pressure drop between A and B due to head loss may be neglected.

The industrial blower sucks in air through the axial opening A with a velocity \(v_{1}\) and discharges it at atmospheric pressure and temperature through the 150-mm-diameter duct B with a velocity \(v_{2}\). The blower handles \(16 \mathrm{\ m}^{3}\) of air per minute with the motor and fan running at 3450 rev /min. If the motor requires 0.32 kW of power under no load (both ducts closed), calculate the power P consumed while air is being pumped.

The feasibility of a one-passenger VTOL (vertical takeoff and landing) craft is under review. The preliminary design calls for a small engine with a high power-to-weight ratio driving an air pump that draws in air through the \(70^{\circ}\) ducts with an inlet velocity v = 40 m /s at a static gage pressure of ?1.8 kPa across the inlet areas totaling \(0.1320 \mathrm{\ m}^{2}\). The air is exhausted vertically down with a velocity u = 420 m /s. For a 90-kg passenger, calculate the maximum net mass m of the machine for which it can take off and hover. (See Table D /1 for air density.)

The helicopter shown has a mass m and hovers in position by imparting downward momentum to a column of air defined by the slipstream boundary shown. Find the downward velocity v given to the air by the rotor at a section in the stream below the rotor, where the pressure is atmospheric and the stream radius is r. Also find the power P required of the engine. Neglect the rotational energy of the air, any temperature rise due to air friction, and any change in air density \(\rho\).

The sprinkler is made to rotate at the constant angular velocity \(\omega\) and distributes water at the volume rate Q. Each of the four nozzles has an exit area A. Water is ejected from each nozzle at an angle \(\phi\) that is measured in the horizontal plane as shown. Write an expression for the torque M on the shaft of the sprinkler necessary to maintain the given motion. For a given pressure and thus flow rate Q, at what speed \(\omega_{0}\) will the sprinkler operate with no applied torque? Let \(\rho\) be the density of water.

The VTOL (vertical takeoff and landing) military aircraft is capable of rising vertically under the action of its jet exhaust, which can be “vectored” from \(\theta \cong 0\) for takeoff and hovering to \(\theta=90^{\circ}\) for forward flight. The loaded aircraft has a mass of 8600 kg. At full takeoff power, its turbo-fan engine consumes air at the rate of 90 kg /s and has an air–fuel ratio of 18. Exhaust-gas velocity is 1020 m /s with essentially atmospheric pressure across the exhaust nozzles. Air with a density of \(1.206 \mathrm{\ kg} / \mathrm{m}^{3}\) is sucked into the intake scoops at a pressure of ?2 kPa (gage) over the total inlet area of \(1.10 \mathrm{\ m}^{2}\). Determine the angle \(\theta\) for vertical takeoff and the corresponding vertical acceleration \(a_{y}\) of the aircraft.

A marine terminal for unloading bulk wheat from a ship is equipped with a vertical pipe with a nozzle at A which sucks wheat up the pipe and transfers it to the storage building. Calculate the x- and y-components of the force R required to change the momentum of the following mass in rounding the bend. Identify all forces applied externally to the bend and mass within it. Air flows through the 14-in.-diameter pipe at the rate of 18 tons per hour under a vacuum of 9 in. of mercury \(\left(p=-4.42 \mathrm{lb} / \mathrm{in} .{ }^{2}\right)\) and carries with it 150 tons of wheat per hour at a speed of 124 ft /sec.

An axial section of the suction nozzle A for a bulk wheat unloader is shown here. The outer pipe is secured to the inner pipe by several longitudinal webs which do not restrict the flow of air. A vacuum of 9 in. of mercury (\(p=-4.42 \mathrm{lb} / \mathrm{in}^{2}\) gage) is maintained in the inner pipe, and the pressure across the bottom of the outer pipe is atmospheric (p = 0). Air at \(0.075 \mathrm{lb} / \mathrm{ft}^{3}\) is drawn in through the space between the pipes at a rate of 18 tons /hr at atmospheric pressure and draws with it 150 tons of wheat per hour up the pipe at a velocity of 124 ft /sec. If the nozzle unit below section A-A weighs 60 lb, calculate the compression C in the connection at A-A.

The valve, which is screwed into the fixed pipe at section A-A, is designed to discharge fresh water at the rate of 340 gal /min into the atmosphere in the x-y plane as shown. Water pressure at A-A is \(150 \mathrm{lb} / \mathrm{in} .^{2}\) gage. The flow area at A-A has a diameter of 2 in., and the diameter of the discharge area at B is 1 in. Neglect the weight of the valve and water within it and compute the shear V, tension F, torsion T, and bending moment M at section A-A. (1 gallon contains \(231 \text { in. }{ }^{3}\))

In the figure is shown a detail of the stationary nozzle diaphragm A and the rotating blades B of a gas turbine. The products of combustion pass through the fixed diaphragm blades at the \(27^{\circ}\) angle and impinge on the moving rotor blades. The angles shown are selected so that the velocity of the gas relative to the moving blade at entrance is at the \(20^{\circ}\) angle for minimum turbulence, corresponding to a mean blade velocity of 315 m /s at a radius of 375 mm. If gas flows past the blades at the rate of 15 kg /s, determine the theoretical power output P of the turbine. Neglect fluid and mechanical friction with the resulting heat-energy loss and assume that all the gases are deflected along the surfaces of the blades with a velocity of constant magnitude relative to the blade.

When the rocket reaches the position in its trajectory shown, it has a mass of 3 Mg and is beyond the effect of the earth’s atmosphere. Gravitational acceleration is \(9.60 \mathrm{\ m} / \mathrm{s}^{2}\). Fuel is being consumed at the rate of 130 kg /s, and the exhaust velocity relative to the nozzle is 600 m/s. Compute the n- and t-components of acceleration of the rocket.

At the instant of vertical launch the rocket expels exhaust at the rate of 220 kg /s with an exhaust velocity of 820 m /s. If the initial vertical acceleration is \(6.80 \mathrm{\ m} / \mathrm{s}^{2}\), calculate the total mass of the rocket and fuel at launch.

The space shuttle, together with its central fuel tank and two booster rockets, has a total mass of \(2.04\left(10^{6}\right)\) kg at liftoff. Each of the two booster rockets produces a thrust of \(11.80\left(10^{6}\right)\) N, and each of the three main engines of the shuttle produces a thrust of \(2.00\left(10^{6}\right)\) N. The specific impulse (ratio of exhaust velocity to gravitational acceleration) for each of the three main engines of the shuttle is 455 s. Calculate the initial vertical acceleration a of the assembly with all five engines operating and find the rate at which fuel is being consumed by each of the shuttle’s three engines.

A small rocket of initial mass \(m_{0}\) is fired vertically upward near the surface of the earth ( g constant). If air resistance is neglected, determine the manner in which the mass m of the rocket must vary as a function of the time t after launching in order that the rocket may have a constant vertical acceleration a, with a constant relative velocity u of the escaping gases with respect to the nozzle.

The mass m of a raindrop increases as it picks up moisture during its vertical descent through still air. If the air resistance to motion of the drop is R and its downward velocity is v, write the equation of motion for the drop and show that the relation \(\Sigma F=d(m v) / d t\) is obeyed as a special case of the variable-mass equation.

A tank truck for washing down streets has a total weight of 20,000 lb when its tank is full. With the spray turned on, 80 lb of water per second issue from the nozzle with a velocity of 60 ft /sec relative to the truck at the \(30^{\circ}\) angle shown. If the truck is to accelerate at the rate of \(2 \mathrm{ft} / \mathrm{sec}^{2}\) when starting on a level road, determine the required tractive force P between the tires and the road when (a) the spray is turned on and (b) the spray is turned off.

A model rocket weighs 1.5 lb just before its vertical launch. Its experimental solid-fuel motor carries 0.1 lb of fuel, has an escape velocity of 3000 ft /sec, and burns the fuel for 0.9 sec. Determine the acceleration of the rocket at launch and its burnout velocity. Neglect aerodynamic drag and state any other assumptions.

The magnetometer boom for a spacecraft consists of a large number of triangular-shaped units which spring into their deployed configuration upon release from the canister in which they were folded and packed prior to release. Write an expression for the force F which the base of the canister must exert on the boom during its deployment in terms of the increasing length x and its time derivatives. The mass of the boom per unit of deployed length is \(\rho\). Treat the supporting base on the spacecraft as a fixed platform and assume that the deployment takes place outside of any gravitational field. Neglect the dimension b compared with x.

Fresh water issues from the two 30-mm-diameter holes in the bucket with a velocity of 2.5 m /s in the directions shown. Calculate the force P required to give the bucket an upward acceleration of \(0.5 \mathrm{\ m} / \mathrm{s}^{2}\) from rest if it contains 20 kg of water at that time. The empty bucket has a mass of 0.6 kg.

The upper end of the open-link chain of length Land mass \(\rho\) per unit length is lowered at a constant speed v by the force P. Determine the reading R of the platform scale in terms of x.

A rocket stage designed for deep-space missions consists of 200 kg of fuel and 300 kg of structure and payload combined. In terms of burnout velocity, what would be the advantage of reducing the structural /payload mass by 1 percent (3 kg) and using that mass for additional fuel? Express your answer in terms of a percent increase in burnout velocity. Repeat your calculation for a 5 percent reduction in the structural /payload mass.

At a bulk loading station, gravel leaves the hopper at the rate of 220 lb /sec with a velocity of 10 ft /sec in the direction shown and is deposited on the moving fl atbed truck. The tractive force between the driving wheels and the road is 380 lb, which overcomes the 200 lb of frictional road resistance. Determine the acceleration a of the truck 4 seconds after the hopper is opened over the truck bed, at which instant the truck has a forward speed of 1.5 mi / hr. The empty weight of the truck is 12,000 lb.

A railroad coal car weighs 54,600 lb empty and carries a total load of 180,000 lb of coal. The bins are equipped with bottom doors which permit discharging coal through an opening between the rails. If the car dumps coal at the rate of 20,000 lb /sec in a downward direction relative to the car, and if frictional resistance to motion is 4 lb per ton of total remaining weight, determine the coupler force P required to give the car an acceleration of \(0.15 \mathrm{\ ft} / \mathrm{sec}^{2}\) in the direction of P at the instant when half the coal has been dumped.

A coil of heavy flexible cable with a total length of 100 m and a mass of 1.2 kg /m is to be laid along a straight horizontal line. The end is secured to a post at A, and the cable peels off the coil and emerges through the horizontal opening in the cart as shown. The cart and drum together have a mass of 40 kg. If the cart is moving to the right with a velocity of 2 m /s when 30 m of cable remain in the drum and the tension in the rope at the post is 2.4 N, determine the force P required to give the cart and drum an acceleration of \(0.3 \mathrm{\ m} / \mathrm{s}^{2}\). Neglect all friction.

By lowering a scoop as it skims the surface of a body of water, the aircraft (nicknamed the “Super Scooper”) is able to ingest 4.5 m3 of fresh water during a 12-second run. The plane then flies to a fire area and makes a massive water drop with the ability to repeat the procedure as many times as necessary. The plane approaches its run with a velocity of 280 km / h and an initial mass of 16.4 Mg. As the scoop enters the water, the pilot advances the throttle to provide an additional 300 hp (223.8 kW) needed to prevent undue deceleration. Determine the initial deceleration when the scooping action starts. (Neglect the difference between the average and the initial rates of water intake.)

A small rocket-propelled vehicle weighs 125 lb, including 20 lb of fuel. Fuel is burned at the constant rate of 2 lb /sec with an exhaust velocity relative to the nozzle of 400 ft /sec. Upon ignition the vehicle is released from rest on the \(10^{\circ}\) incline. Calculate the maximum velocity v reached by the vehicle. Neglect all friction.

The end of a pile of loose-link chain of mass \(\rho\) per unit length is being pulled horizontally along the surface by a constant force P. If the coefficient of kinetic friction between the chain and the surface is \(\mu_{k}\), determine the acceleration a of the chain in terms of x and \(\dot{x}\).

A coal car with an empty mass of 25 Mg is moving freely with a speed of 1.2 m /s under a hopper which opens and releases coal into the moving car at the constant rate of 4 Mg per second. Determine the distance x moved by the car during the time that 32 Mg of coal are deposited in the car. Neglect any frictional resistance to rolling along the horizontal track.

Sand is released from the hopper H with negligible velocity and then falls a distance h to the conveyor belt. The mass flow rate from the hopper is \(m^{\prime}\). Develop an expression for the steady-state belt speed v for the case h = 0. Assume that the sand quickly acquires the belt velocity with no rebound, and neglect friction at the pulleys A and B.

Repeat the previous problem, but now let \(h \neq 0\). Then evaluate your expression for the conditions h = 2 m, L = 10 m, and \(\theta=25^{\circ}\).

The open-link chain of length L and mass \(\rho\) per unit length is released from rest in the position shown, where the bottom link is almost touching the platform and the horizontal section is supported on a smooth surface. Friction at the corner guide is negligible. Determine (a) the velocity \(v_{1}\) of end A as it reaches the corner and (b) its velocity \(v_{2}\) as it strikes the platform. (c) Also specify the total loss Q of energy.

In the figure is shown a system used to arrest the motion of an airplane landing on a field of restricted length. The plane of mass m rolling freely with a velocity \(v_{0}\) engages a hook which pulls the ends of two heavy chains, each of length L and mass \(\rho\) per unit length, in the manner shown. A conservative calculation of the effectiveness of the device neglects the retardation of chain friction on the ground and any other resistance to the motion of the airplane. With these assumptions, compute the velocity v of the airplane at the instant when the last link of each chain is put in motion. Also determine the relation between the displacement x and the time t after contact with the chain. Assume each link of the chain acquires its velocity v suddenly upon contact with the moving links.

The free end of the flexible and inextensible rope of mass \(\rho\) per unit length and total length L is given a constant upward velocity v. Write expressions for P, the force R supporting the fixed end, and the tension \(T_{1}\) in the rope at the loop in terms of x. (For the loop of negligible size, the tension is the same on both sides.)

Replace the rope of Prob. 4 /91 by an open-link chain with the same mass \(\rho\) per unit length. The free end is given a constant upward velocity v. Write expressions for P, the tension \(T_{1}\) at the bottom of the moving part, and the force R supporting the fixed end in terms of x. Also find the energy loss Q in terms of x.

The system of three particles has the indicated particle masses, velocities, and external forces. Determine \(\overline{\mathbf{r}}, \dot{\overline{\mathbf{r}}}, \ddot{\overline{\mathbf{r}}}\), T, \(\mathbf{H}_{O}\), and \(\dot{\mathbf{H}}_{O}\) for this three-dimensional system.

For the particle system of Prob. 4 /93, determine \(\mathbf{H}_{G}\) and \(\dot{\mathbf{H}}_{G}\).

Each of the identical steel balls weighs 4 lb and is fastened to the other two by connecting bars of negligible weight and unequal length. In the absence of friction at the supporting horizontal surface, determine the initial acceleration \(\bar{a}\) of the mass center of the assembly when it is subjected to the horizontal force F = 20 lb applied to the supporting ball. The assembly is initially at rest in the vertical plane. Can you show that a is initially horizontal?

A 2-oz bullet is fired horizontally with a velocity v = 1000 ft /sec into the slender bar of a 3-lb pendulum initially at rest. If the bullet embeds itself in the bar, compute the resulting angular velocity of the pendulum immediately after the impact. Treat the sphere as a particle and neglect the mass of the bar. Why is the linear momentum of the system not conserved?

A large rocket ready for vertical launch has a total mass of \(2.7\left(10^{3}\right)\) Mg. At launch, fuel is burned at the rate of 13 Mg /s with an exhaust velocity of 2400 m /s. Determine the initial acceleration a of the rocket. Assume atmospheric pressure across the exit plane of the exhaust nozzle.

In an operational design test of the equipment of the fire truck, the water cannon is delivering fresh water through its 2-in.-diameter nozzle at the rate of 1400 gal /min at the \(20^{\circ}\) angle. Calculate the total friction force F exerted by the pavement on the tires of the truck, which remains in a fixed position with its brakes locked. (There are \(231 \text { in. }^{3}\) in 1 gal.)

The rocket shown is designed to test the operation of a new guidance system. When it has reached a certain altitude beyond the effective influence of the earth’s atmosphere, its mass has decreased to 2.80 Mg, and its trajectory is \(30^{\circ}\) from the vertical. Rocket fuel is being consumed at the rate of 120 kg /s with an exhaust velocity of 640 m /s relative to the nozzle. Gravitational acceleration is \(9.34 \mathrm{\ m} / \mathrm{s}^{2}\) at its altitude. Calculate the n- and t-components of the acceleration of the rocket.

When only the air of a sand-blasting gun is turned on, the force of the air on a flat surface normal to the stream and close to the nozzle is 20 N. With the nozzle in the same position, the force increases to 30 N when sand is admitted to the stream. If sand is being consumed at the rate of 4.5 kg /min, calculate the velocity v of the sand particles as they strike the surface.

A two-stage rocket is fired vertically up and is above the atmosphere when the first stage burns out and the second stage separates and ignites. The second stage carries 1200 kg of fuel and has an empty mass of 200 kg. Upon ignition the second stage burns fuel at the rate of 5.2 kg /s and has a constant exhaust velocity of 3000 m /s relative to its nozzle. Determine the acceleration of the second stage 60 seconds after ignition and find the maximum acceleration and the time t after ignition at which it occurs. Neglect the variation of g and take it to be \(8.70 \mathrm{\ m} / \mathrm{s}^{2}\) for the range of altitude averaging about 400 km.

Water of density \(\rho\) issues from the nozzle of area A and impinges upon the block of mass m, which is at rest on the rough horizontal surface. If the block is on the verge of tipping rightward, determine the required minimum coefficient of friction \(\mu\) between the block and the surface and the exit velocity v of the water. Water exits the nozzle at atmospheric pressure.

The block of Prob. 4 /102 is now fitted with two small rollers. Determine the normal force under each roller and the initial rightward acceleration of the block if the exit velocity of the water is (a) 75 percent of the tipping velocity and (b) 50 percent of the tipping velocity. Refer to the printed answers for Prob. 4 /102 as needed. (Note: Be sure to use Eq. 4 /9 when summing moments.)

A jet of fresh water under pressure issues from the 3 /4-in.-diameter fixed nozzle with a velocity v = 120 ft /sec and is diverted into the two equal streams. Neglect any energy loss in the streams and compute the force F required to hold the vane in place.

The flexible inextensible rope of length \(\pi r / 2\) and mass \(\rho\) per unit length is attached at A to the fixed quarter-circular guide and allowed to fall from rest in the horizontal position. When the rope comes to rest in the dashed position, the system will have lost energy. Determine the loss \(\Delta Q\) and explain what becomes of the lost energy.

In the static test of a jet engine and exhaust nozzle assembly, air is sucked into the engine at the rate of 30 kg /s and fuel is burned at the rate of 1.6 kg /s. The flow area, static pressure, and axial-flow velocity for the three sections shown are as follows: Sec. A Sec. B Sec. C Flow area, \(\mathrm{m}^{2}\) 0.15 0.16 0.06 Static pressure, kPa ?14 140 14 Axial-fl ow velocity, m /s 120 315 60 Determine the tension T in the diagonal member of the supporting test stand and calculate the force F exerted on the nozzle flange at B by the bolts and gasket to hold the nozzle to the engine housing.

The upper end of the open-link chain of length L and mass \(\rho\) per unit length is released from rest with the lower end just touching the platform of the scale. Determine the expression for the force F read on the scale as a function of the distance x through which the upper end has fallen. (Comment: The chain acquires a free-fall velocity of \(\sqrt{2 g x}\) because the links on the scale exert no force on those above, which are still falling freely. Work the problem in two ways: first, by evaluating the time rate of change of momentum for the entire chain and second, by considering the force F to be composed of the weight of the links at rest on the scale plus the force necessary to divert an equivalent stream of fluid.)

The chain of length L and mass \(\rho\) per unit length is released from rest on the smooth horizontal surface with a negligibly small overhang x to initiate motion. Determine (a) the acceleration a as a function of x, (b) the tension T in the chain at the smooth corner as a function of x, and (c) the velocity v of the last link A as it reaches the corner.

A vertical force P acting on the 2.5-lb cylindrical valve A, shown in section, serves to limit the flow of fresh water from the top of the vertical pipe B of 3-in. inside diameter. Water is fed through the bottom inlet of the pipe. Calculate the force P required to maintain the valve in the position shown under a flow rate of 600 gal /min and a static pressure of \(12 \text { lb/in. }{ }^{2}\) in the water at section C. Recall that 1 gal contains \(231 \text { in. }{ }^{3}\) (Suggestion: Choose the valve and the portion of water above section C as the combined free body.)

The chain of mass \(\rho\) per unit length passes over the small freely turning pulley and is released from rest with only a small imbalance h to initiate motion. Determine the acceleration a and velocity v of the chain and the force R supported by the hook at A, all in terms of h as it varies from essentially zero to H. Neglect the weight of the pulley and its supporting frame and the weight of the small amount of chain in contact with the pulley. (Hint: The force R does not equal two times the equal tensions T in the chain tangent to the pulley.)

The centrifugal pump handles \(20 \mathrm{\ m}^{3}\) of freshwater per minute with inlet and outlet velocities of 18 m /s. The impeller is turned clockwise through the shaft at O by a motor which delivers 40 kW at a pump speed of 900 rev /min. With the pump filled but not turning, the vertical reactions at C and D are each 250 N. Calculate the forces exerted by the foundation on the pump at C and D while the pump is running. The tensions in the connecting pipes at A and B are exactly balanced by the respective forces due to the static pressure in the water. (Suggestion: Isolate the entire pump and water within it between sections A and B and apply the momentum principle to the entire system.)

A rope or hinged-link bicycle-type chain of length L and mass \(\rho\) per unit length is released from rest with x = 0. Determine the expression for the total force R exerted on the fixed platform by the chain as a function of x. Note that the hinged-link chain is a conservative system during all but the last increment of motion.