Solved: A sphere of radius r and mass m is projected along

The rotor of an electric motor has an angular velocity of 3600 rpm when the load and power are cut off. The 50-kg rotor then coasts to rest after 5000 revolutions. Knowing that the kinetic friction of the rotor produces a couple of magnitude \(4 N \cdot m\), determine the centroidal radius of gyration of the rotor.

It is known that 1500 revolutions are required for the 6000-lb flywheel to coast to rest from an angular velocity of 300 rpm. Knowing that the centroidal radius of gyration of the flywheel is 36 in., determine the average magnitude of the couple due to kinetic friction in the bearings.

Two disks of the same material are attached to a shaft as shown. Disk A has a weight of 30 lb and a radius r = 5 in. Disk B is three times as thick as disk A. Knowing that a couple M of magnitude \(15 \ lb \cdot ft\) is to be applied to disk A when the system is at rest, determine the radius nr of disk B if the angular velocity of the system is to be 600 rpm after four revolutions.

Two disks of the same material are attached to a shaft as shown. Disk A is of radius r and has a thickness b, while disk B is of radius nr and thickness 3b. A couple M of constant magnitude is applied when the system is at rest and is removed after the system has executed two revolutions. Determine the value of n which results in the largest final speed for a point on the rim of disk B.

The flywheel of a small punch rotates at 300 rpm. It is known that \(1800 \ \mathrm{ft} \cdot \mathrm{lb}\) of work must be done each time a hole is punched. It is desired that the speed of the flywheel after one punching be not less than 90 percent of the original speed of 300 rpm. (a) Determine the required moment of inertia of the flywheel. (b) If a constant \(25-\mathrm{lb} \cdot \mathrm{ft}\) couple is applied to the shaft of the flywheel, determine the number of revolutions which must occur between each punching, knowing that the initial velocity is to be 300 rpm at the start of each punching.

The flywheel of a punching machine has a mass of 300 kg and a radius of gyration of 600 mm. Each punching operation requires 2500 J of work. (a) Knowing that the speed of the flywheel is 300 rpm just before a punching, determine the speed immediately after the punching. (b) If a constant \(25-\mathrm{N} \cdot \mathrm{m}\) couple is applied to the shaft of the flywheel, determine the number of revolutions executed before the speed is again 300 rpm.

Disk A, of weight 10 lb and radius r = 6 in., is at rest when it is placed in contact with belt BC, which moves to the right with a constant speed v = 40 ft/s. Knowing that \(\mathrm{m}_{k}=0.20\) between the disk and the belt, determine the number of revolutions executed by the disk before it attains a constant angular velocity.

Disk A is of constant thickness and is at rest when it is placed in contact with belt BC, which moves with a constant velocity v. Denoting by \(\mathrm{m}_{k}\) the coefficient of kinetic friction between the disk and the belt, derive an expression for the number of revolutions executed by the disk before it attains a constant angular velocity.

The 10-in.-radius brake drum is attached to a larger flywheel which is not shown. The total mass moment of inertia of the flywheel and drum is \(16 \ \mathrm{lb} \cdot \mathrm{ft} \cdot \mathrm{s}^{2}\) and the coefficient of kinetic friction between the drum and the brake shoe is 0.40. Knowing that the initial angular velocity is 240 rpm clockwise, determine the force which must be exerted by the hydraulic cylinder if the system is to stop in 75 revolutions.

Solve Prob. 17.9, assuming that the initial angular velocity of the flywheel is 240 rpm counterclockwise.

Each of the gears A and B has a mass of 2.4 kg and a radius of gyration of 60 mm, while gear C has a mass of 12 kg and a radius of gyration of 150 mm. A couple M of constant magnitude \(10 \ \mathrm{N} \cdot \mathrm{m}\) is applied to gear C. Determine (a) the number of revolutions of gear C required for its angular velocity to increase from 100 to 450 rpm, (b) the corresponding tangential force acting on gear A.

Solve Prob. 17.11, assuming that the \(10-\mathrm{N} \cdot \mathrm{m}\) couple is applied to gear B.

The gear train shown consists of four gears of the same thickness and of the same material; two gears are of radius r, and the other two are of radius nr. The system is at rest when the couple \(\mathbf{M}_{0}\) is applied to shaft C. Denoting by \(I_{0}\) the moment of inertia of a gear of radius r, determine the angular velocity of shaft A if the couple \(\mathbf{M}_{0}\) is applied for one revolution of shaft C.

The double pulley shown has a mass of 15 kg and a centroidal radius of gyration of 160 mm. Cylinder A and block B are attached to cords that are wrapped on the pulleys as shown. The coefficient of kinetic friction between block B and the surface is 0.2. Knowing that the system is at rest in the position shown when a constant force \(\mathbf{P}=200 \ \mathrm{N}\) is applied to cylinder A, determine (a) the velocity of cylinder A as it strikes the ground, (b) the total distance that block B moves before coming to rest.

Gear A has a mass of 1 kg and a radius of gyration of 30 mm; gear B has a mass of 4 kg and a radius of gyration of 75 mm; gear C has a mass of 9 kg and a radius of gyration of 100 mm. The system is at rest when a couple \(\mathbf{M}_{0}\) of constant magnitude \(4 \ \mathrm{N} \cdot \mathrm{m}\) is applied to gear C. Assuming that no slipping occurs between the gears, determine the number of revolutions required for disk A to reach an angular velocity of 300 rpm.

A slender rod of length l and weight W is pivoted at one end as shown. It is released from rest in a horizontal position and swings freely. (a) Determine the angular velocity of the rod as it passes through a vertical position and determine the corresponding reaction at the pivot. (b) Solve part a for W = 1.8 lb and l = 3 ft.

A slender rod of length l is pivoted about a point C located at a distance b from its center G. It is released from rest in a horizontal position and swings freely. Determine (a) the distance b for which the angular velocity of the rod as it passes through a vertical position is maximum, (b) the corresponding values of its angular velocity and of the reaction at C.

A slender 9-lb rod can rotate in a vertical plane about a pivot at B. A spring of constant k = 30 lb/ft and of unstretched length 6 in. is attached to the rod as shown. Knowing that the rod is released from rest in the position shown, determine its angular velocity after it has rotated through \(90^{\circ}\).

A slender 9-lb rod can rotate in a vertical plane about a pivot at B. A spring of constant k = 30 lb/ft and of unstretched length 6 in. is attached to the rod as shown. Knowing that the rod is released from rest in the position shown, determine its angular velocity after it has rotated through \(90^{\circ}\).

A 160-lb gymnast is executing a series of full-circle swings on the horizontal bar. In the position shown he has a small and negligible clockwise angular velocity and will maintain his body straight and rigid as he swings downward. Assuming that during the swing the centroidal radius of gyration of his body is 1.5 ft, determine his angular velocity and the force exerted on his hands after he has rotated through (a) \(90^{\circ}\), (b) \(180^{\circ}\).

A collar with a mass of 1 kg is rigidly attached at a distance d = 300 mm from the end of a uniform slender rod AB. The rod has a mass of 3 kg and is of length L = 600 mm. Knowing that the rod is released from rest in the position shown, determine the angular velocity of the rod after it has rotated through \(90^{\circ}\).

A collar with a mass of 1 kg is rigidly attached to a slender rod AB of mass 3 kg and length L = 600 mm. The rod is released from rest in the position shown. Determine the distance d for which the angular velocity of the rod is maximum after it has rotated through \(90^{\circ}\).

Two identical slender rods AB and BC are welded together to form an L-shaped assembly. The assembly is pressed against a spring at D and released from the position shown. Knowing that the maximum angle of rotation of the assembly in its subsequent motion is \(90^{\circ}\) counterclockwise, determine the magnitude of the angular velocity of the assembly as it passes through the position where rod AB forms an angle of \(30^{\circ}\) with the horizontal.

The 30-kg turbine disk has a centroidal radius of gyration of 175 mm and is rotating clockwise at a constant rate of 60 rpm when a small blade of weight 0.5 N at point A becomes loose and is thrown off. Neglecting friction, determine the change in the angular velocity of the turbine disk after it has rotated through (a) \(90^{\circ}\), (b) \(270^{\circ}\).

A rope is wrapped around a cylinder of radius r and mass m as shown. Knowing that the cylinder is released from rest, determine the velocity of the center of the cylinder after it has moved downward a distance s.

Solve Prob. 17.25, assuming that the cylinder is replaced by a thin-walled pipe of radius r and mass m.

A 45-lb uniform cylindrical roller, initially at rest, is acted upon by a 20-lb force as shown. Knowing that the body rolls without slipping, determine (a) the velocity of its center G after it has moved 5 ft, (b) the friction force required to prevent slipping.

A small sphere of mass m and radius r is released from rest at A and rolls without sliding on the curved surface to point B where it leaves the surface with a horizontal velocity. Knowing that a = 1.5 m and b = 1.2 m, determine (a) the speed of the sphere as it strikes the ground at C, (b) the corresponding distance c.

The mass center G of a 3-kg wheel of radius R = 180 mm is located at a distance r = 60 mm from its geometric center C. The centroidal radius of gyration of the wheel is \(\bar{k}=90 \ \mathrm{mm}\). As the wheel rolls without sliding, its angular velocity is observed to vary. Knowing that \(\mathrm{V}=8 \ \mathrm{rad} / \mathrm{s}\) in the position shown, determine (a) the angular velocity of the wheel when the mass center G is directly above the geometric center C, (b) the reaction at the horizontal surface at the same instant.

A half section of pipe of mass m Problems and radius r is released from rest in the position shown. Knowing that the pipe rolls without sliding, determine (a) its angular velocity after it has rolled through \(90^{\circ}\), (b) the reaction at the horizontal surface at the same instant. [Hint: Note that GO = 2r/p and that, by the parallel-axis theorem, \(\bar{I}= m r^{2}-m(G O)^{2}\).]

A sphere of mass m and radius r rolls without slipping inside a curved surface of radius R. Knowing that the sphere is released from rest in the position shown, derive an expression for (a) the linear velocity of the sphere as it passes through B, (b) the magnitude of the vertical reaction at that instant.

Two uniform cylinders, each of weight W = 14 lb and radius r = 5 in., are connected by a belt as shown. Knowing that at the instant shown the angular velocity of cylinder B is 30 rad/s clockwise, determine (a) the distance through which cylinder A will rise before the angular velocity of cylinder B is reduced to 5 rad/s, (b) the tension in the portion of belt connecting the two cylinders.

Two uniform cylinders, each of weight W = 14 lb and radius r = 5 in., are connected by a belt as shown. If the system is released from rest, determine (a) the velocity of the center of cylinder A after it has moved through 3 ft, (b) the tension in the portion of belt connecting the two cylinders.

A bar of mass m = 5 kg is held as shown between four disks each of mass \(m^{\prime}=2 \ \mathrm{kg}\) and radius r = 75 mm. Knowing that the forces exerted on the disks are sufficient to prevent slipping and that the bar is released from rest, for each of the cases shown determine the velocity of the bar after it has moved through the distance h.

The 5-kg rod BC is attached by pins to two uniform disks as shown. The mass of the 150-mm-radius disk is 6 kg and that of the 75-mm-radius disk is 1.5 kg. Knowing that the system is released from rest in the position shown, determine the velocity of the rod after disk A has rotated through \(90^{\circ}\).

The motion of the uniform rod AB is guided by small wheels of negligible mass that roll on the surface shown. If the rod is released from rest when \(\mathrm{u}=0\), determine the velocities of A and B when \(\mathrm{u}=30^{\circ}\).

A 5-m-long ladder has a mass of 15 kg and is placed against a house at an angle \(\mathrm{u}=20^{\circ}\). Knowing that the ladder is released from rest, determine the angular velocity of the ladder and the velocity of end A when \(\mathrm{u}=45^{\circ}\). Assume the ladder can slide freely on the horizontal ground and on the vertical wall.

A long ladder of length l, mass m, and centroidal mass moment of inertia \(\bar{I}\) is placed against a house at an angle \(\mathrm{u}=\mathrm{u}_{0}\). Knowing that the ladder is released from rest, determine the angular velocity of the ladder when \(\mathrm{u}=\mathrm{u}_{2}\). Assume the ladder can slide freely on the horizontal ground and on the vertical wall.

The ends of a 9-lb rod AB are constrained to move along slots cut in a vertical plate as shown. A spring of constant k = 3 lb/in. is attached to end A in such a way that its tension is zero when \(\mathrm{u}=0\). If the rod is released from rest when \(\mathrm{u}=50^{\circ}\), determine the angular velocity of the rod and the velocity of end B when \(\mathrm{u}=0\).

The ends of a 9-lb rod AB are constrained to move along slots cut in a vertical plate as shown. A spring of constant k = 3 lb/in. is attached to end A in such a way that its tension is zero when \(\mathrm{u}=0\). If the rod is released from rest when \(\mathrm{u}=0\), determine the angular velocity of the rod and the velocity of end B when \(\mathrm{u}=30^{\circ}\).

The motion of a slender rod of length R is guided by pins at A and B which slide freely in slots cut in a vertical plate as shown. If end B is moved slightly to the left and then released, determine the angular velocity of the rod and the velocity of its mass center (a) at the instant when the velocity of end B is zero, (b) as end B passes through point D.

Each of the two rods shown is of length L = 1 m and has a mass of 5 kg. Point D is connected to a spring of constant k = 20 N/m and is constrained to move along a vertical slot. Knowing that the system is released from rest when rod BD is horizontal and the spring connected to point D is initially unstretched, determine the velocity of point D when it is directly to the right of point \(A^{\prime}\).

The 4-kg rod AB is attached to a collar of negligible mass at A and to a flywheel at B. The flywheel has a mass of 16 kg and a radius of gyration of 180 mm. Knowing that in the position shown the angular velocity of the flywheel is 60 rpm clockwise, determine the velocity of the flywheel when point B is directly below C.

If in Prob. 17.43 the angular velocity of the flywheel is to be the same in the position shown and when point B is directly above C, determine the required value of its angular velocity in the position shown.

The uniform rods AB and BC weigh 2.4 kg and 4 kg, respectively, and the small wheel at C is of negligible weight. If the wheel is moved slightly to the right and then released, determine the velocity of pin B after rod AB has rotated through \(90^{\circ}\).

The uniform rods AB and BC weigh 2.4 kg and 4 kg, respectively, and the small wheel at C is of negligible weight. Knowing that in the position shown the velocity of wheel C is 2 m/s to the right, determine the velocity of pin B after rod AB has rotated through \(90^{\circ}\). .

The 80-mm-radius gear shown has a mass of 5 kg and a centroidal radius of gyration of 60 mm. The 4-kg rod AB is attached to the center of the gear and to a pin at B that slides freely in a vertical slot. Knowing that the system is released from rest when \(\mathrm{u}=60^{\circ}\), determine the velocity of the center of the gear when \(\mathrm{u}=20^{\circ}\).

Knowing that the maximum allowable couple that can be applied to a shaft is \(15.5 \ \mathrm{kip} \cdot \text { in. }\), determine the maximum horsepower that can be transmitted by the shaft at (a) 180 rpm, (b) 480 rpm.

Three shafts and four gears are used to form a gear train which will transmit 7.5 kW from the motor at A to a machine tool at F. (Bearings for the shafts are omitted from the sketch.) Knowing that the frequency of the motor is 30 Hz, determine the magnitude of the couple which is applied to shaft (a) AB, (b) CD, (c) EF.

The shaft-disk-belt arrangement shown is used to transmit 2.4 kW from point A to point D. Knowing that the maximum allowable couples that can be applied to shafts AB and CD are \(25 \ \mathrm{N} \cdot \mathrm{m}\) and \(80 \ \mathrm{N} \cdot \mathrm{m}\), respectively, determine the required minimum speed of shaft AB.

The experimental setup shown is used to measure the power output of a small turbine. When the turbine is operating at 200 rpm, the readings of the two spring scales are 10 and 22 lb, respectively. Determine the power being developed by the turbine.

The rotor of an electric motor has a mass of 25 kg, and it is observed that 4.2 min is required for the rotor to coast to rest from an angular velocity of 3600 rpm. Knowing that kinetic friction produces a couple of magnitude \(1.2 \ \mathrm{N} \cdot \mathrm{m}\), determine the centroidal radius of gyration for the rotor.

A small grinding wheel is attached to the shaft of an electric motor which has a rated speed of 3600 rpm. When the power is turned off, the unit coasts to rest in 70 s. The grinding wheel and rotor have a combined weight of 6 lb and a combined radius of gyration of 2 in. Determine the average magnitude of the couple due to kinetic friction in the bearings of the motor.

A bolt located 50 mm from the center of an automobile wheel is tightened by applying the couple shown for 0.10 s. Assuming that the wheel is free to rotate and is initially at rest, determine the resulting angular velocity of the wheel. The wheel has a mass of 19 kg and has a radius of gyration of 250 mm.

Two disks of the same thickness and same material are attached to a shaft as shown. The 8-lb disk A has a radius \(r_{A}=3 \ \mathrm{in}\)., and disk B has a radius \(r_{B}=4.5 \ \mathrm{in}\). Knowing that a couple M of magnitude \(20 \ \mathrm{lb} \cdot \text { in }\). is applied to disk A when the system is at rest, determine the time required for the angular velocity of the system to reach 960 rpm.

Two disks of the same thickness and same material are attached to a shaft as shown. The 3-kg disk A has a radius \(r_{A}=100 \ \mathrm{mm}\), and disk B has a radius \(r_{B}=125 \ \mathrm{mm}\). Knowing that the angular velocity of the system is to be increased from 200 rpm to 800 rpm during a 3-s interval, determine the magnitude of the couple M that must be applied to disk A.

A disk of constant thickness, initially at rest, is placed in contact with a belt that moves with a constant velocity v. Denoting by \(\mathrm{m}_{k}\) the coefficient of kinetic friction between the disk and the belt, derive an expression for the time required for the disk to reach a constant angular velocity.

Disk A, of weight 5 lb and radius r = 3 in., is at rest when it is placed in contact with a belt which moves at a constant speed v = 50 ft/s. Knowing that \(\mathrm{m}_{k}=0.20\) between the disk and the belt, determine the time required for the disk to reach a constant angular velocity.

A cylinder of radius r and weight W with an initial counterclockwise angular velocity \(V_{0}\) is placed in the corner formed by the floor and a vertical wall. Denoting by \(\mathrm{m}_{k}\) the coefficient of kinetic friction between the cylinder and the wall and the floor, derive an expression for the time required for the cylinder to come to rest.

Two uniform disks and two cylinders are assembled as indicated. Disk A has a mass of 10 kg and disk B has a mass of 6 kg. Knowing that the system is released from rest, determine the time required for cylinder C to have a speed of 0.5 m/s. Disks A and B are bolted together and the cylinders are attached to separate cords wrapped on the disks.

Two uniform disks and two cylinders are assembled as indicated. Disk A has a mass of 10 kg and disk B has a mass of 6 kg. Knowing that the system is released from rest, determine the time required for cylinder C to have a speed of 0.5 m/s. The cylinders are attached to a single cord that passes over the disks. Assume that no slipping occurs between the cord and the disks.

Disk B has an initial angular velocity \(V_{0}\) when it is brought into contact with disk A which is at rest. Show that the final angular velocity of disk B depends only on \(\mathrm{v}_{0}\) and the ratio of the masses \(m_{A}\) and \(m_{B}\) of the two disks.

The 7.5-lb disk A has a radius \(r_{A}=6 \ \mathrm{in}.\) and is initially at rest. The 10-lb disk B has a radius \(r_{B}=8 \ \mathrm{in}.\) and an angular velocity \(\mathrm{V}_{0}\) of 900 rpm when it is brought into contact with disk A. Neglecting friction in the bearings, determine (a) the final angular velocity of each disk, (b) the total impulse of the friction force exerted on disk A.

A tape moves over the two drums shown. Drum A Problems weighs 1.4 lb and has a radius of gyration of 0.75 in., while drum B weighs 3.5 lb and has a radius of gyration of 1.25 in. In the lower portion of the tape the tension is constant and equal to \(T_{A}=0.75 \ \mathrm{lb}\). Knowing that the tape is initially at rest, determine (a) the required constant tension \(T_{B}\) if the velocity of the tape is to be v = 10 ft/s after 0.24 s, (b) the corresponding tension in the portion of the tape between the drums.

Show that the system of momenta for a rigid slab in plane motion reduces to a single vector, and express the distance from the mass center G to the line of action of this vector in terms of the centroidal radius of gyration \(\bar{k}\) of the slab, the magnitude \(\bar{v}\) of the velocity of G, and the angular velocity V.

Show that, when a rigid slab rotates about a fixed axis through O perpendicular to the slab, the system of the momenta of its particles is equivalent to a single vector of magnitude \(m \bar{r} \mathrm{V}\), perpendicular to the line OG, and applied to a point P on this line, called the center of percussion, at a distance \(G P=\bar{k}^{2} / \bar{r}\) from the mass center of the slab.

Show that the sum \(\mathbf{H}_{A}\) of the moments about a point A of the momenta of the particles of a rigid slab in plane motion is equal to \(I_{A} \mathrm{V}\), where \(\mathrm{V}\) is the angular velocity of the slab at the instant considered and \(I_{A}\) the moment of inertia of the slab about A, if and only if one of the following conditions is satisfied: (a) A is the mass center of the slab, (b) A is the instantaneous center of rotation, (c) the velocity of A is directed along a line joining point A and the mass center G.

Consider a rigid slab initially at rest and subjected to an impulsive force \(\mathbf{F}\) contained in the plane of the slab. We define the center of percussion P as the point of intersection of the line of action of \(\mathbf{F}\) with the perpendicular drawn from G. (a) Show that the instantaneous center of rotation C of the slab is located on line GP at a distance \(G C=\bar{k}^{2} / G P\) on the opposite side of G. (b) Show that if the center of percussion were located at C the instantaneous center of rotation would be located at P.

A flywheel is rigidly attached to a 1.5-in.-radius shaft that rolls without sliding along parallel rails. Knowing that after being released from rest the system attains a speed of 6 in./s in 30 s, determine the centroidal radius of gyration of the system.

A wheel of radius r and centroidal radius of gyration \(\bar{k}\) is released from rest on the incline shown at time t = 0. Assuming that the wheel rolls without sliding, determine (a) the velocity of its center at time t, (b) the coefficient of static friction required to prevent slipping.

The double pulley shown has a mass of 3 kg and a radius of gyration of 100 mm. Knowing that when the pulley is at rest, a force P of magnitude 24 N is applied to cord B, determine (a) the velocity of the center of the pulley after 1.5 s, (b) the tension in cord C.

A 9-in.-radius cylinder of weight 18 lb rests on a 6-lb carriage. The system is at rest when a force P of magnitude 2.5 lb is applied as shown for 1.2 s. Knowing that the cylinder rolls without sliding on the carriage and neglecting the mass of the wheels of the carriage, determine the resulting velocity of (a) the carriage, (b) the center of the cylinder.

A 9-in.-radius cylinder of weight 18 lb rests on a 6-lb carriage. The system is at rest when a force P of magnitude 2.5 lb is applied as shown for 1.2 s. Knowing that the cylinder rolls without sliding on the carriage and neglecting the mass of the wheels of the carriage, determine the resulting velocity of (a) the carriage, (b) the center of the cylinder.

Two uniform cylinders, each of mass m = 6 kg and radius r = 125 mm, are connected by a belt as shown. If the system is released from rest when t = 0, determine (a) the velocity of the center of cylinder B at t = 3 s, (b) the tension in the portion of belt connecting the two cylinders.

Two uniform cylinders, each of mass m = 6 kg and radius r = 125 mm, are connected by a belt as shown. Knowing that at the instant shown the angular velocity of cylinder A is 30 rad/s counterclockwise, determine (a) the time required for the angular velocity of cylinder A to be reduced to 5 rad/s, (b) the tension in the portion of belt connecting the two cylinders.

In the gear arrangement shown, gears A and C are attached to rod ABC, which is free to rotate about B, while the inner gear B is fixed. Knowing that the system is at rest, determine the magnitude of the couple M which must be applied to rod ABC, if 2.5 s later the angular velocity of the rod is to be 240 rpm clockwise. Gears A and C weigh 2.5 lb each and may be considered as disks of radius 2 in.; rod ABC weighs 4 lb.

A sphere of radius r and mass m is projected along a rough horizontal surface with the initial velocities shown. If the final velocity of the sphere is to be zero, express (a) the required magnitude of \(\mathrm{V}_{0}\) in terms of \(v_{0}\) and r, (b) the time required for the sphere to come to rest in terms of \(v_{0}\) and the coefficient of kinetic friction \(\mathrm{m}_{k}\).

A bowler projects an 8.5-in.-diameter ball weighing 16 lb along an alley with a forward velocity \(\mathbf{v}_{0}\) of 25 ft/s and a backspin \mathrm{v}_{0} of 9 rad/s. Knowing that the coefficient of kinetic friction between the ball and the alley is 0.10, determine (a) the time \(t_{1}\) at which the ball will start rolling without sliding, (b) the speed of the ball at time \(t_{1}\).

Four rectangular panels, each of length b and height \(\frac{1}{2} b\), are attached with hinges to a circular plate of diameter \(1\bar{2} b\) and held by a wire loop in the position shown. The plate and the panels are made of the same material and have the same thickness. The entire assembly is rotating with an angular velocity \(\mathrm{v}_{0}\) when the wire breaks. Determine the angular velocity of the assembly after the panels have come to rest in a horizontal position.

A 2.5-lb disk of radius 4 in. is attached to the yoke BCD by means of short shafts fitted in bearings at B and D. The 1.5-lb yoke has a radius of gyration of 3 in. about the x axis. Initially the assembly is rotating at 120 rpm with the disk in the plane of the yoke \((\mathrm{u}=0)\). If the disk is slightly disturbed and rotates with respect to the yoke until \(\mathrm{u}=90^{\circ}\), where it is stopped by a small bar at D, determine the final angular velocity of the assembly.

Two 10-lb disks and a small motor are mounted on a 15-lb rectangular platform which is free to rotate about a central vertical spindle. The normal operating speed of the motor is 180 rpm. If the motor is started when the system is at rest, determine the angular velocity of all elements of the system after the motor has attained its normal operating speed. Neglect the mass of the motor and of the belt.

A 3-kg rod of length 800 mm can slide freely in the 240-mm cylinder DE, which in turn can rotate freely in a horizontal plane. In the position shown the assembly is rotating with an angular velocity of magnitude \(\mathrm{v}=40 \ \mathrm{rad} / \mathrm{s}\) and end B of the rod is moving toward the cylinder at a speed of 75 mm/s relative to the cylinder. Knowing that the centroidal mass moment of inertia of the cylinder about a vertical axis is \(0.025 \ \mathrm{kg} \cdot \mathrm{m}^{2}\) and neglecting the effect of friction, determine the angular velocity of the assembly as end B of the rod strikes end E of the cylinder.

A 1.6-kg tube AB can slide freely on rod DE which in turn can rotate freely in a horizontal plane. Initially the assembly is rotating with an angular velocity \(\mathrm{V}=5 \ \mathrm{rad} / \mathrm{s}\) and the tube is held in position by a cord. The moment of inertia of the rod and bracket about the vertical axis of rotation is \(0.30 \ \mathrm{kg} \cdot \mathrm{m}^{2}\) and the centroidal moment of inertia of the tube about a vertical axis is \(0.0025 \ \mathrm{kg} \cdot \mathrm{m}^{2}\). If the cord suddenly breaks, determine (a) the angular velocity of the assembly after the tube has moved to end E, (b) the energy lost during the plastic impact at E.

In the helicopter shown, a vertical tail propeller is used to prevent rotation of the cab as the speed of the main blades is changed. Assuming that the tail propeller is not operating, determine the final angular velocity of the cab after the speed of the main blades has been changed from 180 to 240 rpm. (The speed of the main blades is measured relative to the cab, and the cab has a centroidal moment of inertia of \(650 \ \mathrm{lb} \cdot \mathrm{ft} \cdot \mathrm{s}^{2}\). Each of the four main blades is assumed to be a slender 14-ft rod weighing 55 lb.)

Assuming that the tail propeller in Prob. 17.84 is operating and that the angular velocity of the cab remains zero, determine the final horizontal velocity of the cab when the speed of the main blades is changed from 180 to 240 rpm. The cab weighs 1250 lb and is initially at rest. Also determine the force exerted by the tail propeller if the change in speed takes place uniformly in 12 s.

The circular platform A is fitted with a rim of 200-mm inner radius and can rotate freely about the vertical shaft. It is known that the platform-rim unit has a mass of 5 kg and a radius of gyration of 175 mm with respect to the shaft. At a time when the platform is rotating with an angular velocity of 50 rpm, a 3-kg disk B of radius 80 mm is placed on the platform with no velocity. Knowing that disk B then slides until it comes to rest relative to the platform against the rim, determine the final angular velocity of the platform.

Two 4-kg disks and a small motor are mounted on a 6-kg rectan- Problems gular platform which is free to rotate about a central vertical spindle. The normal operating speed of the motor is 240 rpm. If the motor is started when the system is at rest, determine the angular velocity of all elements of the system after the motor has attained its normal operating speed. Neglect the mass of the motor and of the belt.

The 4-kg rod AB can slide freely inside the 6-kg tube CD. The rod was entirely within the tube (x = 0) and released with no initial velocity relative to the tube when the angular velocity of the assembly was 5 rad/s. Neglecting the effect of friction, determine the speed of the rod relative to the tube when x = 400 mm.

A 1.8-kg collar A and a 0.7-kg collar B can slide without friction on a frame, consisting of the horizontal rod OE and the vertical rod CD, which is free to rotate about its vertical axis of symmetry. The two collars are connected by a cord running over a pulley that is attached to the frame at O. At the instant shown, the velocity \(v_{A}\) of collar A has a magnitude of 2.1 m/s and a stop prevents collar B from moving. The stop is suddenly removed and collar A moves toward E. As it reaches a distance of 0.12 m from O, the magnitude of its velocity is observed to be 2.5 m/s. Determine at that instant the magnitude of the angular velocity of the frame and the moment of inertia of the frame and pulley system about CD.

A 6-lb collar C is attached to a spring and can slide on rod AB, which in turn can rotate in a horizontal plane. The mass moment of inertia of rod AB with respect to end A is \(0.35 \ \mathrm{lb} \cdot \mathrm{ft} \cdot \mathrm{s}^{2}\). The spring has a constant k = 15 lb/in. and an undeformed length of 10 in. At the instant shown the velocity of the collar relative to the rod is zero and the assembly is rotating with an angular velocity of 12 rad/s. Neglecting the effect of friction, determine (a) the angular velocity of the assembly as the collar passes through a point located 7.5 in. from end A of the rod, (b) the corresponding velocity of the collar relative to the rod.

A small 4-lb collar C can slide freely on a thin ring of weight 6 lb and radius 10 in. The ring is welded to a short vertical shaft, which can rotate freely in a fixed bearing. Initially the ring has an angular velocity of 35 rad/s and the collar is at the top of the ring \((\mathrm{u}=0)\) when it is given a slight nudge. Neglecting the effect of friction, determine (a) the angular velocity of the ring as the collar passes through the position \(\mathrm{u}=90^{\circ}\), (b) the corresponding velocity of the collar relative to the ring.

A uniform rod AB, of mass 7 kg and length 1.2 m, is attached to the 11-kg cart C. Knowing that the system is released from rest in the position shown and neglecting friction, determine (a) the velocity of point B as rod AB passes through a vertical position, (b) the corresponding velocity of cart C.

In Prob. 17.82, determine the velocity of rod AB relative to cylinder DE as end B of the rod strikes end E of the cylinder.

In Prob. 17.83, determine the velocity of the tube relative to the rod as the tube strikes end E of the assembly.

The 6-lb steel cylinder A and the 10-lb wooden cart B are at rest in the position shown when the cylinder is given a slight nudge, causing it to roll without sliding along the top surface of the cart. Neglecting friction between the cart and the ground, determine the velocity of the cart as the cylinder passes through the lowest point of the surface at C.

At what height h above its center G should a billiard ball of radius r be struck horizontally by a cue if the ball is to start rolling without sliding?

A bullet weighing 0.08 lb is fired with a horizontal velocity of 1800 ft/s into the lower end of a slender 15-lb bar of length L = 30 in. Knowing that h = 12 in. and that the bar is initially at rest, determine (a) the angular velocity of the bar immediately after the bullet becomes embedded, (b) the impulsive reaction at C, assuming that the bullet becomes embedded in 0.001 s.

In Prob. 17.97, determine (a) the required distance h if the impulsive reaction at C is to be zero, (b) the corresponding angular velocity of the bar immediately after the bullet becomes embedded.

An 16-lb wooden panel is suspended from a pin support at A and is initially at rest. A 4-lb metal sphere is released from rest at B and falls into a hemispherical cup C attached to the panel at a point located on its top edge. Assuming that the impact is perfectly plastic, determine the velocity of the mass center G of the panel immediately after the impact.

A 16-lb wooden panel is suspended from a pin support at A and is initially at rest. A 4-lb metal sphere is released from rest at \(B^{\prime}\) and falls into a hemispherical cup \(C^{\prime}\) attached to the panel at the same level as the mass center G. Assuming that the impact is perfectly plastic, determine the velocity of the mass center G of the panel immediately after the impact.

A 45-g bullet is fired with a velocity of 400 m/s at \(\mathrm{u}=30^{\circ}\) into a 9-kg square panel of side b = 200 mm. Knowing that h = 150 mm and that the panel is initially at rest, determine (a) the velocity of the center of the panel immediately after the bullet becomes embedded, (b) the impulsive reaction at A, assuming that the bullet becomes embedded in 2 ms.

A 45-g bullet is fired with a velocity of 400 m/s at \(\mathrm{u}=5^{\circ}\) into a 9-kg square panel of side b = 200 mm. Knowing that the panel is initially at rest, determine (a) the required distance h if the horizontal component of the impulsive reaction at A is to be zero, (b) the corresponding velocity of the center of the panel immediately after the bullet becomes embedded.

Two uniform rods, each of mass m, form the L-shaped rigid body ABC which is initially at rest on the frictionless horizontal surface when hook D of the carriage E engages a small pin at C. Knowing that the carriage is pulled to the right with a constant velocity \(\mathbf{v}_{0}\), determine immediately after the impact (a) the angular velocity of the body, (b) the velocity of corner B. Assume that the velocity of the carriage is unchanged and that the impact is perfectly plastic.

The uniform slender rod AB of weight 5 lb and length 30 in. forms an angle \(\mathrm{b}=30^{\circ}\) with the vertical as it strikes the smooth corner shown with a vertical velocity \(\mathbf{v}_{1}\) of magnitude 8 ft/s and no angular velocity. Assuming that the impact is perfectly plastic, determine the angular velocity of the rod immediately after the impact.

A bullet weighing 0.08 lb is fired with a horizontal velocity of 1800 ft/s into the 15-lb wooden rod AB of length L = 30 in. The rod, which is initially at rest, is suspended by a cord of length L = 30 in. Determine the distance h for which, immediately after the bullet becomes embedded, the instantaneous center of rotation of the rod is point C.

A uniform sphere of radius r rolls down the incline shown without slipping. It hits a horizontal surface and, after slipping for a while, it starts rolling again. Assuming that the sphere does not bounce as it hits the horizontal surface, determine its angular velocity and the velocity of its mass center after it has resumed rolling.

A uniformly loaded rectangular crate is released from rest in the position shown. Assuming that the floor is sufficiently rough to prevent slipping and that the impact at B is perfectly plastic, determine the smallest value of the ratio a/b for which corner A will remain in contact with the floor.

A bullet of mass m is fired with a horizontal velocity \(\mathbf{v}_{0}\) and at a height \(h=\frac{1}{2} R\) into a wooden disk of much larger mass M and radius R. The disk rests on a horizontal plane and the coefficient of friction between the disk and the plane is finite. (a) Determine the linear velocity \(\overline {\mathrm{v}}_{1}\) and the angular velocity \(\mathrm{v}_{1}\) of the disk immediately after the bullet has penetrated the disk. (b) Describe the ensuing motion of the disk and determine its linear velocity after the motion has become uniform.

Determine the height h at which the bullet of Prob. 17.108 should be fired (a) if the disk is to roll without sliding immediately after impact, (b) if the disk is to slide without rolling immediately after impact.

A uniform slender bar of length L = 200 mm and mass m = 0.5 kg is supported by a frictionless horizontal table. Initially the bar is spinning about its mass center G with a constant angular speed \(\mathrm{v}_{1}=6 \ \mathrm{rad} / \mathrm{s}\). Suddenly latch D is moved to the right and is struck by end A of the bar. Knowing that the coefficient of restitution between A and D is e = 0.6, determine the angular velocity of the bar and the velocity of its mass center immediately after the impact.

A uniform slender rod of length L is dropped onto rigid supports at A and B. Since support B is slightly lower than support A, the rod strikes A with a velocity \(\overline{\mathbf{v}}_{1}\) before it strikes B. Assuming perfectly elastic impact at both A and B, determine the angular velocity of the rod and the velocity of its mass center immediately after the rod (a) strikes support A, (b) strikes support B, (c) again strikes support A.

The slender rod AB of length L forms an angle b with the vertical as it strikes the frictionless surface shown with a vertical velocity \(\overline{\mathbf{V}}_{1}\) and no angular velocity. Assuming that the impact is perfectly plastic, derive an expression for the angular velocity of the rod immediately after the impact.

The slender rod AB of length L = 1 m forms an angle \(\mathrm{b}=30^{\circ}\) with the vertical as it strikes the frictionless surface shown with a vertical velocity \(\overline{\mathbf{v}}_{1}=2 \ \mathrm{m} / \mathrm{s}\) and no angular velocity. Knowing that the coefficient of restitution between the rod and the ground is e = 0.8, determine the angular velocity of the rod immediately after the impact.

The trapeze/lanyard air drop (t/LAD) launch is a proposed innovative method for airborne launch of a payload-carrying rocket. The release sequence involves several steps as shown in (1) where the payload rocket is shown at various instances during the launch. To investigate the first step of this process, where the rocket body drops freely from the carrier aircraft until the 2-m lanyard stops the vertical motion of B, a trial rocket is tested as shown in (2). The rocket can be considered a uniform \(1 \times 7-\mathrm{m}\) rectangle with a mass of 4000 kg. Knowing that the rocket is released from rest and falls vertically 2 m before the lanyard becomes taut, determine the angular velocity of the rocket immediately after the lanyard is taut.

The uniform rectangular block shown is moving along a frictionless surface with a velocity \(\overline{\mathbf{v}}_{1}\) when it strikes a small obstruction at B. Assuming that the impact between corner A and obstruction B is perfectly plastic, determine the magnitude of the velocity \(\overline{\mathbf{v}}_{1}\) for which the maximum angle u through which the block will rotate will be \(30^{\circ}\).

A slender rod of length L Problems and mass m is released from rest in the position shown. It is observed that after the rod strikes the vertical surface it rebounds to form an angle of \(30^{\circ}\) with the vertical. (a) Determine the coefficient of restitution between knob K and the surface. (b) Show that the same rebound can be expected for any position of knob K.

A slender rod of mass m and length L is released from rest in the position shown and hits edge D. Assuming perfectly plastic impact at D, determine for b = 0.6L, (a) the angular velocity of the rod immediately after the impact, (b) the maximum angle through which the rod will rotate after the impact.

A uniformly loaded square crate is released from rest with its corner D directly above A; it rotates about A until its corner B strikes the floor, and then rotates about B. The floor is sufficiently rough to prevent slipping and the impact at B is perfectly plastic. Denoting by \(\mathrm{V}_{0}\) the angular velocity of the crate immediately before B strikes the floor, determine (a) the angular velocity of the crate immediately after B strikes the floor, (b) the fraction of the kinetic energy of the crate lost during the impact, (c) the angle u through which the crate will rotate after B strikes the floor.

A 1-oz bullet is fired with a horizontal velocity of 750 mi/h into the 18-lb wooden beam AB. The beam is suspended from a collar of negligible mass that can slide along a horizontal rod. Neglecting friction between the collar and the rod, determine the maximum angle of rotation of the beam during its subsequent motion.

For the beam of Prob. 17.119, determine the velocity of the 1-oz bullet for which the maximum angle of rotation of the beam will be \(90^{\circ}\).

The plank CDE has a mass of 15 kg and rests on a small pivot at D. The 55-kg gymnast A is standing on the plank at C when the 70-kg gymnast B jumps from a height of 2.5 m and strikes the plank at E. Assuming perfectly plastic impact and that gymnast A is standing absolutely straight, determine the height to which gymnast A will rise.

Solve Prob. 17.121, assuming that the gymnasts change places so that gymnast A jumps onto the plank while gymnast B stands at C.

A small plate B is attached to a cord that is wrapped around a uniform 8-lb disk of radius R = 9 in. A 3-lb collar A is released from rest and falls through a distance h = 15 in. before hitting plate B. Assuming that the impact is perfectly plastic and neglecting the weight of the plate, determine immediately after the impact (a) the velocity of the collar, (b) the angular velocity of the disk.

Solve Prob. 17.123, assuming that the coefficient of restitution between A and B is 0.8.

Two identical slender rods may swing freely from the pivots shown. Rod A is released from rest in a horizontal position and swings to a vertical position, at which time the small knob K strikes rod B which was at rest. If \(h=\frac{1}{2} l\) and \(e=\frac{1}{2}\), determine (a) the angle through which rod B will swing, (b) the angle through which rod A will rebound.

A 2-kg solid sphere of radius r = 40 mm is dropped from a height h = 200 mm and lands on a uniform slender plank AB of mass 4 kg and length L = 500 mm which is held by two inextensible cords. Knowing that the impact is perfectly plastic and that the sphere remains attached to the plank at a distance a = 40 mm from the left end, determine the velocity of the sphere immediately after impact. Neglect the thickness of the plank.

Member ABC has a mass of 2.4 kg and is attached to a pin support at B. An 800-g sphere D strikes the end of member ABC with a vertical velocity \(\mathbf{v}_{1}\) of 3 m/s. Knowing that L = 750 mm and that the coefficient of restitution between the sphere and member ABC is 0.5, determine immediately after the impact (a) the angular velocity of member ABC, (b) the velocity of the sphere.

Member ABC has a mass of 2.4 kg and is attached to a pin support at B. An 800-g sphere D strikes the end of member ABC with a vertical velocity \(\mathbf{v}_{1}\) of 3 m/s. Knowing that L = 750 mm and that the coefficient of restitution between the sphere and member ABC is 0.5, determine immediately after the impact (a) the angular velocity of member ABC, (b) the velocity of the sphere.

Sphere A of mass \(m_{A}=2 \ \mathrm{kg}\) and radius r = 40 mm rolls without slipping with a velocity \(\overline{\mathbf{v}}_{1}=2 \ \mathrm{m} / \mathrm{s}\) on a horizontal surface when it hits squarely a uniform slender bar B of mass is \(m_{B}=0.5 \ \mathrm{kg}\) and length L = 100 mm that is standing on end and is at rest. Denoting by \(\mathrm{m}_{k}\) the coefficient of kinetic friction between the sphere and the horizontal surface, neglecting friction between the sphere and the bar, and knowing the coefficient of restitution between A and B is 0.1, determine the angular velocities of the sphere and the bar immediately after the impact.

A large 3-lb sphere with a radius r = 3 in. is thrown into a light basket at the end of a thin, uniform rod weighing 2 lb and length L = 10 in. as shown. Immediately before the impact the angular velocity of the rod is 3 rad/s counterclockwise and the velocity of the sphere is 2 ft/s down. Assume the sphere sticks in the basket. Determine after the impact (a) the angular velocity of the bar and sphere, (b) the components of the reactions at A.

A small rubber ball of radius r is thrown against a rough floor with a velocity \(\overline{\mathbf{v}}_{A}\) of magnitude \(v_{0}\) and a backspin \(\mathrm{V}_{A}\) of magnitude \(\mathrm{v}_{0}\). It is observed that the ball bounces from A to B, then from B to A, then from A to B, etc. Assuming perfectly elastic impact, determine the required magnitude \(\mathrm{v}_{0}\) of the backspin in terms of \(\bar{v}_{0}\) and r.

Sphere A of mass m and radius r rolls without slipping with a velocity \(\bar{v}_{1}\) on a horizontal surface when it hits squarely an identical sphere B that is at rest. Denoting by \(\mathrm{m}_{k}\) the coefficient of kinetic friction between the spheres and the surface, neglecting friction between the spheres, and assuming perfectly elastic impact, determine (a) the linear and angular velocities of each sphere immediately after the impact, (b) the velocity of each sphere after it has started rolling uniformly.

In a game of pool, ball A is rolling without slipping with a velocity \(\overline{\mathbf{v}}_{0}\) as it hits obliquely ball B, which is at rest. Denoting by r the radius of each ball and by \(\mathrm{m}_{k}\) the coefficient of kinetic friction between a ball and the table, and assuming perfectly elastic impact, determine (a) the linear and angular velocity of each ball immediately after the impact, (b) the velocity of ball B after it has started rolling uniformly.

Each of the bars AB and BC is of length L = 400 mm and mass m = 1.2 kg. Determine the angular velocity of each bar immediately after the impulse \(\mathbf{Q} \Delta t=(1.5 \ \mathrm{N} \cdot \mathrm{s}) \mathbf{i}\) is applied at C.

A uniform disk of constant thickness and initially at rest is placed in contact with the belt shown, which moves at a constant speed v = 80 ft/s. Knowing that the coefficient of kinetic friction between the disk and the belt is 0.15, determine (a) the number of revolutions executed by the disk before it reaches a constant angular velocity, (b) the time required for the disk to reach that constant angular velocity.

The 8-in.-radius brake drum is attached to a larger flywheel that is not shown. The total mass moment of inertia of the flywheel and drum is \(14 \ \mathrm{lb} \cdot \mathrm{ft} \cdot \mathrm{s}^{2}\) and the coefficient of kinetic friction between the drum and the brake shoe is 0.35. Knowing that the initial angular velocity of the flywheel is 360 rpm counterclockwise, determine the vertical force P that must be applied to the pedal C if the system is to stop in 100 revolutions.

A \(6 \times 8 \text {-in }\). rectangular plate is suspended by pins at A and B. The pin at B is removed and the plate swings freely about pin A. Determine (a) the angular velocity of the plate after it has rotated through \(90^{\circ}\), (b) the maximum angular velocity attained by the plate as it swings freely.

The gear shown has a radius R = 150 mm and a radius of gyration \(\bar{k}=125 \ \mathrm{mm}\). The gear is rolling without sliding with a velocity \(\overline{\mathbf{v}}_{1}\) of magnitude 3 m/s when it strikes a step of height h = 75 mm. Because the edge of the step engages the gear teeth, no slipping occurs between the gear and the step. Assuming perfectly plastic impact, determine (a) the angular velocity of the gear immediately after the impact, (b) the angular velocity of the gear after it has rotated to the top of the step.

A uniform slender rod is placed at corner B and is given a slight clockwise motion. Assuming that the corner is sharp and becomes slightly embedded in the end of the rod, so that the coefficient of static friction at B is very large, determine (a) the angle b through which the rod will have rotated when it loses contact with the corner, (b) the corresponding velocity of end A.

The motion of the slender 250-mm rod AB is guided by pins at A and B that slide freely in slots cut in a vertical plate as shown. Knowing that the rod has a mass of 2 kg and is released from rest when \(\mathrm{u}=0\), determine the reactions at A and B when \(\mathrm{u}=90^{\circ}\).

A 35-g bullet B is fired horizontally with a velocity of 400 m/s into the side of a 3-kg square panel suspended from a pin at A. Knowing that the panel is initially at rest, determine the components of the reaction at A after the panel has rotated \(45^{\circ}\).

Two panels A and B are attached with hinges to a rectangular plate and held by a wire as shown. The plate and the panels are made of the same material and have the same thickness. The entire assembly is rotating with an angular velocity \(\mathrm{V}_{0}\) when the wire breaks. Determine the angular velocity of the assembly after the panels have come to rest against the plate.

Disks A and B are made of the same material and are of the same thickness; they can rotate freely about the vertical shaft. Disk B is at rest when it is dropped onto disk A, which is rotating with an angular velocity of 500 rpm. Knowing that disk A has a mass of 8 kg, determine (a) the final angular velocity of the disks, (b) the change in kinetic energy of the system.

A square block of mass m is falling with a velocity \(\overline{\mathbf{v}}_{1}\) when it strikes a small obstruction at B. Knowing that the coefficient of restitution for the impact between corner A and the obstruction B is e = 0.5, determine immediately after the impact (a) the angular velocity of the block, (b) the velocity of its mass center G.

A 3-kg bar AB is attached by a pin at D to a 4-kg square plate, which can rotate freely about a vertical axis. Knowing that the angular velocity of the plate is 120 rpm when the bar is vertical, determine (a) the angular velocity of the plate after the bar has swung into a horizontal position and has come to rest against pin C, (b) the energy lost during the plastic impact at C.

A 1.8-lb javelin DE impacts a 10-lb slender rod ABC with a horizontal velocity of \(\mathbf{v}_{0}=30 \ \mathrm{ft} / \mathrm{s}\) as shown. Knowing that the javelin becomes embedded into the end of the rod at point C and does not penetrate very far into it, determine immediately after the impact (a) the angular velocity of the rod ABC, (b) the components of the reaction at B. Assume the javelin and the rod move as a single rigid body after the impact.