The rod of mass m and length L is released from rest without rotating. When it falls a

Determine the angular momentum of the 100-kg disk or rod about point G and about point O.

Determine the angular impulse about point O for t = 3 s.

The 60-kg wheel has a radius of gyration about its center O of \(k_{O}=300 \ \mathrm{mm}\). If it is subjected to a couple moment of \(M=\left(3 t^{2}\right) \ \mathrm{N} \cdot \mathrm{m}\), where t is in seconds, determine the angular velocity of the wheel when t = 4 s, starting from rest.

The 300-kg wheel has a radius of gyration about its mass center O of \(k_{O} = 400 \ mm\). If the wheel is subjected to a couple moment of \(M=300 \ \mathrm{N} \cdot \mathrm{m}\), determine its angular velocity 6 s after it starts from rest and no slipping occurs. Also, determine the friction force that the ground applies to the wheel.

If rod OA of negligible mass is subjected to the couple moment \(M=9 \ \mathrm{N} \cdot \mathrm{m}\), determine the angular velocity of the 10-kg inner gear t = 5 s after it starts from rest. The gear has a radius of gyration about its mass center of \(k_{A} = 100 \ mm\), and it rolls on the fixed outer gear, B. Motion occurs in the horizontal plane.

Gears A and B of mass 10 kg and 50 kg have radii of gyration about their respective mass centers of \(k_{A} = 80 \ mm\) and \(k_{B} = 150 \ mm\). If gear A is subjected to the couple moment \(M = 10 \ N \cdot m\) when it is at rest, determine the angular velocity of gear B when t = 5 s.

The 50-kg spool is subjected to a horizontal force of P = 150 N. If the spool rolls without slipping, determine its angular velocity 3 s after it starts from rest. The radius of gyration of the spool about its center of mass is \(k_{G} = 175 \ mm\).

The reel has a weight of 150 lb and a radius of gyration about its center of gravity of \(k_{G} = 1.25 \ ft\). If it is subjected to a torque of \(M = 25 \ lb \cdot ft\), and starts from rest when the torque is applied, determine its angular velocity in 3 seconds. The coefficient of kinetic friction between the reel and the horizontal plane is \(\mu_{k} = 0.15\).

The rigid body (slab) has a mass m and rotates with an angular velocity \(\mathbf{\omega}\) about an axis passing through the fixed point O. Show that the momenta of all the particles composing the body can be represented by a single vector having a magnitude \(m v_{G}\) and acting through point P, called the center of percussion, which lies at a distance \(r_{P / G}=k_{G}^{2} / r_{G / O}\) from the mass center G. Here \(k_{G}\) is the radius of gyration of the body, computed about an axis perpendicular to the plane of motion and passing through G.

At a given instant, the body has a linear momentum \(\mathbf{L}=m \mathbf{v}_{G}\) and an angular momentum \(\mathbf{H}_{G}=I_{G} \boldsymbol{\omega}\) computed about its mass center. Show that the angular momentum of the body computed about the instantaneous center of zero velocity IC can be expressed as \(\mathbf{H}_{l C}=I_{I C} \boldsymbol{\omega}\), where \(I_{I C}\) represents the body’s moment of inertia computed about the instantaneous axis of zero velocity. As shown, the IC is located at a distance \(r_{G / I C}\) away from the mass center G.

Show that if a slab is rotating about a fixed axis perpendicular to the slab and passing through its mass center G, the angular momentum is the same when computed about any other point P.

The 40-kg disk is rotating at \(\boldsymbol {\omega} = 100 \ rad/s\). When the force P is applied to the brake as indicated by the graph. If the coefficient of kinetic friction at B is \(\mu_{k} = 0.3\), determine the time t needed to stay the disk from rotating. Neglect the thickness of the brake.

The impact wrench consists of a slender 1-kg rod AB which is 580 mm long, and cylindrical end weights at A and B that each have a diameter of 20 mm and a mass of 1 kg. This assembly is free to turn about the handle and socket, which are attached to the lug nut on the wheel of a car. If the rod AB is given an angular velocity of 4 rad/s and it strikes the bracket C on the handle without rebounding, determine the angular impulse imparted to the lug nut.

The airplane is traveling in a straight line with a speed of 300 km/h, when the engines A and B produce a thrust of \(T_{A} = 40 \ kN\) and \(T_{B} = 20 \ kN\), respectively. Determine the angular velocity of the airplane in t = 5 s. The plane has a mass of 200 Mg, its center of mass is located at G, and its radius of gyration about G is \(k_{G} = 15 \ m\).

The double pulley consists of two wheels which are attached to one another and turn at the same rate. The pulley has a mass of 15 kg and a radius of gyration of \(k_{O} = 110 \ mm\). If the block at A has a mass of 40 kg, determine the speed of the block in 3 s after a constant force of 2 kN is applied to the rope wrapped around the inner hub of the pulley. The block is originally at rest.

The assembly weighs 10 lb and has a radius of gyration \(k_{G} = 0.6 \ ft\) about its center of mass G. The kinetic energy of the assembly is \(31 \ ft \cdot lb\) when it is in the position shown. If it rolls counterclockwise on the surface without slipping, determine its linear momentum at this instant.

The disk has a weight of 10 lb and is pinned at its center O. If a vertical force of P = 2 lb is applied to the cord wrapped around its outer rim, determine the angular velocity of the disk in four seconds starting from rest. Neglect the mass of the cord.

The 30-kg gear A has a radius of gyration about its center of mass O of \(k_{O} = 125 \ mm\). If the 20-kg gear rack B is subjected to a force of P = 200 N, determine the time required for the gear to obtain an angular velocity of 20 rad/s, starting from rest. The contact surface between the gear rack and the horizontal plane is smooth.

The pulley has a weight of 8 lb and may be treated as a thin disk. A cord wrapped over its surface is subjected to forces \(T_{A} = 4 \ lb\) and \(T_{B} = 5 \ lb\). Determine the angular velocity of the pulley when t = 4 s if it starts from rest when t = 0. Neglect the mass of the cord.

The 40-kg roll of paper rests along the wall where the coefficient of kinetic friction is \(\mu_{k} = 0.2\). If a vertical force of P = 40 N is applied to the paper, determine the angular velocity of the roll when t = 6 s starting from rest. Neglect the mass of the unraveled paper and take the radius of gyration of the spool about the axle O to be \(k_{O} = 80 \ mm\).

The slender rod has a mass m and is suspended at its end A by a cord. If the rod receives a horizontal blow giving it an impulse I at its bottom B, determine the location y of the point P about which the rod appears to rotate during the impact.

The rod of length L and mass m lies on a smooth horizontal surface and is subjected to a force P at its end A as shown. Determine the location d of the point about which the rod begins to turn, i.e, the point that has zero velocity.

A 4-kg disk A is mounted on arm BC, which has a negligible mass. If a torque of \(M=\left(5 e^{0.5 t}\right) \ \mathrm{N} \cdot \mathrm{m}\), where t is in seconds, is applied to the arm at C, determine the angular velocity of BC in 2 s starting from rest. Solve the problem assuming that (a) the disk is set in a smooth bearing at B so that it moves with curvilinear translation, (b) the disk is fixed to the shaft BC, and (c) the disk is given an initial freely spinning angular velocity of \(\boldsymbol {\omega}_{D}=\{-80 \mathbf{k}\} \ \mathrm{rad} / \mathrm{s}\) prior to application of the torque.

The frame of a tandem drum roller has a weight of 4000 lb excluding the two rollers. Each roller has a weight of 1500 lb and a radius of gyration about its axle of 1.25 ft. If a torque of \(M = 300 \ lb \cdot ft\) is supplied to the rear roller A, determine the speed of the drum roller 10 s later, starting from rest.

The 100-lb wheel has a radius of gyration of \(k_{G} = 0.75 \ ft\). If the upper wire is subjected to a tension of T = 50 lb, determine the velocity of the center of the wheel in 3 s, starting from rest. The coefficient of kinetic friction between the wheel and the surface is \(\mu_{k} = 0.1\).

The 4-kg slender rod rests on a smooth floor. If it is kicked so as to receive a horizontal impulse \(I = 8 \ N \cdot s\) at point A as shown, determine its angular velocity and the speed of its mass center.

The double pulley consists of two wheels which are attached to one another and turn at the same rate. The pulley has a mass of 15 kg and a radius of gyration \(k_{O} = 110 \ mm\). If the block at A has a mass of 40 kg, determine the speed of the block in 3 s after a constant force F = 2 kN is applied to the rope wrapped around the inner hub of the pulley. The block is originally at rest. Neglect the mass of the rope.

The 100-kg spool is resting on the inclined surface for which the coefficient of kinetic friction is \(\mu_{k} = 0.1\). Determine the angular velocity of the spool when t = 4 s after it is released from rest. The radius of gyration about the mass center is \(k_{G} = 0.25 \ m\).

The spool has a weight of 30 lb and a radius of gyration \(k_{O} = 0.45 \ ft\). A cord is wrapped around its inner hub and the end subjected to a horizontal force P = 5 lb. Determine the spool’s angular velocity in 4 s starting from rest. Assume the spool rolls without slipping.

The two gears A and B have weights and radii of gyration of \(W_{A}=15 \ \mathrm{lb}, \quad k_{A}=0.5 \ \mathrm{ft}\) and \(W_{B}=10 \ \mathrm{lb}, \quad k_{B}=0.35 \ \mathrm{ft}\), respectively. If a motor transmits a couple moment to gear B of \(M=2\left(1-e^{-0.5 t}\right) \ \mathrm{lb} \cdot \mathrm{ft}\), where t is in seconds, determine the angular velocity of gear A in t = 5 s, starting from rest.

The hoop (thin ring) has a mass of 5 kg and is released down the inclined plane such that it has a backspin \(\omega = 8 \ rad/s\) and its center has a velocity \(v_{G} = 3 \ m/s\) as shown. If the coefficient of kinetic friction between the hoop and the plane is \(\mu_{k} = 0.6\), determine how long the hoop rolls before it stops slipping.

The 30-kg gear is subjected to a force of P = (20t) N, where t is in seconds. Determine the angular velocity of the gear at t = 4 s, starting from rest. Gear rack B is fixed to the horizontal plane, and the gear’s radius of gyration about its mass center O is \(k_{O} = 125 \ mm\).

The 30-lb flywheel A has a radius of gyration about its center of 4 in. Disk B weighs 50 lb and is coupled to the flywheel by means of a belt which does not slip at its contacting surfaces. If a motor supplies a counterclockwise torque to the flywheel of \(M = (50t) \ lb \cdot ft\), where t is in seconds, determine the time required for the disk to attain an angular velocity of 60 rad/s starting from rest.

If the shaft is subjected to a torque of \(M = (15t^{2}) \ N \cdot m\), where t is in seconds, determine the angular velocity of the assembly when t = 3 s, starting from rest. Rods AB and BC each have a mass of 9 kg.

The double pulley consists of two wheels which are attached to one another and turn at the same rate. The pulley has a mass of 15 kg and a radius of gyration of \(k_{O} = 110 \ mm\). If the block at A has a mass of 40 kg and the container at B has a mass of 85 kg, including its contents, determine the speed of the container when t = 3 s after it is released from rest.

The crate has a mass \(m_{c}\). Determine the constant speed \(v_{0}\) it acquires as it moves down the conveyor. The rollers each have a radius of r, mass m, and are spaced d apart. Note that friction causes each roller to rotate when the crate comes in contact with it.

The turntable T of a record player has a mass of 0.75 kg and a radius of gyration \(k_{z} = 125 \ mm\). It is turning freely at \(\omega_{T} = 2 \ rad/s\) when a 50-g record (thin disk) falls on it. Determine the final angular velocity of the turntable just after the record stops slipping on the turntable.

The 10-g bullet having a velocity of 800 m/s is fired into the edge of the 5-kg disk as shown. Determine the angular velocity of the disk just after the bullet becomes embedded into its edge. Also, calculate the angle \(\theta\) the disk will swing when it stops. The disk is originally at rest. Neglect the mass of the rod AB.

The 10-g bullet having a velocity of 800 m/s is fired into the edge of the 5-kg disk as shown. Determine the angular velocity of the disk just after the bullet becomes embedded into its edge. Also, calculate the angle \(\theta\) the disk will swing when it stops. The disk is originally at rest. The rod AB has a mass of 3 kg.

The circular disk has a mass m and is suspended at A by the wire. If it receives a horizontal impulse I at its edge B, determine the location y of the point P about which the disk appears to rotate during the impact.

The 80-kg man is holding two dumbbells while standing on a turntable of negligible mass, which turns freely about a vertical axis. When his arms are fully extended, the turntable is rotating with an angular velocity of 0.5 rev/s. Determine the angular velocity of the man when he retracts his arms to the position shown. When his arms are fully extended, approximate each arm as a uniform 6-kg rod having a length of 650 mm, and his body as a 68-kg solid cylinder of 400-mm diameter. With his arms in the retracted position, assume the man is an 80-kg solid cylinder of 450-mm diameter. Each dumbbell consists of two 5-kg spheres of negligible size.

The platform swing consists of a 200-lb flat plate suspended by four rods of negligible weight. When the swing is at rest, the 150-lb man jumps off the platform when his center of gravity G is 10 ft from the pin at A. This is done with a horizontal velocity of 5 ft/s, measured relative to the swing at the level of G. Determine the angular velocity he imparts to the swing just after jumping off.

The 2-kg rod ACB supports the two 4-kg disks at its ends. If both disks are given a clockwise angular velocity \(\left(\omega_{A}\right)_{1}=\left(\omega_{B}\right)_{1}=5 \ \mathrm{rad} / \mathrm{s}\) while the rod is held stationary and then released, determine the angular velocity of the rod after both disks have stopped spinning relative to the rod due to frictional resistance at the pins A and B. Motion is in the horizontal plane. Neglect friction at pin C.

The satellite has a mass of 200 kg and a radius of gyration about z axis of \(k_{z} = 0.1 \ m\), excluding the two solar panels A and B. Each solar panel has a mass of 15 kg and can be approximated as a thin plate. If the satellite is originally spinning about the z axis at a constant rate \(\omega_{z} = 0.5 \ rad/s\) when \(\theta = 90^{\circ}\), determine the rate of spin if both panels are raised and reach the upward position, \(\theta = 0^{\circ}\), at the same instant.

Disk A has a weight of 20 lb. An inextensible cable is attached to the 10-lb weight and wrapped around the disk. The weight is dropped 2 ft before the slack is taken up. If the impact is perfectly elastic, i.e., e = 1, determine the angular velocity of the disk just after impact.

The plank has a weight of 30 lb, center of gravity at G, and it rests on the two sawhorses at A and B. If the end D is raised 2 ft above the top of the sawhorses and is released from rest, determine how high end C will rise from the top of the sawhorses after the plank falls so that it rotates clockwise about A, strikes and pivots on the sawhorse at B, and rotates clockwise off the sawhorse at A.

The 12-kg rod AB is pinned to the 40-kg disk. If the disk is given an angular velocity \(\omega_{D} = 100 \ rad/s\) while the rod is held stationary, and the assembly is then released, determine the angular velocity of the rod after the disk has stopped spinning relative to the rod due to frictional resistance at the bearing B. Motion is in the horizontal plane. Neglect friction at the pin A.

A thin rod of mass m has an angular velocity \(\boldsymbol {\omega}_{0}\) while rotating on a smooth surface. Determine its new angular velocity just after its end strikes and hooks onto the peg and the rod starts to rotate about P without rebounding. Solve the problem (a) using the parameters given, (b) setting m = 2 kg, \(\omega_{0} = 4 \ rad/s\), l = 1.5 m.

Tests of impact on the fixed crash dummy are conducted using the 300-lb ram that is released from rest at \(\theta = 30^{\circ}\), and allowed to fall and strike the dummy at \(\theta = 90^{\circ}\). If the coefficient of restitution between the dummy and the ram is e = 0.4, determine the angle \(\theta\) to which the ram will rebound before momentarily coming to rest.

The vertical shaft is rotating with an angular velocity of 3 rad/s when \(\theta = 0^{\circ}\). If a force F is applied to the collar so that \(\theta = 90^{\circ}\), determine the angular velocity of the shaft. Also, find the work done by force F. Neglect the mass of rods GH and EF and the collars I and J. The rods AB and CD each have a mass of 10 kg.

The mass center of the 3-lb ball has a velocity of \((v_{G})_{1} = 6 \ ft/s\) when it strikes the end of the smooth 5-lb slender bar which is at rest. Determine the angular velocity of the bar about the z axis just after impact if e = 0.8.

The pendulum consists of a slender 2-kg rod AB and 5-kg disk. It is released from rest without rotating. When it falls 0.3 m, the end A strikes the hook S, which provides a permanent connection. Determine the angular velocity of the pendulum after it has rotated \(90^{\circ}\). Treat the pendulum’s weight during impact as a nonimpulsive force.

The 10-lb block is sliding on the smooth surface when the corner D hits a stop block S. Determine the minimum velocity v the block should have which would allow it to tip over on its side and land in the position shown. Neglect the size of S. Hint: During impact consider the weight of the block to be nonimpulsive.

Determine the height h at which a billiard ball of mass m must be struck so that no frictional force develops between it and the table at A. Assume that the cue C only exerts a horizontal force P on the ball.

The pendulum consists of a 15-kg solid ball and 6-kg rod. If it is released from rest when \(\theta_{1} = 90^{\circ}\), determine the angle \(\theta_{2}\) after the ball strikes the wall, rebounds, and the pendulum swings up to the point of momentary rest. Take e = 0.6.

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

The hammer consists of a 10-kg solid cylinder C and 6-kg uniform slender rod AB. If the hammer is released from rest when \(\theta = 90^{\circ}\) and strikes the 30-kg block D when \(\theta = 0^{\circ}\), determine the velocity of block D and the angular velocity of the hammer immediately after the impact. The coefficient of restitution between the hammer and the block is e = 0.6.

The 20-kg disk strikes the step without rebounding. Determine the largest angular velocity \(\omega_{1}\) the disk can have and not lose contact with the step, A.

The solid ball of mass m is dropped with a velocity \(\mathbf{v}_{1}\) onto the edge of the rough step. If it rebounds horizontally off the step with a velocity \(\mathbf{v}_{2}\), determine the angle \(\theta\) at which contact occurs. Assume no slipping when the ball strikes the step. The coefficient of restitution is e.

The wheel has a mass of 50 kg and a radius of gyration of 125 mm about its center of mass G. Determine the minimum value of the angular velocity \(\boldsymbol {\omega}_{1}\) of the wheel, so that it strikes the step at A without rebounding and then rolls over it without slipping.

The wheel has a mass of 50 kg and a radius of gyration of 125 mm about its center of mass G. If it rolls without slipping with an angular velocity of \(\boldsymbol {\omega}_{1} = 5 \ rad/s\) before it strikes the step at A, determine its angular velocity after it rolls over the step. The wheel does not lose contact with the step when it strikes it.

The rod of mass m and length L is released from rest without rotating. When it falls a distance L, the end A strikes the hook S, which provides a permanent connection. Determine the angular velocity \(\omega\) of the rod after it has rotated \(90^{\circ}\). Treat the rod’s weight during impact as a nonimpulsive force.

The 15-lb rod AB is released from rest in the vertical position. If the coefficient or restitution between the floor and the cushion at B is e = 0.7, determine how high the end of the rod rebounds after impact with the floor.

A ball having a mass of 8 kg and initial speed of \(v_{1} = 0.2 \ m/s\) rolls over a 30-mm-long depression. Assuming that the ball rolls off the edges of contact first A, then B, without slipping, determine its final velocity \(\mathbf{v}_{2}\) when it reaches the other side.

A solid ball with a mass m is thrown on the ground such that at the instant of contact it has an angular velocity \(\boldsymbol {\omega}_{1}\) and velocity components \(\left(\mathbf{v}_{G}\right)_{x 1}\) and \(\left(\mathbf{v}_{G}\right)_{y 1}\) as shown. If the ground is rough so no slipping occurs, determine the components of the velocity of its mass center just after impact. The coefficient of restitution is e.

The pendulum consists of a 10-lb solid ball and 4-lb rod. If it is released from rest when \(\theta_{0} = 0^{\circ}\), determine the angle \(\theta_{1}\) of rebound after the ball strikes the wall and the pendulum swings up to the point of momentary rest. Take e = 0.6.

The soil compactor moves forward at constant velocity by supplying power to the rear wheels. Use appropriate numerical data for the wheel, roller, and body and calculate the angular momentum of this system about point A at the ground, point B on the rear axle, and point G, the center of gravity for the system.

The swing bridge opens and closes by turning \(90^{\circ}\) using a motor located under the center of the deck at A that applies a torque M to the bridge. If the bridge was supported at its end B, would the same torque open the bridge at the same time, or would it open slower or faster? Explain your answer using numerical values and an impulse and momentum analysis. Also, what are the benefits of making the bridge have the variable depth as shown?

Why is it necessary to have the tail blade B on the helicopter that spins perpendicular to the spin of the main blade A? Explain your answer using numerical values and an impulse and momentum analysis.

The amusement park ride consists of two gondolas A and B, and counterweights C and D that swing in opposite directions. Using realistic dimensions and mass, calculate the angular momentum of this system for any angular position of the gondolas. Explain through analysis why it is a good idea to design this system to have counterweights with each gondola.

The cable is subjected to a force of P = (10t 2) lb. where t is in seconds. Determine the angular velocity of the spool 3 s after P is applied, starting from rest. The spool has a weight of 150 lb and a radius of gyration of 1.25 ft about its center, O.

The space capsule has a mass of 1200 kg and a moment of inertia IG = 900 kg # m2 about an axis passing through G and directed perpendicular to the page. If it is traveling forward with a speed vG = 800 m>s and executes a turn by means of two jets, which provide a constant thrust of 400 N for 0.3 s, determine the capsules angular velocity just after the jets are turned off.

The tire has a mass of 9 kg and a radius of gyration kO = 225 mm. If it is released from rest and rolls down the plane without slipping, determine the speed of its center O when t = 3 s.

The wheel having a mass of 100 kg and a radius of gyration about the z axis of kz = 300 mm, rests on the smooth horizontal plane. If the belt is subjected to a force of P = 200 N, determine the angular velocity of the wheel and the speed of its center of mass O, three seconds after the force is applied.

The spool has a weight of 30 lb and a radius of gyration kO = 0.65 ft. If a force of 40 lb is applied to the cord at A, determine the angular velocity of the spool in t = 3 s starting from rest. Neglect the mass of the pulley and cord.

Spool B is at rest and spool A is rotating at 6 rad>s when the slack in the cord connecting them is taken up. If the cord does not stretch, determine the angular velocity of each spool immediately after the cord is jerked tight. The spools A and B have weights and radii of gyration WA = 30 lb, kA = 0.8 ft, WB = 15 lb, kB = 0.6 ft, respectively.

A thin disk of mass m has an angular velocity V1 while rotating on a smooth surface. Determine its new angular velocity just after the hook at its edge strikes the peg P and the disk starts to rotate about P without rebounding

The space satellite has a mass of 125 kg and a moment of inertia \(I_z = 0.940\ kg \cdot m^2\), excluding the four solar panels A, B, C, and D. Each solar panel has a mass of 20 kg and can be approximated as a thin plate. If the satellite is originally spinning about the z axis at a constant rate \(\omega_z = 0.5\ rad/s\) when \(\theta = 90^{\circ}\), determine the rate of spin if all the panels are raised and reach the upward position, \(\theta = 0^{\circ}\), at the same instant.