A uniform torus and a cylindrical ring, each solidand of mass m, are released from rest

The right-angle bar with equal legs weighs 6 lb and is freely hinged to the vertical plate at C. The bar is prevented from rotating by the two pegs A and B fixed to the plate. Determine the acceleration a of the plate for which no force is exerted on the bar by either peg A or B.

In Prob. 6 /1, if the plate is given a horizontal acceleration a = 2g, calculate the force exerted on the bar by either peg A or B.

The driver of a pickup truck accelerates from rest to a speed of 45 mi/hr over a horizontal distance of 225 ft with constant acceleration. The truck is hauling an empty 500-lb trailer with a uniform 60-lb gate hinged at O and held in the slightly tilted position by two pegs, one on each side of the trailer frame at A. Determine the maximum shearing force developed in each of the two pegs during the acceleration.

A passenger car of an overhead monorail system is driven by one of its two small wheels A or B. Select the one for which the car can be given the greater acceleration without slipping the driving wheel and compute the maximum acceleration if the effective coefficient of friction is limited to 0.25 between the wheels and the rail. Neglect the small mass of the wheels.

The uniform box of mass m slides down the rough incline. Determine the location d of the effective normal force N. The effective normal force is located at the centroid of the nonuniform pressure distribution which the incline exerts on the bottom surface of the block.

The uniform slender bar of mass m and length L is held in the position shown by the stop at A. What acceleration a will cause the normal force acting on the roller at B to become (a) one-half of the static value, (b) one-fourth of the static value, and (c) zero?

The homogeneous crate of mass m is mounted on small wheels as shown. Determine the maximum force P which can be applied without overturning the crate about (a) its lower front edge with h = b and (b) its lower back edge with h = 0.

The frame is made from uniform rod which has a mass \(\rho\) per unit length. A smooth recessed slot constrains the small rollers at A and B to travel horizontally. Force P is applied to the frame through a cable attached to an adjustable collar C. Determine the magnitudes and directions of the normal forces which act on the rollers if (a) h = 0.3L, (b) h = 0.5L, and (c) h = 0.9L. Evaluate your results for \(\rho=2\) kg /m, L = 500 mm, and P = 60 N. What is the acceleration of the frame in each case?

A uniform slender rod rests on a car seat as shown. Determine the deceleration a for which the rod will begin to tip forward. Assume that friction at B is sufficient to prevent slipping.

Determine the value of P which will cause the homogeneous cylinder to begin to roll up out of its rectangular recess. The mass of the cylinder is m and that of the cart is M. The cart wheels have negligible mass and friction.

The uniform 5-kg bar AB is suspended in a vertical position from an accelerating vehicle and restrained by the wire BC. If the acceleration is a = 0.6g, determine the tension T in the wire and the magnitude of the total force supported by the pin at A.

If the collar P is given a constant acceleration a = 3g to the right, the pendulum will assume a steady state deflection \(\theta=30^{\circ}\). Determine the stiffness \(k_{T}\) of the torsional spring which will allow this to happen. The torsional spring is undeformed when the pendulum is in the vertical position.

If the collar P of the pendulum of Prob. 6 /12 is given a constant acceleration a = 5g, what will be the steady-state deflection of the pendulum from the vertical? Use the value \(k_{T}=7 m g L\).

The uniform 30-kg bar OB is secured to the accelerating frame in the \(30^{\circ}\) position from the horizontal by the hinge at O and roller at A. If the horizontal acceleration of the frame is \(a=20 \mathrm{\ m} / \mathrm{s}^{2}\), compute the force \(F_{A}\) on the roller and the x- and y-components of the force supported by the pin at O.

The bicyclist applies the brakes as he descends the \(10^{\circ}\) incline. What deceleration a would cause the dangerous condition of tipping about the front wheel A? The combined center of mass of the rider and bicycle is at G.

The right-angle bar acts to control the maximum acceleration of an experimental vehicle by depressing the spring-loaded limit switch A with a vertical force. If a 60-percent reduction in the force applied to the button at A (relative to the static value) results in the switch being tripped, determine the maximum acceleration a which the vehicle is permitted to experience.

The 1650-kg car has its mass center at G. Calculate the normal forces \(N_{A}\) and \(N_{B}\) between the road and the front and rear pairs of wheels under conditions of maximum acceleration. The mass of the wheels is small compared with the total mass of the car. The coefficient of static friction between the road and the rear driving wheels is 0.80.

The four-wheel-drive all-terrain vehicle has a mass of 300 kg with center of mass \(G_{2}\). The driver has a mass of 85 kg with center of mass \(G_{1}\). If all four wheels are observed to spin momentarily as the driver attempts to go forward, what is the forward acceleration of the driver and ATV? The coefficient of friction between the tires and the ground is 0.40.

A cleated conveyor belt transports solid homogeneous cylinders up a 15° incline. The diameter of each cylinder is half its height. Determine the maximum acceleration which the belt may have without tipping the cylinders as it starts.

The thin hoop of negligible mass and radius r contains a homogeneous semicylinder of mass m which is rigidly attached to the hoop and positioned such that its diametral face is vertical. The assembly is centered on the top of a cart of mass M which rolls freely on the horizontal surface. If the system is released from rest, what x-directed force P must be applied to the cart to keep the hoop and semicylinder stationary with respect to the cart, and what is the resulting acceleration a of the cart? Motion takes place in the x-y plane. Neglect the mass of the cart wheels and any friction in the wheel bearings. What is the requirement on the coefficient of static friction between the hoop and cart?

Determine the magnitude P and direction \(\theta\) of the force required to impart a rearward acceleration \(a=5 \mathrm{ft} / \mathrm{sec}^{2}\) to the loaded wheelbarrow with no rotation from the position shown. The combined weight of the wheelbarrow and its load is 500 lb with center of gravity at G. Compare the normal force at B under acceleration with that for static equilibrium in the position shown. Neglect the friction and mass of the wheel.

The mine skip has a loaded mass of 2000 kg and is attached to the towing vehicle by the light hinged link CD. If the towing vehicle has an acceleration of \(3 \mathrm{~m} / \mathrm{s}^{2}\), calculate the corresponding reactions under the small wheels at A and B.

The block A and attached rod have a combined mass of 60 kg and are confined to move along the 60° guide under the action of the 800-N applied force. The uniform horizontal rod has a mass of 20 kg and is welded to the block at B. Friction in the guide is negligible. Compute the bending moment M exerted by the weld on the rod at B.

The homogeneous rectangular plate weighs 40 lb and is supported in the vertical plane by the light parallel links shown. If a couple M = 80 lb-ft is applied to the end of link AB with the system initially at rest, calculate the force supported by the pin at C as the plate lifts off its support with \(\theta=30^{\circ}\).

A jet transport with a landing speed of 200 km/h reduces its speed to 60 km/h with a negative thrust R from its jet thrust reversers in a distance of 425 m along the runway with constant deceleration. The total mass of the aircraft is 140 Mg with mass center at G. Compute the reaction N under the nose wheel B toward the end of the braking interval and prior to the application of mechanical braking. At the lower speed, aerodynamic forces on the aircraft are small and may be neglected.

The 1300-lb homogeneous plate is suspended from the overhead carriage by the two parallel steel cables. What acceleration a will cause the tensions in the two cables to be equal? What is the resulting steady-state deflection \(\theta\) of the cables from the vertical? Evaluate your results for the case where b = 2h.

The uniform L-shaped bar pivots freely at point P of the slider, which moves along the horizontal rod. Determine the steady-state value of the angle \(\theta\) if (a) a = 0 and (b) a = g/2. For what value of a would the steady-state value of \(\theta\) be zero?

The 30,000-lb concrete pipe section is being transported on a flatbed truck. Five inextensible cables are passed across the top of the pipe and tightened securely to the flatbed with an initial tension of 2000 lb. What is the maximum deceleration a which the truck can experience if the pipe is to remain stationary relative to the truck? The coefficient of static friction between the concrete and the flatbed is 0.80, and that between the cables and the concrete is 0.75.

Determine the maximum counterweight W for which the loaded 4000-lb coal car will not overturn about the rear wheels B. Neglect the mass of all pulleys and wheels. (Note that the tension in the cable at C is not 2W.)

The 1800-kg rear-wheel-drive car accelerates forward at a rate of g /2. If the modulus of each of the rear and front springs is 35 kN/m, estimate the resulting momentary nose-up pitch angle \(\theta\). (This upward pitch angle during acceleration is called squat, while the downward pitch angle during braking is called dive!) Neglect the unsprung mass of the wheels and tires. (Hint: Begin by assuming a rigid vehicle.)

The experimental Formula One race car is traveling at 300 km / h when the driver begins braking to investigate the behavior of the extreme-grip tires. An accelerometer in the car records a maximum deceleration of 4g when both the front and rear tires are on the verge of slipping. The car and driver have a combined mass of 690 kg with mass center G. The horizontal drag acting on the car at this speed is 4 kN and may be assumed to pass through the mass center G. The downforce acting over the body of the car at this speed is 13 kN. For simplicity, assume that 35% of this force acts directly over the front wheels, 40% acts directly over the rear wheels, and the remaining portion acts at the mass center. What is the necessary coefficient of friction \(\mu\) between the tires and the road for this condition? Compare your results with those for passenger car tires.

The uniform 225-lb crate is supported by the thin homogeneous 40-lb platform BF and light support links whose motion is controlled by the hydraulic cylinder CD. If the cylinder is extending at a constant rate of 6 in. /sec when \(\theta=75^{\circ}\), determine the magnitudes of the forces supported by the pins at B and F. Additionally, determine the total friction force acting on the crate. The crate is centered on the 6-ft platform, and friction is sufficient to keep the crate motionless relative to the platform. (Hint: Be careful with the location of the resultant normal force beneath the crate.)

Two pulleys are fastened together to form an integral unit. At a certain instant, the indicated belt tensions act on the unit and the unit is turning counterclockwise. Determine the angular acceleration of the unit for this instant if the moment due to friction in the bearing at O is 2.5 N?m.

The uniform 20-kg slender bar is pivoted at O and swings freely in the vertical plane. If the bar is released from rest in the horizontal position, calculate the initial value of the force R exerted by the bearing on the bar an instant after release.

The figure shows an overhead view of a hydraulically-operated gate. As fluid enters the piston side of the cylinder near A, the rod at B extends causing the gate to rotate about a vertical axis through O. For a 2-in.-diameter piston, what fluid pressure p will give the gate an initial counterclockwise angular acceleration of 4 rad /sec? The radius of gyration about O for the 500-lb gate is \(k_{O}=38 \text { in }\).

The uniform 100-kg beam is freely hinged about its upper end A and is initially at rest in the vertical position with \(\theta=0\). Determine the initial angular acceleration of the beam and the magnitude \(F_{A}\) of the force supported by the pin at A due to the application of the force P = 300 N on the attached cable.

The motor M is used to hoist the 12,000-lb stadium panel (centroidal radius of gyration \(\bar{k}=6.5 \mathrm{ft}\)) into position by pivoting the panel about its corner A. If the motor is capable of producing 5000 lb-ft of torque, what pulley diameter d will give the panel an initial counterclockwise angular acceleration of \(1.5 \mathrm{deg} / \mathrm{sec}^{2}\)? Neglect all friction.

A momentum wheel for dynamics-class demonstrations is shown. It is basically a bicycle wheel modified with rim band-weighting, handles, and a pulley for cord startup. The heavy rim band causes the radius of gyration of the 7-lb wheel to be 11 in. If a steady 10-lb pull T is applied to the cord, determine the angular acceleration of the wheel. Neglect bearing friction.

Each of the two drums and connected hubs of 8-in. radius weighs 200 lb and has a radius of gyration about its center of 15 in. Calculate the angular acceleration of each drum. Friction in each bearing is negligible.

Determine the angular acceleration and the force on the bearing at O for (a) the narrow ring of mass m and (b) the fl at circular disk of mass m immediately after each is released from rest in the vertical plane with OC horizontal.

The uniform 5-kg portion of a circular hoop is released from rest while in the position shown where the torsional spring of stiffness \(k_{T}=15 \mathrm{~N} \cdot \mathrm{m} / \mathrm{rad}\) has been twisted 90° clockwise from its undeformed position. Determine the magnitude of the pin force at O at the instant of release. Motion takes place in a vertical plane and the hoop radius is r = 150 mm.

The 30-in. slender bar weighs 20 lb and is mounted on a vertical shaft at O. If a torque M = 100 lb-in. is applied to the bar through its shaft, calculate the horizontal force R on the bearing as the bar starts to rotate.

The half ring of mass m and radius r is welded to a small horizontal shaft mounted in a bearing as shown. Neglect the mass of the shaft and determine the angular acceleration of the ring when a torque M is applied to the shaft.

The uniform plate of mass m is released from rest while in the position shown. Determine the initial angular acceleration \(\alpha\) of the plate and the magnitude of the force supported by the pin at O. The axis of rotation is horizontal.

The uniform slender bar AB has a mass of 8 kg and swings in a vertical plane about the pivot at A. If \(\dot{\theta}=2 \mathrm{rad} / \mathrm{s}\) when \(\theta=30^{\circ}\), compute the force supported by the pin at A at that instant.

The uniform 16.1-lb slender bar is hinged about a horizontal axis through O and released from rest in the horizontal position. Determine the distance b from the mass center to O which will result in an initial angular acceleration of \(\text { 16.1 } \mathrm{rad} / \mathrm{sec}^{2}\), and find the force R on the bar at O just after release.

The uniform quarter-circular sector of mass m is released from rest with one straight edge vertical as shown. Determine the initial angular acceleration and the horizontal and vertical components of the reaction at the ideal pivot at O.

The 15-kg uniform steel plate is freely hinged about the horizontal z-axis. Calculate the force supported by each of the bearings at A and B an instant after the plate is released from rest while in the horizontal position shown.

The square frame is composed of four equal lengths of uniform slender rod, and the ball attachment at O is suspended in a socket (not shown). Beginning from the position shown, the assembly is rotated 45° about axis A-A and released. Determine the initial angular acceleration of the frame. Repeat for a 45° rotation about axis B-B. Neglect the small mass, offset, and friction of the ball.

A uniform torus and a cylindrical ring, each solid and of mass m, are released from rest in the positions shown. Determine the magnitude of the pin reaction at O and the angular acceleration of each body an instant after release. Neglect friction in the pivot at O for each case.

A reel of flexible power cable is mounted on the dolly, which is fixed in position. There are 200 ft of cable weighing 0.436 lb per foot of length wound on the reel at a radius of 15 in. The empty spool weighs 62 lb and has a radius of gyration about its axis of 12 in. A tension T of 20 lb is required to overcome frictional resistance to turning. Calculate the angular acceleration \(\alpha\) of the reel if a tension of 40 lb is applied to the free end of the cable.

The uniform bar of mass m is supported by the smooth pin at O and is connected to the cylinder of mass \(m_{1}\) by the light cable which passes over the light pulley at C. If the system is released from rest while in the position shown, determine the tension in the cable. Use the values m = 30 kg, \(m_{1}=20 \mathrm{~kg}\), and L = 6 m.

An air table is used to study the elastic motion of flexible spacecraft models. Pressurized air escaping from numerous small holes in the horizontal surface provides a supporting air cushion which largely eliminates friction. The model shown consists of a cylindrical hub of radius r and four appendages of length l and small thickness t. The hub and the four appendages all have the same depth d and are constructed of the same material of density \(\rho\). Assume that the spacecraft is rigid and determine the moment M which must be applied to the hub to spin the model from rest to an angular velocity \(\omega\) in a time period of \(\tau\) seconds. (Note that for a spacecraft with highly flexible appendages, the moment must be judiciously applied to the rigid hub to avoid undesirable large elastic deflections of the appendages.)

A vibration test is run to check the design adequacy of bearings A and B. The unbalanced rotor and attached shaft have a combined mass of 2.8 kg. To locate the mass center, a torque of \(0.660 \mathrm{~N} \cdot \mathrm{m}\) is applied to the shaft to hold it in equilibrium in a position rotated 90° from that shown. A constant torque \(M=1.5 \mathrm{~N} \cdot \mathrm{m}\) is then applied to the shaft, which reaches a speed of 1200 rev /min in 18 revolutions starting from rest. (During each revolution the angular acceleration varies, but its average value is the same as for constant acceleration.) Determine (a) the radius of gyration k of the rotor and shaft about the rotation axis, (b) the force F which each bearing exerts on the shaft immediately after M is applied, and (c) the force R exerted by each bearing when the speed of 1200 rev /min is reached and M is removed. Neglect any frictional resistance and the bearing forces due to static equilibrium.

The solid cylindrical rotor B has a mass of 43 kg and is mounted on its central axis C-C. The frame A rotates about the fixed vertical axis O-O under the applied torque \(M=30 \mathrm{~N} \cdot \mathrm{m}\). The rotor may be unlocked from the frame by withdrawing the locking pin P. Calculate the angular acceleration \(\alpha\) of the frame A if the locking pin is (a) in place and (b) withdrawn. Neglect all friction and the mass of the frame.

The right-angle body is made of uniform slender bar of mass m and length L. It is released from rest while in the position shown. Determine the initial angular acceleration \(\alpha\) of the body and the magnitude of the force supported by the pivot at O.

Each of the two grinding wheels has a diameter of 6 in., a thickness of 3 /4 in., and a specific weight of \(425 \mathrm{lb} / \mathrm{ft}^{3}\). When switched on, the machine accelerates from rest to its operating speed of 3450 rev /min in 5 sec. When switched off, it comes to rest in 35 sec. Determine the motor torque and frictional moment, assuming that each is constant. Neglect the effects of the inertia of the rotating motor armature.

The uniform slender bar is released from rest in the horizontal position shown. Determine the value of x for which the angular acceleration is a maximum, and determine the corresponding angular acceleration \(\alpha\). .

The assembly from Prob. 3 /219 is repeated here with the following additional information. The 2-kg collar at C has an outer diameter of 80 mm and is press fitted to the light 50-mm-diameter shaft. Each spoke has a mass of 1.5 kg and carries a 3-kg sphere with a radius of 40 mm attached to its end. The pulley at D has a mass of 5 kg with a centroidal radius of gyration of 60 mm. If a tension T = 20 N is applied to the end of the securely wrapped cable with the assembly initially at rest, determine the initial angular acceleration of the assembly. Neglect friction in the bearings at A and B and state any assumptions.

The right-angle plate is formed from a fl at plate having a mass per unit area and is welded to the horizontal shaft mounted in the bearing at O. If the shaft is free to rotate, determine the initial angular acceleration \(\alpha\) of the plate when it is released from rest with the upper surface in the horizontal plane. Also determine the y- and z-components of the resultant force on the shaft at O.

The semicircular disk of mass m and radius r is released from rest at \(\theta=0\) and rotates freely in the vertical plane about its fixed bearing at O. Derive expressions for the n- and t-components of the force F on the bearing as functions of \(\theta\).

The uniform steel I-beam has a mass of 300 kg and is supported in the vertical plane as shown. Calculate the force R supported by the pin at O for the condition immediately after the support at B is suddenly removed. The mass of the bracket on the left end is small and may be neglected. Also treat the beam as a uniform slender bar.

The gear train shown operates in a horizontal plane and is used to transmit motion to the rack D of mass \(m_{D}\). If an input torque M is applied to gear A, what will be the resulting acceleration a of the unloaded rack? (The mechanism which it normally drives has been disengaged.) Gear C is keyed to the same shaft as gear B. Gears A, B, and C have pitch diameters \(d_{A}\), \(d_{B}\), and \(d_{C}\), and centroidal mass moments of inertia \(I_{A}\), \(I_{B}\), and \(I_{C}\), respectively. All friction is negligible.

The uniform semicircular ring of mass m = 2.5 kg and mean radius r = 200 mm is mounted on spokes of negligible mass and pivoted about a horizontal axis through O. If the ring is released from rest in the position \(\theta=30^{\circ}\), determine the force R supported by the bearing O just after release.

The link B weighs 0.80 lb with center of mass 2.20 in. from O-O and has a radius of gyration about O-O of 2.76 in. The link is welded to the steel tube and is free to rotate about the fixed horizontal shaft at O-O. The tube weighs 1.84 lb. If the tube is released from rest with the link in the horizontal position, calculate the initial angular acceleration \(\alpha\) of the assembly and the corresponding reaction O exerted by the shaft on the link.

A flexible cable 60 meters long with a mass of 0.160 kg per meter of length is wound around the reel. With y = 0, the weight of the 4-kg cylinder is required to start turning the reel to overcome friction in its bearings. Determine the downward acceleration a in meters per second squared of the cylinder as a function of y in meters. The empty reel has a mass of 16 kg with a radius of gyration about its bearing of 200 mm.

The uniform 72-ft mast weighs 600 lb and is hinged at its lower end to a fixed support at O. If the winch C develops a starting torque of 900 lb-ft, calculate the total force supported by the pin at O as the mast begins to lift off its support at B. Also find the corresponding angular acceleration \(\alpha\) of the mast. The cable at A is horizontal, and the mass of the pulleys and winch is negligible.

The robotic device consists of the stationary pedestal OA, arm AB pivoted at A, and arm BC pivoted at B. The rotation axes are normal to the plane of the figure. Estimate (a) the moment \(M_{A}\) applied to arm AB required to rotate it about joint A at \(4 \mathrm{rad} / \mathrm{s}^{2}\) counterclockwise from the position shown with joint B locked and (b) the moment \(M_{B}\) applied to arm BC required to rotate it about joint B at the same rate with joint A locked. The mass of arm AB is 25 kg and that of BC is 4 kg, with the stationary portion of joint A excluded entirely and the mass of joint B divided equally between the two arms. Assume that the centers of mass \(G_{1}\) and \(G_{2}\) are in the geometric centers of the arms and model the arms as slender rods.

For the beam described in Prob. 6 /62, determine the maximum angular velocity \(\omega\) reached by the beam as it rotates in the vertical plane about the bearing at O. Also determine the corresponding force R supported by the pin at O for this condition. Again, treat the beam as a slender 300-kg bar and neglect the mass of the supporting bracket.

The uniform parabolic plate has a mass per unit area of \(225 \mathrm{~kg} / \mathrm{m}^{2}\). If the plate is released from rest while in the horizontal position shown, determine the initial angular acceleration \(\alpha\) about the bearing axis AB.

The uniform slender bar rests on a smooth horizontal surface when a force F is applied normal to the bar at point A. Point A is observed to have an initial acceleration \(a_{A}\) of \(20 \mathrm{~m} / \mathrm{s}^{2}\), and the bar has a corresponding angular acceleration \(\alpha\) of \(18 \mathrm{rad} / \mathrm{s}^{2}\). Determine the distance b.

The 64.4-lb solid circular disk is initially at rest on the horizontal surface when a 3-lb force P, constant in magnitude and direction, is applied to the cord wrapped securely around its periphery. Friction between the disk and the surface is negligible. Calculate the angular velocity \(\omega\) of the disk after the 3-lb force has been applied for 2 seconds and find the linear velocity v of the center of the disk after it has moved 3 feet from rest.

A long cable of length L and mass \(\rho\) per unit length is wrapped around the periphery of a spool of negligible mass. One end of the cable is fixed, and the spool is released from rest in the position shown. Find the initial acceleration a of the center of the spool.

The uniform semicircular rod of mass m is lying motionless on the smooth horizontal surface when the force F is applied at B as shown. Determine the resulting initial acceleration of point A.

Repeat Prob. 6 /74, except that the direction of the applied force has been changed as shown in the figure.

The spacecraft is spinning with a constant angular velocity about the z-axis at the same time that its mass center O is traveling with a velocity \(v_{O}\) in the y-direction. If a tangential hydrogen-peroxide jet is fi red when the craft is in the position shown, determine the expression for the absolute acceleration of point A on the spacecraft rim at the instant the jet force is F. The radius of gyration of the craft about the z-axis is k, and its mass is m.

The body consists of a uniform slender bar and a uniform disk, each of mass m/2. It rests on a smooth surface. Determine the angular acceleration \(\alpha\) and the acceleration of the mass center of the body when the force P = 6 N is applied as shown. The value of the mass m of the entire body is 1.2 kg.

Determine the angular acceleration of each of the two wheels as they roll without slipping down the inclines. For wheel A investigate the case where the mass of the rim and spokes is negligible and the mass of the bar is concentrated along its centerline. For wheel B assume that the thickness of the rim is negligible compared with its radius so that all of the mass is concentrated in the rim. Also specify the minimum coefficient of static friction \(\mu_{s}\) required to prevent each wheel from slipping.

The solid homogeneous cylinder is released from rest on the ramp. If \(\theta=40^{\circ}, \mu_{s}=0.30, \text { and } \mu_{k}=0.20\), determine the acceleration of the mass center G and the friction force exerted by the ramp on the cylinder.

The 30-kg spool of outer radius \(r_{o}=450 \mathrm{mm}\) has a centroidal radius of gyration \(\bar{k}=275 \mathrm{~mm}\) and a central shaft of radius \(r_{i}=200 \mathrm{~mm}\). The spool is at rest on the incline when a tension T = 300 N is applied to the end of a cable which is wrapped securely around the central shaft as shown. Determine the acceleration of the spool center G and the magnitude and direction of the friction force acting at the interface of the spool and incline. The friction coefficients there are \(\mu_{s}=0.45 \text { and } \mu_{k}=0.30\). The tension T is applied parallel to the incline and the angle \(\theta=20^{\circ}\).

Repeat Prob. 6 /80 for the case where the cable configuration has been changed as shown in the figure.

The fairing which covers the spacecraft package in the nose of the booster rocket is jettisoned when the rocket is in space where gravitational attraction is negligible. A mechanical actuator moves the two halves slowly from the closed position I to position II at which point the fairings are released to rotate freely about their hinges at O under the influence of the constant acceleration a of the rocket. When position III is reached, the hinge at O is released and the fairings drift away from the rocket. Determine the angular velocity \(\omega\) of the fairing at the 90° position. The mass of each fairing is m with center of mass at G and radius of gyration \(k_{O}\) about O.

The uniform heavy bar AB of mass m is moving on its light end rollers along the horizontal with a velocity v when end A passes point C and begins to move on the curved portion of the path with radius r. Determine the force exerted by the path on roller A immediately after it passes C.

The uniform slender bar of mass m and total length L is released from rest in the position shown. Determine the force supported by the small roller at A and the acceleration of roller A along the smooth guide. Evaluate your results for \(\theta=15^{\circ}\).

During a test, the car travels in a horizontal circle of radius R and has a forward tangential acceleration a. Determine the lateral reactions at the front and rear wheel pairs if (a) the car speed v = 0 and (b) the speed v ? 0. The car mass is m and its polar moment of inertia (about a vertical axis through G) is \(\bar{I}\). Assume that \(R \gg d\).

The system of Prob. 6 /20 is repeated here. If the hoop- and semicylinder-assembly is centered on the top of the stationary cart and the system is released from rest, determine the initial acceleration a of the cart and the angular acceleration of the hoop and semicylinder. Friction between the hoop and cart is suffi cient to prevent slip. Motion takes place in the x-y plane. Neglect the mass of the cart wheels and any friction in the wheel bearings.

The 9-ft steel beam weighs 280 lb and is hoisted from rest where the tension in each of the cables is 140 lb. If the hoisting drums are given initial angular accelerations \(\alpha_{1}=4 \mathrm{rad} / \mathrm{sec}^{2} \text { and } \alpha_{2}=6 \mathrm{rad} / \mathrm{sec}^{2}\), calculate the corresponding tensions \(T_{A} \text { and } T_{B}\) in the cables. The beam may be treated as a slender bar.

The system is released from rest with the cable taut, and the homogeneous cylinder does not slip on the rough incline. Determine the angular acceleration of the cylinder and the minimum coefficient \(\mu_{s}\) of friction for which the cylinder will not slip.

The mass center G of the 20-lb wheel is off center by 0.50 in. If G is in the position shown as the wheel rolls without slipping through the bottom of the circular path of 6-ft radius with an angular velocity \(\omega\) of 10 rad /sec, compute the force P exerted by the path on the wheel. (Be careful to use the correct mass-center acceleration.)

End A of the uniform 5-kg bar is pinned freely to the collar, which has an acceleration \(a=4 \mathrm{~m} / \mathrm{s}^{2}\) along the fixed horizontal shaft. If the bar has a clockwise angular velocity \(\omega=2 \mathrm{rad} / \mathrm{s}\) as it swings past the vertical, determine the components of the force on the bar at A for this instant.

The uniform rectangular panel of mass m is moving to the right when wheel B drops off the horizontal support rail. Determine the resulting angular acceleration and the force \(T_{A}\) in the strap at A immediately after wheel B rolls off the rail. Neglect friction and the mass of the small straps and wheels.

The truck, initially at rest with a solid cylindrical roll of paper in the position shown, moves forward with a constant acceleration a. Find the distance s which the truck goes before the paper rolls off the edge of its horizontal bed. Friction is sufficient to prevent slipping.

The crank OA rotates in the vertical plane with a constant clockwise angular velocity \(\omega_{0}\) of 4.5 rad/s. For the position where OA is horizontal, calculate the force under the light roller B of the 10-kg slender bar AB.

The uniform rectangular 300-lb plate is held in the horizontal position by two cables each of length L = 3 ft. If the cable at A suddenly breaks, calculate the tension \(T_{B}\) in the cable at B an instant after the break occurs.

The rectangular plate of Prob. 6 /94 is repeated here. The cables at A and B are now attached to a 50-lb trolley which is constrained to move in the horizontal guide. If the cable at A suddenly breaks, calculate the tension \(T_{B}\) in the cable at B an instant after the break occurs and the acceleration \(a_{T}\) of the trolley in the guide. Neglect friction and the mass of the trolley wheels.

The robotic device of Prob. 6 /68 is repeated here. Member AB is rotating about joint A with a counterclockwise angular velocity of 2 rad/s, and this rate is increasing at \(4 \mathrm{rad} / \mathrm{s}^{2}\). Determine the moment \(M_{B}\)\) exerted by arm AB on arm BC if joint B is held in a locked condition. The mass of arm BC is 4 kg, and the arm may be treated as a uniform slender rod

Small ball-bearing rollers mounted on the ends of the slender bar of mass m and length l constrain the motion of the bar in the horizontal x-y slots. If a couple M is applied to the bar initially at rest at \(\theta=45^{\circ}\), determine the forces exerted on the rollers at A and B as the bar starts to move.

The assembly consisting of a uniform slender bar (mass m /5) and a rigidly attached uniform disk (mass 4m /5) is freely pinned to point O on the collar that in turn slides on the fixed horizontal guide. The assembly is at rest when the collar is given a sudden acceleration a to the left as shown. Determine the initial angular acceleration of the assembly.

The yo-yo has a mass m and a radius of gyration k about its center O. The cord has a maximum length y = L and is wound around the small inner hub of radius r with its end secured to a point on the hub. If the yo-yo is released from the position y = 0 with a downward velocity \(v_{O}\) of its center O, determine the tension T in the cord and the acceleration a of its center during its downward and upward motions. Also find the maximum downward velocity v of its center.

The uniform 12-ft pole is hinged to the truck bed and released from the vertical position as the truck starts from rest with an acceleration of \(3 \mathrm{ft} / \mathrm{sec}^{2}\). If the acceleration remains constant during the motion of the pole, calculate the angular velocity \(\omega\) of the pole as it reaches the horizontal position.

The uniform bar of mass m is constrained by the light rollers which move in the smooth guide, which lies in a vertical plane. If the bar is released from rest, what is the force at each roller an instant after release? Use the values m = 18 kg and r = 150 mm.

A bowling ball with a circumference of 27 in. weighs 14 lb and has a centroidal radius of gyration of 3.28 in. If the ball is released with a velocity of 20 ft/sec but with no angular velocity as it touches the alley floor, compute the distance traveled by the ball before it begins to roll without slipping. The coefficient of friction between the ball and the floor is 0.20.

The fi gure shows the edge view of a uniform concrete slab with a mass of 12 Mg. The slab is being hoisted slowly by the winch D with cable attached to the dolly. At the position \(\theta=60^{\circ}\), the distance x from the fixed ground position to the dolly is equal to the length L = 4 m of the slab. If the hoisting cable should break at this position, determine the initial acceleration \(a_{A}\) of the small dolly, whose mass is negligible, and the initial tension T in the fixed cable. End A of the slab will not slip on the dolly.

In a study of head injury against the instrument panel of a car during sudden or crash stops where lap belts without shoulder straps or airbags are used, the segmented human model shown in the figure is analyzed. The hip joint O is assumed to remain fixed relative to the car, and the torso above the hip is treated as a rigid body of mass m freely pivoted at O. The center of mass of the torso is at G with the initial position of OG taken as vertical. The radius of gyration of the torso about O is \(k_{O}\). If the car is brought to a sudden stop with a constant deceleration a, determine the speed v relative to the car with which the model’s head strikes the instrument panel. Substitute the values m = 50 kg, \(\bar{r}=450 \mathrm{~mm}\), r = 800 mm, kO = 550 mm, \(\theta=45^{\circ}\), and a = 10g and compute v.

The 25-mm-thick uniform steel plate is being pulled very slowly into position by the cable connected to a winch. If the cable breaks at the instant represented, what are the force under each roller and the acceleration \(\mathbf{a}_{G}\) of the center of mass? Neglect friction and the mass of the rollers, and consider the rollers small. Evaluate your results for a = 3 m, b = 6 m, and r = 6 m. (Note: Arc BC is of radius r, the same as arc BD.)

The connecting rod AB of a certain internal-combustion engine weighs 1.2 lb with mass center at G and has a radius of gyration about G of 1.12 in. The piston and piston pin A together weigh 1.80 lb. The engine is running at a constant speed of 3000 rev/min, so that the angular velocity of the crank is \(3000(2 \pi) / 60=100 \pi \mathrm{rad} / \mathrm{sec}\). Neglect the weights of the components and the force exerted by the gas in the cylinder compared with the dynamic forces generated and calculate the magnitude of the force on the piston pin A for the crank angle \(\theta=90^{\circ}\). (Suggestion: Use the alternative moment relation, Eq. 6/3, with B as the moment center.)

The truck carries a 1500-mm-diameter spool of cable with a mass of 0.75 kg per meter of length. There are 150 turns on the full spool. The empty spool has a mass of 140 kg with radius of gyration of 530 mm. The truck alone has a mass of 2030 kg with mass center at G. If the truck starts from rest with an initial acceleration of 0.2g, determine (a) the tension T in the cable where it attaches to the wall and (b) the normal reaction under each pair of wheels. Neglect the rotational inertia of the truck wheels.

The four-bar mechanism lies in a vertical plane and is controlled by crank OA which rotates counterclockwise at a steady rate of 60 rev /min. Determine the torque M which must be applied to the crank at O when the crank angle \(\theta=45^{\circ}\). The uniform coupler AB has a mass of 7 kg, and the masses of crank OA and the output arm BC may be neglected.

Repeat the analysis of Prob. 6/108 with the added information that the mass of crank OA is 1.2 kg and the mass of the output arm BC is 1.8 kg. Each of these bars may be considered uniform for this analysis.

The Ferris wheel at an amusement park has an even number n of gondolas, each freely pivoted at its point of support on the wheel periphery. Each gondola has a loaded mass m, a radius of gyration k about its point of support A, and a mass center a distance h from A. The wheel structure has a moment of inertia \(I_{O}\) about its bearing at O. Determine an expression for the tangential force F which must be transmitted to the wheel periphery at C in order to give the wheel an initial angular acceleration starting from rest. Suggestion: Analyze the gondolas in pairs A and B. Be careful not to assume that the initial angular acceleration of the gondolas is the same as that of the wheel. (Note: An American engineer named George Washington Gale Ferris, Jr., created a giant amusement-wheel ride for the World’s Columbian Exposition in Chicago in 1893. The wheel was 250 ft in diameter with 36 gondolas, each of which carried up to 60 passengers. Fully loaded, the wheel and gondolas had a mass of 1200 tons. The ride was powered by a 1000-hp steam engine.)

The slender rod of mass m and length l has a particle (negligible radius, mass 2m) attached to its end. If the body is released from rest when in the position shown, determine its angular velocity as it passes the vertical position.

The log is suspended by the two parallel 5-m cables and used as a battering ram. At what angle \(\theta\) should the log be released from rest in order to strike the object to be smashed with a velocity of 4 m /s?

The assembly is constructed of homogeneous slender rod which has a mass per unit length, and it rotates freely about a horizontal axis through the pivot at O. If the assembly is nudged from the starting position shown, determine its angular velocity after it has rotated (a) 45°, (b) 90°, and (c) 180°.

The velocity of the 8-kg cylinder is 0.3 m /s at a certain instant. What is its speed v after dropping an additional 1.5 m? The mass of the grooved drum is 12 kg, its centroidal radius of gyration is \(\bar{k}=210 \mathrm{~mm}\), and the radius of its groove is \(r_{i}=200 \mathrm{~mm}\). The frictional moment at O is a constant 3 N?m.

The 32.2-lb wheel is released from rest and rolls on its hubs without slipping. Calculate the speed v of the center O of the wheel after it has moved a distance x = 10 ft down the incline. The radius of gyration of the wheel about O is 5 in.

The uniform semicircular bar of radius r = 75 mm and mass m = 3 kg rotates freely about a horizontal axis through the pivot O. The bar is initially held in position 1 against the action of the torsional spring and then suddenly released. Determine the spring stiffness \(k_{T}\) which will give the bar a counterclockwise angular velocity \(\omega=4 \mathrm{rad} / \mathrm{s}\) when it reaches position 2, at which the spring is undeformed.

The homogeneous rectangular crate weighs 250 lb and is supported in the horizontal position by the cable at A and the corner hinge at O. If the cable at A is suddenly released, calculate the angular velocity \(\omega\) of the crate just before it strikes the 30° incline. Does the weight of the crate influence the results, other quantities unchanged?

The 24-lb disk is rigidly attached to the 7-lb bar OA, which is pivoted freely about a horizontal axis through point O. If the system is released from rest in the position shown, determine the angular velocity of the bar and the magnitude of the pin reaction at O after the bar has rotated 90°.

The two wheels of Prob. 6 /78, shown again here, represent two extreme conditions of distribution of mass. For case A all of the mass m is assumed to be concentrated in the center of the hoop in the axial bar of negligible diameter. For case B all of the mass m is assumed to be concentrated in the rim. Determine the speed of the center of each hoop after it has traveled a distance x down the incline from rest. The hoops roll without slipping.

The 15-kg slender bar OA is released from rest in the vertical position and compresses the spring of stiffness k = 20 kN/m as the horizontal position is passed. Determine the proper setting of the spring, by specifying the distance h, which will result in the bar having an angular velocity \(\omega=4 \mathrm{rad} / \mathrm{s}\) as it crosses the horizontal position. What is the effect of x on the dynamics of the problem?

The light circular hoop of radius r contains a quarter-circular sector of mass m and is initially at rest on the horizontal surface. A couple M is applied to the hoop, which rolls without slipping along the horizontal surface. Determine the velocity v of the hoop after it has rolled through onehalf of a revolution. Evaluate for mg = 6 lb, r = 9 in., and M = 4 lb-ft.

A steady 5-lb force is applied normal to the handle of the hand-operated grinder. The gear inside the housing with its shaft and attached handle together weigh 3.94 lb and have a radius of gyration about their axis of 2.85 in. The grinding wheel with its attached shaft and pinion (inside housing) together weigh 1.22 lb and have a radius of gyration of 2.14 in. If the gear ratio between gear and pinion is 4:1, calculate the speed N of the grinding wheel after 6 complete revolutions of the handle starting from rest.

The fi gure shows an impact tester used in studying material response to shock loads. The 60-lb pendulum is released from rest and swings through the vertical with negligible resistance. At the bottommost point of the ensuing motion, the pendulum strikes a notched material specimen A. After impact with the specimen, the pendulum swings upward to a height \(\bar{h}^{\prime}=3.17 \mathrm{ft}\). If the impact-energy capacity of the pendulum is 300 ft-lb, determine the change in the angular velocity of the pendulum during the interval from just before to just after impact with the specimen. The center of mass distance \(\bar{r}\) from O and the radius of gyration kO of the pendulum about O are both 35.5 in. (Note: Positioning the center of mass directly at the radius of gyration eliminates shock loads on the bearing at O and extends the life of the tester significantly.)

The uniform rectangular plate is released from rest in the position shown. Determine the maximum angular velocity \(\omega\) during the ensuing motion. Friction at the pivot is negligible

The 50-kg flywheel has a radius of gyration \(\bar{k}=0.4 \mathrm{~m}\) about its shaft axis and is subjected to the torque \(M=2\left(1-e^{-0.1 \theta}\right) \mathrm{N} \cdot \mathrm{m}\), where \(\theta\) is in radians. If the flywheel is at rest when \(\theta=0\), determine its angular velocity after 5 revolutions.

The 20-kg wheel has an eccentric mass which places the center of mass G a distance \(\bar{r}=70 \mathrm{mm}\) away from the geometric center O. A constant couple \(M=6 \mathrm{N} \cdot \mathrm{m}\) is applied to the initially stationary wheel, which rolls without slipping along the horizontal surface and enters the curve of radius R = 600 mm. Determine the normal force under the wheel just before it exits the curve at C. The wheel has a rolling radius r = 100 mm and a radius of gyration \(k_{O}=65 \mathrm{~mm}\).

The fi gure shows the cross section AB of a garage door which is a rectangular 2.5-m by 5-m panel of uniform thickness with a mass of 200 kg. The door is supported by the struts of negligible mass and hinged at O. Two spring-and-cable assemblies, one on each side of the door, control the movement. When the door is in the horizontal open position, each spring is unextended. If the door is given a slight imbalance from the open position and allowed to fall, determine the value of the spring constant k for each spring which will limit the angular velocity of the door to 1.5 rad /s when edge B strikes the floor.

The uniform 40-lb bar with attached 12-lb wheels is released from rest when \(\theta=60^{\circ}\). If the wheels roll without slipping on the horizontal and vertical surfaces, determine the angular velocity of the bar when \(\theta=45^{\circ}\). Each wheel has a centroidal radius of gyration of 4.5 inches.

A 1200-kg flywheel with a radius of gyration of 400 mm has its speed reduced from 5000 to 3000 rev /min during a 2-min interval. Calculate the average power supplied by the flywheel. Express your answer both in kilowatts and in horsepower.

The wheel consists of a 4-kg rim of 250-mm radius with hub and spokes of negligible mass. The wheel is mounted on the 3-kg yoke OA with mass center at G and with a radius of gyration about O of 350 mm. If the assembly is released from rest in the horizontal position shown and if the wheel rolls on the circular surface without slipping, compute the velocity of point A when it reaches \(A^{\prime}\).

The uniform slender bar ABC weighs 6 lb and is initially at rest with end A bearing against the stop in the horizontal guide. When a constant couple M = 72 lb-in. is applied to end C, the bar rotates causing end A to strike the side of the vertical guide with a velocity of 10 ft /sec. Calculate the loss of energy \(\Delta E\) due to friction in the guides and rollers. The mass of the rollers may be neglected.

The torsional spring at A has a stiffness \(k_{T}=10 \mathrm{N} \cdot \mathrm{m} / \mathrm{rad}\) and is undeformed when the uniform 10-kg bars OA and AB are in the vertical position and overlap. If the system is released from rest with \(\theta=60^{\circ}\), determine the angular velocity of wheel B when \(\theta=30^{\circ}\). The 6-kg wheel at B has a centroidal radius of gyration of 50 mm and is observed to roll without slipping on the horizontal surface.

The uniform slender bar of mass m pivots freely about a horizontal axis through O. If the bar is released from rest in the horizontal position shown where the spring is unstretched, it is observed to rotate a maximum of 30° clockwise. The spring constant k = 200 N/m and the distance b = 200 mm. Determine (a) the mass m of the bar and (b) the angular velocity \(\omega\) of the bar when the angular displacement is 15° clockwise from the release position.

The system is released from rest when the angle \(\theta=90^{\circ}\). Determine the angular velocity of the uniform slender bar when \(\theta\) equals 60°. Use the values \(m_{1}=1 \mathrm{~kg}, m_{2}=1.25 \mathrm{~kg}\), and b = 0.4 m.

The homogeneous torus and cylindrical ring are released from rest and roll without slipping down the incline. Determine an expression for the velocity difference \(v_{\text {diff }}\) which develops between the two objects during the ensuing motion as a function of the distance x they have traveled down the incline. Assume that the masses roll straight down the incline and evaluate your expression for the case where a = 0.2R. Which object is in the lead, and does the relative size of a ever alter the finishing order?

The uniform 12-lb disk pivots freely about a horizontal axis through O. A 4-lb slender bar is fastened to the disk as shown. If the system is nudged from rest while in the position shown, determine its angular velocity \(\omega\) after it has rotated 180°.

Under active development is the storage of energy in high-speed rotating disks where friction is effectively eliminated by encasing the rotor in an evacuated enclosure and by using magnetic bearings. For a 10-kg rotor with a radius of gyration of 90 mm rotating initially at 80 000 rev /min, calculate the power P which can be extracted from the rotor by applying a constant 2.10-N?m retarding torque (a) when the torque is first applied and (b) at the instant when the torque has been applied for 120 seconds.

For the pivoted slender rod of length l, determine the distance x for which the angular velocity will be a maximum as the bar passes the vertical position after being released in the horizontal position shown. State the corresponding angular velocity.

The wheel has mass m and a centroidal radius of gyration \(\bar{k}\) and rolls without slipping up the incline under the action of a force P. The force is applied to the end of a cord which is wrapped securely around the inner hub of the wheel as shown. Determine the speed \(v_{O}\) of the wheel center O after the wheel center has traveled a distance d up the incline. The wheel is at rest when the force P is first applied.

The 8-kg crank OA, with mass center at G and radius of gyration about O of 0.22 m, is connected to the 12-kg uniform slender bar AB. If the linkage is released from rest in the position shown, compute the velocity v of end B as OA swings through the vertical.

The sheave of 400-mm radius has a mass of 50 kg and a radius of gyration of 300 mm. The sheave and its 100-kg load are suspended by the cable and the spring, which has a stiffness of 1.5 kN /m. If the system is released from rest with the spring initially stretched 100 mm, determine the velocity of O after it has dropped 50 mm.

Motive power for the experimental 10-Mg bus comes from the energy stored in a rotating flywheel which it carries. The flywheel has a mass of 1500 kg and a radius of gyration of 500 mm and is brought up to a maximum speed of 4000 rev /min. If the bus starts from rest and acquires a speed of 72 km / h at the top of a hill 20 m above the starting position, compute the reduced speed N of the flywheel. Assume that 10 percent of the energy taken from the flywheel is lost. Neglect the rotational energy of the wheels of the bus. The 10-Mg mass includes the flywheel.

The homogeneous solid semicylinder is released from rest in the position shown. If friction is suffi-cient to prevent slipping, determine the maximum angular velocity reached by the cylinder as it rolls on the horizontal surface.

The figure shows the side view of a door to a storage compartment. As the 40-kg uniform door is opened, the light rod slides through the collar at C and compresses the spring of stiffness k. With the door closed (\(\theta=0\)), a constant force P = 225 N is applied to the end of the door via a cable. If the door has a clockwise angular velocity of 1 rad /s as the position \(\theta=60^{\circ}\) is passed, determine (a) the stiffness of the spring and (b) the angular velocity of the door as it passes the position \(\theta=45^{\circ}\). Neglect all friction and the mass of the pulleys at C and D. The spring is uncompressed when the door is vertical, and b = 1.25 m.

Reconsider the door of Prob. 6 /144. If the door is in the closed vertical position when a constant input force P = 225 N is applied through the end of the cable, determine the maximum angle \(\theta_{\max }\) reached by the door before it comes to a stop. Plot the angular velocity of the door over this period and determine the maximum angular velocity of the door along with the corresponding angle \(\theta\) at which it occurs. The uniform door has a mass m = 40 kg, dimension b = 1.25 m, and the spring constant is k = 2280 N /m.

A small experimental vehicle has a total mass m of 500 kg including wheels and driver. Each of the four wheels has a mass of 40 kg and a centroidal radius of gyration of 400 mm. Total frictional resistance R to motion is 400 N and is measured by towing the vehicle at a constant speed on a level road with engine disengaged. Determine the power output of the engine for a speed of 72 km / h up the 10-percent grade (a) with zero acceleration and (b) with an acceleration of \(3 \mathrm{~m} / \mathrm{s}^{2}\). (Hint: Power equals the time rate of increase of the total energy of the vehicle plus the rate at which frictional work is overcome.)

The two slender bars each of mass m and length b are pinned together and move in the vertical plane. If the bars are released from rest in the position shown and move together under the action of a couple M of constant magnitude applied to AB, determine the velocity of A as it strikes O.

The open square frame is constructed of four identical slender rods, each of length b. If the frame is released from rest in the position shown, determine the speed of corner A (a) after A has dropped a distance b and (b) after A has dropped a distance 2b. The small wheels roll without friction in the slots of the vertical surface.

The load of mass m is supported by the light parallel links and the fi xed stop A. Determine the initial angular acceleration \(\alpha\) of the links due to the application of the couple M to one end as shown.

The uniform slender bar of mass m is shown in its equilibrium confi guration before the force P is applied. Compute the initial angular acceleration of the bar upon application of P.

The two uniform slender bars are hinged at O and supported on the horizontal surface by their end rollers of negligible mass. If the bars are released from rest in the position shown, determine their initial angular acceleration \(\alpha\) as they collapse in the vertical plane. (Suggestion: Make use of the instantaneous center of zero velocity in writing the expression for dT.)

Links A and B each weigh 8 lb, and bar C weighs 12 lb. Calculate the angle \(\theta\) assumed by the links if the body to which they are pinned is given a steady horizontal acceleration a of \(4 \mathrm{ft} / \mathrm{sec}^{2}\).

The mechanism shown moves in the vertical plane. The vertical bar AB weighs 10 lb, and each of the two links weighs 6 lb with mass center at G and with a radius of gyration of 10 in. about its bearing (O or C). The spring has a stiffness of 15 lb /ft and an unstretched length of 18 in. If the support at D is suddenly withdrawn, determine the initial angular acceleration of the links. v

The load of mass m is given an upward acceleration a from its supported rest position by the application of the forces P. Neglect the mass of the links compared with m and determine the initial acceleration a. v

The cargo box of the food-delivery truck for aircraft servicing has a loaded mass m and is elevated by the application of a couple M on the lower end of the link which is hinged to the truck frame. The horizontal slots allow the linkage to unfold as the cargo box is elevated. Determine the upward acceleration of the box in terms of h for a given value of M. Neglect the mass of the links. v

The sliding block is given a horizontal acceleration to the right that is slowly increased to a steady value a. The attached pendulum of mass m and mass center G assumes a steady angular deflection \(\theta\). The torsion spring at O exerts a moment \(M=k_{T} \theta\) on the pendulum to oppose the angular deflection. Determine the torsional stiffness \(k_{T}\) that will allow a steady deflection \(\theta\). v

Each of the uniform bars OA and OB has a mass of 2 kg and is freely hinged at O to the vertical shaft, which is given an upward acceleration a = g /2. The links which connect the light collar C to the bars have negligible mass, and the collar slides freely on the shaft. The spring has a stiffness k = 130 N /m and is uncompressed for the position equivalent to \(\theta=0\). Calculate the angle \(\theta\) assumed by the bars under conditions of steady acceleration. v

The linkage consists of the two slender bars and moves in the horizontal plane under the influence of force P. Link OC has a mass m and link AC has a mass 2m. The sliding block at B has negligible mass. Without dismembering the system, determine the initial angular acceleration \(\alpha\) of the links as P is applied at A with the links initially at rest. (Suggestion: Replace P by its equivalent force-couple system.)

The portable work platform is elevated by means of the two hydraulic cylinders articulated at points C. The pressure in each cylinder produces a force F. The platform, man, and load have a combined mass m, and the mass of the linkage is small and may be neglected. Determine the upward acceleration a of the platform and show that it is independent of both b and \(\theta\).

Each of the three identical uniform panels of a segmented industrial door has mass m and is guided in the tracks (one shown dashed). Determine the horizontal acceleration a of the upper panel under the action of the force P. Neglect any friction in the guide rollers.

The mechanical tachometer measures the rotational speed N of the shaft by the horizontal motion of the collar B along the rotating shaft. This movement is caused by the centrifugal action of the two 12-oz weights A, which rotate with the shaft. Collar C is fixed to the shaft. Determine the rotational speed N of the shaft for a reading \(\beta=15^{\circ}\). The stiffness of the spring is 5 lb /in., and it is uncompressed when \(\theta=0 \text { and } \beta=0\). Neglect the weights of the links.

A planetary gear system is shown, where the gear teeth are omitted from the figure. Each of the three identical planet gears A, B, and C has a mass of 0.8 kg, a radius r = 50 mm, and a radius of gyration of 30 mm about its center. The spider E has a mass of 1.2 kg and a radius of gyration about O of 60 mm. The ring gear D has a radius R = 150 mm and is fixed. If a torque M = 5 N?m is applied to the shaft of the spider at O, determine the initial angular acceleration \(\alpha\) of the spider.

The sector and attached wheels are released from rest in the position shown in the vertical plane. Each wheel is a solid circular disk weighing 12 lb and rolls on the fixed circular path without slipping. The sector weighs 18 lb and is closely approximated by one-fourth of a solid circular disk of 16-in. radius. Determine the initial angular acceleration of the sector.

The aerial tower shown is designed to elevate a workman in a vertical direction. An internal mechanism at B maintains the angle between AB and BC at twice the angle \(\theta\) between BC and the ground. If the combined mass of the man and the cab is 200 kg and if all other masses are neglected, determine the torque M applied to BC at C and the torque \(M_{B}\) in the joint at B required to give the cab an initial vertical acceleration of \(1.2 \mathrm{~m} / \mathrm{s}^{2}\) when it is started from rest in the position \(\theta=30^{\circ}\).

The uniform arm OA has a mass of 4 kg, and the gear D has a mass of 5 kg with a radius of gyration about its center of 64 mm. The large gear B is fixed and cannot rotate. If the arm and small gear are released from rest in the position shown in the vertical plane, calculate the initial angular acceleration of OA.

The vehicle is used to transport supplies to and from the bottom of the 25-percent grade. Each pair of wheels, one at A and the other at B, has a mass of 140 kg with a radius of gyration of 150 mm. The drum C has a mass of 40 kg and a radius of gyration of 100 mm. The total mass of the vehicle is 520 kg. The vehicle is released from rest with a restraining force T of 500 N in the control cable which passes around the drum and is secured at D. Determine the initial acceleration a of the vehicle. The wheels roll without slipping.

A person who walks through the revolving door exerts a 90-N horizontal force on one of the four door panels and keeps the 15° angle constant relative to a line which is normal to the panel. If each panel is modeled by a 60-kg uniform rectangular plate which is 1.2 m in length as viewed from above, determine the final angular velocity \(\omega\) of the door if the person exerts the force for 3 seconds. The door is initially at rest and friction may be neglected.

The 75-kg flywheel has a radius of gyration about its shaft axis of \(\bar{k}=0.50 \mathrm{m}) and is subjected to the torque \(M=10\left(1-e^{-t}\right) \mathrm{N} \cdot \mathrm{m}\), where t is in seconds. If the flywheel is at rest at time t = 0, determine its angular velocity \(\omega\) at t = 3 s.

Determine the angular momentum of the earth about the center of the sun. Assume a homogeneous earth and a circular earth orbit of radius \(149.6\left(10^{6}\right) \mathrm{km}\). Consult Table D /2 of Appendix D for other needed information. Comment on the relative contributions of the terms \(\bar{I} \omega \text { and } m \bar{v} d\).

The frame of mass m is welded together from uniform slender rods. The frame is released from rest in the upper position shown and constrained to fall vertically by two light rollers which travel along the smooth slots. The roller at A catches in the support at O without rebounding and serves as a hinge for the frame as it rotates thereafter. Determine the angular velocity of the frame an instant after the roller at A engages the support at O.

The frictional moment \(M_{f}\) acting on a rotating turbine disk and its shaft is given by \(M_{f}=k \omega^{2}\) where \(\omega\) is the angular velocity of the turbine. If the source of power is cut off while the turbine is running with an angular velocity \(\omega_{0}\), determine the time t for the speed of the turbine to drop to one-half of its initial value. The moment of inertia of the turbine disk and shaft is I.

The cable drum has a mass of 800 kg with radius of gyration of 480 mm about its center O and is mounted in bearings on the 1200-kg carriage. The carriage is initially moving to the left with a speed of 1.5 m /s, and the drum is rotating counterclockwise with an angular velocity of 3 rad /s when a constant horizontal tension T = 400 N is applied to the cable at time t = 0. Determine the velocity v of the carriage and the angular velocity \(\omega\) of the drum when t = 10 s. Neglect the mass of the carriage wheels.

The man is walking with speed \(v_{1}=1.2 \mathrm{~m} / \mathrm{s}\) to the right when he trips over a small floor discontinuity. Estimate his angular velocity \(\omega\) just after the impact. His mass is 76 kg with center-of-mass height h = 0.87 m, and his mass moment of inertia about the ankle joint O is \(66 \mathrm{~kg} \cdot \mathrm{m}^{2}\), where all are properties of the portion of his body above O; i.e., both the mass and moment of inertia do not include the foot.

The 15-kg wheel with 150-mm outer radius and 115-mm centroidial radius of gyration is rolling without slipping down the 15° incline at a speed of 2 m /s when a tension T = 30 N is applied to a cable wrapped securely around an inner hub with a radius of 100 mm. Determine the time t required for the wheel to come to a stop if the tension is applied first in configuration (a) and then in configuration (b). The wheel is observed to roll without slipping throughout the entire motion for both cases.

A uniform slender bar of mass M and length L is translating on the smooth horizontal x-y plane with a velocity \(v_{M}\) when a particle of mass m traveling with a velocity \(v_{m}\) as shown strikes and becomes embedded in the bar. Determine the fi nal linear and angular velocities of the bar with its embedded particle.

The homogeneous circular cylinder of mass m and radius R carries a slender rod of mass m /2 attached to it as shown. If the cylinder rolls on the surface without slipping with a velocity \(v_{O}\) of its center O, determine the angular momenta \(H_{G}\) and \(H_{O}\) of the system about its center of mass G and about O for the instant shown.

The grooved pulley of mass m is acted on by a constant force F through a cable which is wrapped securely around the exterior of the pulley. The pulley supports a cylinder of mass M which is attached to the end of a cable which is wrapped securely around an inner hub. If the system is stationary when the force F is first applied, determine the upward velocity of the supported mass after 3 seconds. Use the values m = 40 kg, M = 10 kg, \(r_{o}=225 \mathrm{mm}, r_{i}=150 \mathrm{mm}, k_{O}=160 \mathrm{mm}\), and F = 75 N. Assume no mechanical interference for the indicated time frame and neglect friction in the bearing at O. What is the time-averaged value of the force in the cable which supports the 10-kg mass?

The wad of clay of mass m is initially moving with a horizontal velocity \(v_{1}\) when it strikes and sticks to the initially stationary uniform slender bar of mass M and length L. Determine the fi nal angular velocity of the combined body and the x-component of the linear impulse applied to the body by the pivot O during the impact.

The 30-lb uniform plate has the indicated velocities at corners A and B. What is the angular momentum of the plate about the mass center G

The plate of Prob. 6 /179 is repeated here where the coordinates of corner C are established. The plate is falling freely in the x-y vertical plane. What is the linear momentum of the plate and its angular momentum about point O? Additionally, determine the time rate of change of the angular momentum of the plate about point O.

Just after leaving the platform, the diver’s fully extended 80-kg body has a rotational speed of 0.3 rev /s about an axis normal to the plane of the trajectory. Estimate the angular velocity N later in the dive when the diver has assumed the tuck position. Make reasonable assumptions concerning the mass moment of inertia of the body in each confi guration.

The device shown is a simplified model of an amusement-park ride in which passengers are rotated about the vertical axis of the central post at an angular speed \(\Omega\) while sitting in a pod which is capable of rotating the occupants 360° about the longitudinal axis of the connecting arm attached to the central collar. Determine the percent increase n in angular velocity between configurations (a) and (b), where the passenger pod has rotated 90° about the connecting arm. For the model, m = 1.2 kg, r = 75 mm, l = 300 mm, and L = 650 mm. The post and connecting arms rotate freely about the z-axis at an initial angular speed \(\Omega=120 \mathrm{rev} /\mathrm{min}\) and have a combined mass moment of inertia about the z-axis of \(30\left(10^{-3}\right) \mathrm{kg} \cdot \mathrm{m}^{2}\).

The uniform concrete block, which weighs 171 lb and falls from rest in the horizontal position shown, strikes the fixed corner A and pivots around it with no rebound. Calculate the angular velocity \(\omega\) of the block immediately after it hits the corner and the percentage loss n of energy due to the impact.

Two small variable-thrust jets are actuated to keep the spacecraft angular velocity about the z-axis constant at \(\omega_{0}=1.25 \mathrm{rad} / \mathrm{s}\) as the two telescoping booms are extended from \(r_{1}=1.2 \mathrm{m}\) to \(r_{2}=4.5 \mathrm{m}\) at a constant rate over a 2-min period. Determine the necessary thrust T for each jet as a function of time where t = 0 is the time when the telescoping action is begun. The small 10-kg experiment modules at the ends of the booms may be treated as particles, and the mass of the rigid booms is negligible.

The body composed of slender rods of mass \(\rho\) per unit length is lying motionless on the smooth horizontal surface when a linear impluse \(\int P d t\) is applied as shown. Determine the velocity \(\mathbf{v}_{B}\) of corner B immediately following the application of the impulse if \(l=500 \mathrm{mm}, \rho=3 \mathrm{kg} / \mathrm{m}, \text { and } \int P d t=8 \mathrm{N} \cdot \mathrm{s}\).

Each of the two 300-mm uniform rods A has a mass of 1.5 kg and is hinged at its end to the rotating base B. The 4-kg base has a radius of gyration of 40 mm and is initially rotating freely about its vertical axis with a speed of 300 rev /min and with the rods latched in the vertical positions. If the latches are released and the rods assume the horizontal positions, calculate the new rotational speed N of the assembly.

The phenomenon of vehicle “tripping” is investigated here. The sport-utility vehicle is sliding sideways with speed \(v_{1}\) and no angular velocity when it strikes a small curb. Assume no rebound of the right-side tires and estimate the minimum speed \(v_{1}\) which will cause the vehicle to roll completely over to its right side. The mass of the SUV is 2300 kg and its mass moment of inertia about a longitudinal axis through the mass center G is \(900 \mathrm{kg} \cdot \mathrm{m}^{2}\).

The slender bar of mass m and length l is released from rest in the horizontal position shown. If point A of the bar becomes attached to the pivot at B upon impact, determine the angular velocity \(\omega\) of the bar immediately after impact in terms of the distance x. Evaluate your expression for x = 0, l /2, and l.

Child A weighs 75 lb and is sitting at rest relative to the merry-go-round which is rotating counterclockwise with an angular velocity \(\Omega=15 \mathrm{rev} / \mathrm{min}\). Child B, with a weight of 65 lb, runs toward the merry-go-round with a speed v = 12 ft /sec and jumps onto the edge. Compute the angular velocity \(\Omega^{\prime}\) of the merry-go-round when child B has just jumped aboard the merry-go-round and is standing in position (a). If child B moves to position (b), what is the new angular velocity of the merry-go-round? For simplicity, model the inertia of child A as a 20-in.-diameter uniform disk (horizontal) and model child B as a 14-in.-diameter uniform and upright cylinder. The merry-go-round itself has a centroidal mass moment of inertia \(I_{O}=110 \text { lb-ft-sec }{ }^{2}\).

The system is initially rotating freely with angular velocity \(\omega_{1}=10 \mathrm{rad} / \mathrm{s}\) when the inner rod A is centered lengthwise within the hollow cylinder B as shown in the figure. Determine the angular velocity of the system (a) if the inner rod A has moved so that a length b /2 is protruding from the cylinder, (b) just before the rod leaves the cylinder, and (c) just after the rod leaves the cylinder. Neglect the moment of inertia of the vertical support shafts and friction in the two bearings. Both bodies are constructed of the same uniform material. Use the values b = 400 mm and r = 20 mm, and refer to the results of Prob. B /34 as needed.

The homogeneous sphere of mass m and radius r is projected along the incline of angle \(\theta\) with an initial speed \(v_{0}\) and no angular velocity (\(\omega_{0}=0\)). If the coefficient of kinetic friction is \(\mu_{k}\), determine the time duration t of the period of slipping. In addition, state the velocity v of the mass center G and the angular velocity \(\omega\) at the end of the period of slipping.

The homogeneous sphere of Prob. 6 /191 is placed on the incline with a clockwise angular velocity \(\omega_{0}\) but no linear velocity of its center (\(v_{0}=0\)). Determine the time duration t of the period of slipping. In addition, state the velocity v of the mass center G and angular velocity \(\omega\) at the end of the period of slipping.

The 100-lb platform rolls without slipping along the 10° incline on two pairs of 16-in.-diameter wheels. Each pair of wheels with attached axle weighs 25 lb and has a centroidal radius of gyration of 5.5 in. The platform has an initial speed of 3 ft /sec down the incline when a tension T is applied through a cable attached to the platform. If the platform acquires a speed of 3 ft /sec up the incline after the tension has been applied for 8 seconds, what is the average value of the tension in the cable?

The uniform slender bar of mass m and length l has no angular velocity as end A strikes the ground against the stop with no rebound. If \(\alpha=15^{\circ}\), what is the minimum magnitude of the initial velocity \(\mathbf{v}_{1}\) for which the bar will rotate about A to the vertical position?

The 165-lb ice skater with arms extended horizontally spins about a vertical axis with a rotational speed of 1 rev /sec. Estimate his rotational speed N if he fully retracts his arms, bringing his hands very close to the centerline of his body. As a reasonable approximation, model the extended arms as uniform slender rods, each of which is 27 in. long and weighs 15 lb. Model the torso as a solid 135-lb cylinder 13 in. in diameter. Treat the man with arms retracted as a solid 165-lb cylinder of 13-in. diameter. Neglect friction at the skate–ice interface.

In the rotating assembly shown, arm OA and the attached motor housing B together weigh 10 lb and have a radius of gyration about the z-axis of 7 in. The motor armature and attached 5-in.- radius disk have a combined weight of 15 lb and a radius of gyration of 4 in. about their own axis. The entire assembly is free to rotate about the z-axis. If the motor is turned on with OA initially at rest, determine the angular speed N of OA when the motor has reached a speed of 300 rev /min relative to arm OA.

The motor at B supplies a constant torque M which is applied to a 375-mm-diameter internal drum around which is wound the cable shown. This cable then wraps around an 80-kg pulley attached to a 125-kg cart carrying 600 kg of rock. The motor is able to bring the loaded cart to a cruising speed of 1.5 m /s in 3 seconds. What torque M is the motor able to supply, and what is the average value of the tension in each side of the cable which is wrapped around the pulley at O during the speed-up period? The cable does not slip on the pulley and the centroidal radius of gyration of the pulley is 450 mm. What is the power output of the motor when the cart reaches its cruising speed?

The body of the spacecraft weighs 322 lb on earth and has a radius of gyration about its z-axis of 1.5 ft. Each of the two solar panels may be treated as a uniform flat plate weighing 16.1 lb. If the spacecraft is rotating about its z-axis at the angular rate of 1.0 rad /sec with \(\theta=0\), determine the angular rate \(\omega\) after the panels are rotated to the position \(\theta=\pi / 2\) by an internal mechanism. Neglect the small momentum change of the body about the y-axis.

A 55-kg dynamics instructor is demonstrating the principles of angular momentum to her class. She stands on a freely rotating platform with her body aligned with the vertical platform axis. With the platform not rotating, she holds a modified bicycle wheel so that its axis is vertical. She then turns the wheel axis to a horizontal orientation without changing the 600-mm distance from the centerline of her body to the wheel center, and her students observe a platform rotation rate of 30 rev /min. If the rim-weighted wheel has a mass of 10 kg and a centroidal radius of gyration \(\bar{k}=300 \mathrm{mm}\), and is spinning at a fairly constant rate of 250 rev /min, estimate the mass moment of inertia I of the instructor (in the posture shown) about the vertical platform axis.

The 8-lb slotted circular disk has a radius of gyration about its center O of 6 in. and initially is rotating freely about a fixed vertical axis through O with a speed \(N_{1}=600 \mathrm{rev} / \mathrm{min}\). The 2-lb uniform slender bar A is initially at rest relative to the disk in the centered slot position as shown. A slight disturbance causes the bar to slide to the end of the slot where it comes to rest relative to the disk. Calculate the new angular speed \(N_{2}\) of the disk, assuming the absence of friction in the shaft bearing at O. Does the presence of any friction in the slot affect the final result?

The gear train shown starts from rest and reaches an output speed of \(\omega_{C}=240 \mathrm{rev} / \mathrm{min}\) in 2.25 s. Rotation of the train is resisted by a constant 150 N?m moment at the output gear C. Determine the required input power to the 86% efficient motor at A just before the final speed is reached. The gears have masses \(m_{A}=6 \mathrm{kg}, m_{B}=10 \mathrm{kg} \text {, and } m_{C}=24 \mathrm{kg}\), pitch diameters \(d_{A}=120 \mathrm{mm}, d_{B}=160 \mathrm{mm}\), and \(d_{C}=240 \mathrm{mm}\), and centroidal radii of gyration \(k_{A}=48 \mathrm{mm}, k_{B}=64 \mathrm{mm}, \text { and } k_{C}=96 \mathrm{mm}\).

The uniform cylinder is rolling without slip with a velocity v along the horizontal surface when it overtakes a ramp traveling with speed \(v_{0}\). Determine an expression for the speed \(v_{\prime}\) which the cylinder has relative to the ramp immediately after it rolls up onto the ramp. Finally, determine the percentage n of cylinder kinetic energy lost if \(\text { (a) } \theta=10^{\circ} \text { and } v_{0}=0.25 v \text { and }(b) \theta=10^{\circ} \text { and } v_{0}=0.5v\). Assume that the clearance between the ramp and the ground is essentially zero, that the mass of the ramp is very large, and that the cylinder does not slip on the ramp.

A frozen-juice can rests on the horizontal rack of a freezer door as shown. With what maximum angular velocity ? can the door be “slammed” shut against its seal and not dislodge the can? Assume that the can rolls without slipping on the corner of the rack, and neglect the dimension d compared with the 500-mm distance.

The 30-kg wheel has a radius of gyration about its center of 75 mm and is rotating clockwise at the rate of 300 rev /min when it is released onto the incline with no velocity of its center O. While the wheel is slipping, it is observed that the center O remains in a fixed position. Determine the coefficient of kinetic friction \(\mu_{k}\) and the time t during which slipping occurs. Also determine the velocity v of the center 4 seconds after the wheel has stopped slipping.

The mass m is traveling with speed v when it strikes the corner of the plate of mass M. If the mass sticks to the side of the plate, determine the maximum angle \(\theta\) reached by the plate. Use the values m = 500 g, M = 20 kg, v = 30 m /s, b = 400 mm, and h = 800 mm.

The 5-kg bar is released from rest while in the position shown and its end rollers travel in the vertical-plane slot shown. If the speed of roller A is 3.25 m /s as it passes point C, determine the work done by friction on the system over this portion of the motion. The bar has a length l = 700 mm.

The body of mass m = 1.2 kg is lying motionless on the smooth horizontal surface when an impulse \(\int P d t=6 \mathrm{N} \cdot \mathrm{s}\) is applied as shown. Determine the linear momentum G and the angular momentum \(\mathrm{H}_{G}\) of the body about the mass center G immediately following the application of the impulse.

The preliminary design of a unit for automatically reducing the speed of a freely rotating assembly is shown. Initially the unit is rotating freely about a vertical axis through O at a speed of 600 rev /min with the arms secured in the positions shown by AB. When the arms are released, they swing outward and become latched in the dashed positions shown. The disk has a mass of 30 kg with a radius of gyration of 90 mm about O. Each arm has a length of 160 mm and a mass of 0.84 kg and may be treated as a uniform slender rod. Determine the new speed N of rotation and calculate the loss \(|\Delta E|\) of energy of the system. Would the results be affected by either the direction of rotation or the sequence of release of the rods?

The nose-wheel assembly is raised by the application of a torque M to link BC through the shaft at B. The arm and wheel AO have a combined weight of 100 lb with center of mass at G, and a centroidal radius of gyration of 14 in. If the angle \(\theta=30^{\circ}\), determine the torque M necessary to rotate link AO with a counterclockwise angular velocity of 10 deg /sec that is increasing at the rate of 5 deg /sec every second. Additionally, determine the total force supported by the pin at A. The mass of links BC and CD may be neglected for this analysis.

Each of the solid circular disk wheels has a mass of 2 kg, and the inner solid cylinder has a mass of 3 kg. The disks and cylinder are mounted on the small central shaft so that each can rotate independently of the other with negligible friction in the bearings. Calculate the acceleration of the center of the wheels when the 20-N force is applied as shown. The coefficients of friction between the wheels and the horizontal surface are \(\mu_{k}=0.30 \text { and } \mu_{s}=0.40\).

The dump truck carries \(5 \mathrm{m}^{3}\) of dirt with a density of \(1600 \mathrm{~kg} / \mathrm{m}^{3}\), and the elevating mechanism rotates the dump about the pivot A at a constant angular rate of 4 deg /s. The mass center of the dump and load is at G. Determine the maximum power P required during the tilting of the load.

Four identical slender rods each of mass m are welded at their ends to form a square, and the corners are then welded to a light metal hoop of radius r. If the rigid assembly of rods and hoop is allowed to roll down the incline, determine the minimum value of the coeffi cient of static friction which will prevent slipping.

The gear train shown operates in a horizontal plane at a steady speed and receives 6 hp from a motor at A to move rack D against a 4500-pound load L. At what speed will the rack move if gears A, B, and C have pitch diameters \(d_{A}=12 \mathrm{in}\)., \(d_{B}=24 \mathrm{in}\)., and \(d_{C}=12 \mathrm{in}\).? Gear C is keyed to the same shaft as gear B and friction in the bearings is negligible.

The uniform slender bar weighs 60 lb and is released from rest in the near-vertical position shown, where the spring of stiffness 10 lb /ft is unstretched. Calculate the speed with which end A strikes the horizontal surface.

The body of mass m is supported by feet of negligible size and is at rest relative to the tilted surface, which is contained within an experimental vehicle. If the vehicle is brought to rest with a slowly increasing deceleration a, determine the ratio h /b which will just allow the body to tip forward about the front feet before it slides forward relative to the vehicle. For this ratio, what is the deceleration a at which tipping occurs?

Repeat Prob. 6 /215 for the case of a slowly increasing forward acceleration a. Determine the ratio h /b which will allow the package to tip backward about the rear feet before it slides backward relative to the vehicle. In terms of this ratio, what is the acceleration a?

The small block of mass m slides along the radial slot of the disk while the disk rotates in the horizontal plane about its center O. The block is released from rest relative to the disk and moves outward with an increasing velocity \(\dot{r}\) along the slot as the disk turns. Determine the expression in terms of r and \(\dot{r}\) for the torque M that must be applied to the disk to maintain a constant angular velocity \(\omega\) of the disk.

The forklift truck with center of mass at \(G_{1}\) has a weight of 3200 lb including the vertical mast. The fork and load have a combined weight of 1800 lb with center of mass at \(G_{2}\). The roller guide at B is capable of supporting horizontal force only, whereas the connection at C, in addition to supporting horizontal force, also transmits the vertical elevating force. If the fork is given an upward acceleration which is sufficient to reduce the force under the rear wheels at A to zero, calculate the corresponding reaction at B.

A space telescope is shown in the figure. One of the reaction wheels of its attitude-control system is spinning as shown at 10 rad /s, and at this speed the friction in the wheel bearing causes an internal moment of \(10^{-6} \mathrm{N} \cdot \mathrm{m}\). Both the wheel speed and the friction moment may be considered constant over a time span of several hours. If the mass moment of inertia of the entire spacecraft about the x-axis is \(150\left(10^{3}\right) \mathrm{kg} \cdot \mathrm{m}^{2}\), determine how much time passes before the line of sight of the initially stationary spacecraft drifts by 1 arcsecond, which is 1 /3600 degree. All other elements are fixed relative to the spacecraft, and no torquing of the reaction wheel shown is performed to correct the attitude drift. Neglect external torques.

The uniform semicircular plate is at rest on the smooth horizontal surface when the force F is applied at B. Determine the coordinates of the point P in the plate which has zero initial acceleration.

The slender rod of mass \(m_{1}\) and length L has a movable slider of mass \(m_{2}\) which can be tightened at any location x along the rod. The assembly is initially falling in translation with speed \(v_{1}\). A small peg on the left end of the rod becomes engaged in the receptacle. Determine the angular velocity \(\omega_{2}\) of the body just after impact. For the condition \(m_{2}=m_{1} / 2\), determine the maximum value of \(\omega_{2}\) and the corresponding value of x. Plot 2 versus x/L for this mass condition.

The 6-lb pendulum with mass center at G is pivoted at A to the fixed support CA. It has a radius of gyration of 17 in. about O-O and swings through an amplitude \(\theta=60^{\circ}). For the instant when the pendulum is in the extreme position, calculate the moments \(M_{x}\), \(M_{y}\), and \(M_{z}\) applied by the base support to the column at C.

The uniform 40-lb bar with attached 12-lb wheels is released from rest in the orientation shown. The wheels have a centroidal radius of gyration of 4.5 in., and the coefficients of static and kinetic friction between the wheels and the horizontal and vertical surfaces are \(\mu_{s}=0.65 \text { and } \mu_{k}=0.50\). Friction may be neglected in the pins connecting the wheels to the bar. Determine the acceleration components of the mass center of the bar at the instant of release.

The four-bar mechanism operates in a horizontal plane. At the instant illustrated, \(\theta=30^{\circ}\) and crank OA has a constant counterclockwise angular velocity of 3 rad /s. Determine the required magnitude of the couple M necessary to drive the system at this instant. Member BCD has a mass of 8 kg with a radius of gyration of 450 mm about point C. The mass of crank OA and connecting link AB may be neglected for this analysis.

If the 1.2-kg uniform slender bar is released from rest in the position \(\theta=0\) where the spring is unstretched, determine and plot its angular velocity as a function of \(\theta\) over the range \(0 \leq \theta \leq \theta_{\max }\), where \(\theta_{\max}\) is the value of \(\theta\) at which the bar momentarily comes to rest. The value of the spring constant k is 100 N /m, and friction can be neglected. State the maximum angular speed and the value of \(\theta\) at which it occurs.

The steel I-beam is to be transported by the overhead trolley to which it is hinged at O. If the trolley starts from rest with \(\theta=\dot{\theta}=0\) and is given a constant horizontal acceleration \(a=2 \mathrm{~m} / \mathrm{s}^{2}\), find the maximum values of \(\dot{\theta} \text { and } \theta\). The magnitude of the initial swing would constitute a shop safety consideration.

The uniform power pole of mass m and length L is hoisted into a vertical position with its lower end supported by a fixed pivot at O. The guy wires supporting the pole are accidentally released, and the pole falls to the ground. Plot the x- and y-components of the force exerted on the pole at O in terms of \(\theta\) from 0 to 90°. Can you explain why \(O_{y}\) increases again after going to zero?

The compound pendulum is composed of a uniform slender rod of length l and mass 2m to which is fastened a uniform disk of diameter l /2 and mass m. The body pivots freely about a horizontal axis through O. If the pendulum has a clockwise angular velocity of 3 rad /s when \(\theta=0\) at time t = 0, determine the time t at which the pendulum passes the position \(\theta=90^{\circ}\). The pendulum length l = 0.8 m.

The uniform 100-kg beam AB is hanging initially at rest with \(\theta=0\) when the constant force P = 300 N is applied to the cable. Determine (a) the maximum angular velocity reached by the beam with the corresponding angle and (b) the maximum angle \(\theta_{\max}\) reached by the beam.

For a train traveling at 160 km /h around a horizontal curve of radius 1.9 km, calculate the elevation angle \(\beta\) of the track so that passengers will feel only a force normal to their seats and the rails will exert no side thrust against the wheels, as indicated in part (a) of the figure. An experimental train rounds this same curve at a speed of 260 km /h with cars which are automatically tilted an angle with respect to the rails, as shown in part (b) of the figure. This angle reduces the side thrust F felt by the passengers. Determine the tilt angle \(\theta\) required to limit F to 30 percent of the side thrust they would feel if both \(\theta\) and \(\beta\) were zero.

The 60-ft telephone pole of essentially uniform diameter is being hoisted into the vertical position by two cables attached at B as shown. The end O rests on a fixed support and cannot slip. When the pole is nearly vertical, the fitting at B suddenly breaks, releasing both cables. When the angle \(\theta\) reaches 10°, the speed of the upper end A of the pole is 4.5 ft /sec. From this point, calculate the time t which the workman would have to get out of the way before the pole hits the ground. With what speed \(v_{A}\) does end A hit the ground?

The four-bar mechanism of Prob. 6 /108 is repeated here. The coupler AB has a mass of 7 kg, and the masses of crank OA and the output arm BC may be neglected. Determine and plot the torque M which must be applied to the crank at O in order to keep the speed of the crank steady at 60 rev /min over the range \(0 \leq \theta \leq 2 \pi\). What is the largest magnitude M which occurs during this motion, and at what angle \(\theta\) does it occur? Additionally, plot the magnitude of the force on each pin over this range and state the maximum magnitude of each pin force along with the corresponding crank angle \(\theta\) at which it occurs.

Reconsider the mechanism of Prob. 6 /232. If crank OA now starts from rest and acquires a speed of 60 rev /min in one complete revolution with constant angular acceleration, determine and plot the torque M which must be applied to the crank over the range \(0 \leq \theta \leq 2 \pi\). What is the largest magnitude M which occurs during this motion, and at what angle \(\theta\) does it occur? Additionally, plot the magnitude of the force in each pin over this range and state the maximum magnitude of each pin force along with the corresponding crank angle \(\theta\) at which it occurs.

Reconsider the basic mechanism of Prob. 6 /232, only now the mass of the crank OA is 1.2 kg and that of the uniform output arm BC is 1.8 kg. For simplicity, treat the crank OA as uniform. Determine and plot the torque M which must be applied to the crank at O in order to keep the speed of the crank steady at 60 rev /min over the range \(0 \leq \theta \leq 2 \pi\). What is the largest magnitude M which occurs during this motion, and at what angle \(\theta\) does it occur? Additionally, plot the magnitude of the force on each pin over this range and state the maximum magnitude of each pin force along with the corresponding crank angle \(\theta\) at which it occurs.