A drum of 4-in. radius is attached to a disk of 8-in.

A conveyor system is fitted with vertical panels, and a 15-in. rod AB weighing 5 lb is lodged between two panels as shown. If the rod is to remain in the position shown, determine the maximum allowable acceleration of the system.

A conveyor system is fitted with vertical panels, and a 15-in. rod AB weighing 5 lb is lodged between two panels as shown. Knowing that the acceleration of the system is \(3\ \mathrm{ft}/\mathrm{s}^2\) to the left, determine (a) the force exerted on the rod at C, (b) the reaction at B.

Knowing that the coefficient of static friction between the tires and the road is 0.80 for the automobile shown, determine the maximum possible acceleration on a level road, assuming (a) four-wheel drive, (b) rear-wheel drive, (c) front-wheel drive.

The motion of the 2.5-kg rod AB is guided by two small wheels which roll freely in horizontal slots. If a force P of magnitude 8 N is applied at B, determine (a) the acceleration of the rod, (b) the reactions at A and B.

A uniform rod BC of mass 4 kg is connected to a collar A by a 250-mm cord AB. Neglecting the mass of the collar and cord, determine (a) the smallest constant acceleration \(\mathbf{a}_{A}\) for which the cord and the rod will lie in a straight line, (b) the corresponding tension in the cord.

A 2000-kg truck is being used to lift a 400-kg boulder B that is on a 50-kg pallet A. Knowing the acceleration of the rear-wheel-drive truck is \(1\mathrm{\ m}/\mathrm{s}^2\), determine (a) the reaction at each of the front wheels, (b) the force between the boulder and the pallet.

The support bracket shown is used to transport a cylindrical can from one elevation to another. Knowing that \(\mathrm{m}_{s}=0.25\) between the can and the bracket, determine (a) the magnitude of the upward acceleration a for which the can will slide on the bracket, (b) the smallest ratio h/d for which the can will tip before it slides.

Solve Prob. 16.7, assuming that the acceleration a of the bracket is directed downward.

A 20-kg cabinet is mounted on casters that allow it to move freely (m = 0) on the floor. If a 100-N force is applied as shown, determine (a) the acceleration of the cabinet, (b) the range of values of h for which the cabinet will not tip.

Solve Prob. 16.9, assuming that the casters are locked and slide on the rough floor \(\left(\mathrm{m}_{k}=0.25\right)\).

A completely filled barrel and its contents have a combined mass of 90 kg. A cylinder C is connected to the barrel at a height h = 550 mm as shown. Knowing \(\mathrm{m}_{s}=0.40\) and \(\mathrm{m}_{k}=0.35\), determine the maximum mass of C so the barrel will not tip.

A 40-kg vase has a 200-mm-diameter base and is being moved using a 100-kg utility cart as shown. The cart moves freely (m = 0) on the ground. Knowing the coefficient of static friction between the vase and the cart is \(\mathrm{m}_{s}=0.4\), determine the maximum force F that can be applied if the vase is not to slide or tip.

The retractable shelf shown is supported by two identical linkage-and-spring systems; only one of the systems is shown. A 20-kg machine is placed on the shelf so that half of its weight is supported by the system shown. If the springs are removed and the system is released from rest, determine (a) the acceleration of the machine, (b) the tension in link AB. Neglect the weight of the shelf and links.

A uniform rectangular plate has a mass of 5 kg and is held in position by three ropes as shown. Knowing that \(\mathrm{u}=30^{\circ}\), determine, immediately after rope CF has been cut, (a) the acceleration of the plate, (b) the tension in ropes AD and BE.

A uniform rectangular plate has a mass of 5 kg and is held in position by three ropes as shown. Determine the largest value of u for which both ropes AD and BE remain taut immediately after rope CF has been cut.

Three bars, each of mass 3 kg, are welded together and pin-connected to two links BE and CF. Neglecting the weight of the links, determine the force in each link immediately after the system is released from rest.

Members ACE and DCB are each 600 mm long and are connected by a pin at C. The mass center of the 10-kg member AB is located at G. Determine (a) the acceleration of AB immediately after the system has been released from rest in the position shown, (b) the corresponding force exerted by roller A on member AB. Neglect the weight of members ACE and DCB.

The 15-lb rod BC connects a disk centered at A to crank CD. Knowing that the disk is made to rotate at the constant speed of 180 rpm, determine for the position shown the vertical components of the forces exerted on rod BC by pins at B and C.

The triangular weldment ABC is guided by two pins that slide freely in parallel curved slots of radius 6 in. cut in a vertical plate. The weldment weighs 16 lb and its mass center is located at point G. Knowing that at the instant shown the velocity of each pin is 30 in./s downward along the slots, determine (a) the acceleration of the weldment, (b) the reactions at A and B.

The coefficients of friction between the 30-lb block and the 5-lb platform BD are \(\mathrm{m}_{s}=0.50\) and \(\mathrm{m}_{k}=0.40\). Determine the accelerations of the block and of the platform immediately after wire AB has been cut.

Draw the shear and bending-moment diagrams for the vertical rod AB of Prob. 16.16.

Draw the shear and bending-moment diagrams for the connecting rod BC of Prob. 16.18.

For a rigid slab in translation, show that the system of the effective forces consists of vectors \(\left(\Delta m_{i}\right) \overline{\mathbf{a}}\) attached to the various particles of the slab, where \(\overline{\mathbf{a}}\) is the acceleration of the mass center G of the slab. Further show, by computing their sum and the sum of their moments about G, that the effective forces reduce to a single vector \(m \overline{\mathbf{a}}\) attached at G.

For a rigid slab in centroidal rotation, show that the system of the effective forces consists of vectors \(-\left(\Delta m_{i}\right) \mathrm{v}^{2} \mathbf{r}_{i}^{\prime} \text { and }\left(\Delta m_{i}\right)\left(A \times \mathbf{r}_{i}^{\prime}\right)\) attached to the various particles \(P_{i}\) of the slab, where V and A are the angular velocity and angular acceleration of the slab, and where \(\mathbf{r}_{i}^{\prime}\) denotes the position vector of the particle \(P_{i}\) relative to the mass center G of the slab. Further show, by computing their sum and the sum of their moments about G, that the effective forces reduce to a couple \(\overline{I}\mathrm{A}\).

The rotor of an electric motor has an angular velocity of 3600 rpm when the load and power are cut off. The 50-kg rotor, which has a centroidal radius of gyration of 180 mm, then coasts to rest. Knowing that kinetic friction results in a couple of magnitude \(3.5\mathrm{\ N}\cdot\mathrm{m}\) exerted on the rotor, determine the number of revolutions that the rotor executes before coming to rest.

It takes 10 min for a 6000-lb flywheel to coast to rest from an angular velocity of 300 rpm. Knowing that the radius of gyration of the flywheel is 36 in., determine the average magnitude of the couple due to kinetic friction in the bearings.

The 8-in.-radius brake drum is attached to a larger flywheel that is not shown. The total mass moment of inertia of the drum and the flywheel is \(14\mathrm{\ lb}\cdot\mathrm{ft}\cdot\mathrm{s}^2\) and the coefficient of kinetic friction between the drum and the brake shoe is 0.35. Knowing that the angular velocity of the flywheel is 360 rpm counterclockwise when a force P of magnitude 75 lb is applied to the pedal C, determine the number of revolutions executed by the flywheel before it comes to rest.

Solve Prob. 16.27, assuming that the initial angular velocity of the flywheel is 360 rpm clockwise.

The 100-mm-radius brake drum is attached to a flywheel which is not shown. The drum and flywheel together have a mass of 300 kg and a radius of gyration of 600 mm. The coefficient of kinetic friction between the brake band and the drum is 0.30. Knowing that a force P of magnitude 50 N is applied at A when the angular velocity is 180 rpm counterclockwise, determine the time required to stop the flywheel when a = 200 mm and b = 160 mm.

The 180-mm-radius disk is at rest when it is placed in contact with a belt moving at a constant speed. Neglecting the weight of the link AB and knowing that the coefficient of kinetic friction between the disk and the belt is 0.40, determine the angular acceleration of the disk while slipping occurs.

Solve Prob. 16.30, assuming that the direction of motion of the belt is reversed.

In order to determine the mass moment of inertia of a flywheel of radius 600 mm, a 12-kg block is attached to a wire that is wrapped around the flywheel. The block is released and is observed to fall 3 m in 4.6 s. To eliminate bearing friction from the computation, a second block of mass 24 kg is used and is observed to fall 3 m in 3.1 s. Assuming that the moment of the couple due to friction remains constant, determine the mass moment of inertia of the flywheel.

The flywheel shown has a radius of 20 in., a weight of 250 lb, and a radius of gyration of 15 in. A 30-lb block A is attached to a wire that is wrapped around the flywheel, and the system is released from rest. Neglecting the effect of friction, determine (a) the acceleration of block A, (b) the speed of block A after it has moved 5 ft.

Each of the double pulleys shown has a mass moment of inertia of \(15\mathrm{\ lb}\cdot\mathrm{ft}\cdot\mathrm{s}^2\) and is initially at rest. The outside radius is 18 in., and the inner radius is 9 in. Determine (a) the angular acceleration of each pulley, (b) the angular velocity of each pulley after point A on the cord has moved 10 ft.

Each of the gears A and B has a mass of 9 kg and has a radius of gyration of 200 mm; gear C has a mass of 3 kg and has a radius of gyration of 75 mm. If a couple M of constant magnitude 5 N-m is applied to gear C, determine (a) the angular acceleration of gear A, (b) the tangential force which gear C exerts on gear A.

Solve Prob. 16.35, assuming that the couple M is applied to disk A.

Gear A weighs 1 lb and has a radius of gyration of 1.3 in; gear B weighs 6 lb and has a radius of gyration of 3 in.; gear C weighs 9 lb and has a radius of gyration of 4.3 in. Knowing a couple M of constant magnitude of \(40\ \mathrm{lb}\cdot\mathrm{in}\) is applied to gear A, determine (a) the angular acceleration of gear C, (b) the tangential force which gear B exerts on gear C.

Disks A Problems and B are bolted together, and cylinders D and E are attached to separate cords wrapped on the disks. A single cord passes over disks B and C. Disk A weighs 20 lb and disks B and C each weigh 12 lb. Knowing that the system is released from rest and that no slipping occurs between the cords and the disks, determine the acceleration (a) of cylinder D, (b) of cylinder E.

A belt of negligible mass passes between cylinders A and B and is pulled to the right with a force P. Cylinders A and B weigh, respectively, 5 and 20 lb. The shaft of cylinder A is free to slide in a vertical slot and the coefficients of friction between the belt and each of the cylinders are \(\mathrm{m}_{s}=0.50\) and \(\mathrm{m}_{k}=0.40\). For P = 3.6 lb, determine (a) whether slipping occurs between the belt and either cylinder, (b) the angular acceleration of each cylinder.

Solve Prob. 16.39 for P = 2.00 lb.

Disk A has a mass of 6 kg and an initial angular velocity of 360 rpm clockwise; disk B has a mass of 3 kg and is initially at rest. The disks are brought together by applying a horizontal force of magnitude 20 N to the axle of disk A. Knowing that \(m_{k}=0.15\) between the disks and neglecting bearing friction, determine (a) the angular acceleration of each disk, (b) the final angular velocity of each disk.

Solve Prob. 16.41, assuming that initially disk A is at rest and disk B has an angular velocity of 360 rpm clockwise.

Disk A has a mass \(m_A=4\mathrm{\ kg}\), a radius \(r_A=300\mathrm{\ mm}\), and an initial angular velocity \(\mathrm{V}_0=300\mathrm{\ rpm}\) clockwise. Disk B has a mass \(m_B=1.6\mathrm{\ kg}\), a radius \(r_B=180\mathrm{\ mm}\), and is at rest when it is brought into contact with disk A. Knowing that \(m_{k}=0.35\) between the disks and neglecting bearing friction, determine (a) the angular acceleration of each disk, (b) the reaction at the support C.

Disk B is at rest when it is brought into contact with disk A, which has an initial angular velocity \(\mathrm{V}_{0}\). (a) Show that the final angular velocities of the disks are independent of the coefficient of friction \(m_{k}\) between the disks as long as \(\mathrm{m}_{k} \neq 0\). (b) Express the final angular velocity of disk A in terms of \(\mathrm{v}_{0}\) and the ratio of the masses of the two disks \(m_{A} / m_{B}\).

Cylinder A has an initial angular velocity of 720 rpm clockwise, and cylinders B and C are initially at rest. Disks A and B each weigh 5 lb and have radius r = 4 in. Disk C weighs 20 lb and has a radius of 8 in. The disks are brought together when C is placed gently onto A and B. Knowing that \(\mathrm{m}_{k}=0.25\) between A and C and no slipping occurs between B and C, determine (a) the angular acceleration of each disk, (b) the final angular velocity of each disk.

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

For a rigid slab in plane motion, show that the system of the effective forces consists of vectors \(\left(\Delta m_{i}\right) \overline{\mathbf{a}}\), \(-\left(\Delta m_{i}\right) \mathrm{v}^{2} \mathbf{r}_{i}^{\prime}\), and \(\left(\Delta m_{i}\right)\left(A \times \mathbf{r}_{i}^{\prime}\right)\) attached to the various particles \(P_{i}\) of the slab, where \(\bar{\mathbf{a}}\) is the acceleration of the mass center G of the slab, V is the angular velocity of the slab, A is its angular acceleration, and \(\mathbf{r}_{i}^{\prime}\) denotes the position vector of the particle \(P_{i}\), relative to G. Further show, by computing their sum and the sum of their moments about G, that the effective forces reduce to a vector \(m \bar{\mathbf{a}}\) attached at G and a couple \(\bar{I}\mathrm{A}\).

A uniform slender rod AB rests on a frictionless horizontal surface, and a force P of magnitude 0.25 lb is applied at A in a direction perpendicular to the rod. Knowing that the rod weighs 1.75 lb, determine (a) the acceleration of point A, (b) the acceleration of point B, (c) the location of the point on the bar that has zero acceleration.

(a) In Prob. 16.48, determine the point of the rod AB at which the force P should be applied if the acceleration of point B is to be zero. (b) Knowing that P = 0.25 lb, determine the corresponding acceleration of point A.

A force P of magnitude 3 N is applied to a tape wrapped around a thin hoop of mass 2.4 kg. Knowing that the body rests on a frictionless horizontal surface, determine the acceleration of (a) point A, (b) point B.

A force P is applied to a tape wrapped around a uniform disk that rests on a frictionless horizontal surface. Show that for each \(360^{\circ}\) rotation of the disk the center of the disk will move a distance pr.

A 250-lb satellite has a radius of gyration of 24 in. with respect to the y axis and is symmetrical with respect to the zx plane. Its orientation is changed by firing four small rockets A, B, C, and D, each of which produces a 4-lb thrust T directed as shown. Determine the angular acceleration of the satellite and the acceleration of its mass center G (a) when all four rockets are fired, (b) when all rockets except D are fired.

A rectangular plate of mass 5 kg is suspended from four vertical wires, and a force P of magnitude 6 N is applied to corner C as shown. Immediately after P is applied, determine the acceleration of (a) the midpoint of edge BC, (b) corner B.

A uniform slender L-shaped bar ABC is at rest on a horizontal surface when a force P of magnitude 4 N is applied at point A. Neglecting friction between the bar and the surface and knowing that the mass of the bar is 2 kg, determine (a) the initial angular acceleration of the bar, (b) the initial acceleration of point B.

By pulling on the string of a yo-yo, a person manages to make the yo-yo spin, while remaining at the same elevation above the floor. Denoting the mass of the yo-yo by m, the radius of the inner drum on which the string is wound by r, and the centroidal radius of gyration of the yo-yo by \(\bar{k}\), determine the angular acceleration of the yo-yo.

The 80-g yo-yo shown has a centroidal radius of gyration of 30 mm. The radius of the inner drum on which a string is wound is 6 mm. Knowing that at the instant shown the acceleration of the center of the yo-yo is \(1\mathrm{\ m}/\mathrm{s}^2\) upward, determine (a) the required tension T in the string, (b) the corresponding angular acceleration of the yo-yo.

A 6-lb sprocket wheel has a centroidal radius of gyration of 2.75 in. and is suspended from a chain as shown. Determine the acceleration of points A and B of the chain, knowing that \(T_{\mathrm{A}}=3\ \mathrm{lb}\) and \(T_{\mathrm{B}}=4\ \mathrm{lb}\).

The steel roll shown has a mass of 1200 kg, a centroidal radius of gyration of 150 mm, and is lifted by two cables looped around its shaft. Knowing that for each cable \(T_A=3100\mathrm{\ N}\) and \(T_B=3300\mathrm{\ N}\), determine (a) the angular acceleration of the roll, (b) the acceleration of its mass center.

The steel roll shown has a mass of 1200 kg, has a centroidal radius of gyration of 150 mm, and is lifted by two cables looped around its shaft. Knowing that at the instant shown the acceleration of the roll is \(150\mathrm{\ mm}/\mathrm{s}^2\) downward and that for each cable \(T_A=3000\mathrm{\ N}\), determine (a) the corresponding tension \(T_B\), (b) the angular acceleration of the roll.

A 15-ft beam weighing 500 lb is lowered by means of two cables unwinding from overhead cranes. As the beam approaches the ground, the crane operators apply brakes to slow the unwinding motion. Knowing that the deceleration of cable A is \(20\mathrm{\ ft}/\mathrm{s}^2\) and the deceleration of cable B is \(2\mathrm{\ ft}/\mathrm{s}^2\), determine the tension in each cable.

A 15-ft beam weighing 500 lb is lowered by means of two cables unwinding from overhead cranes. As the beam approaches the ground, the crane operators apply brakes to slow the unwinding motion. Knowing that the deceleration of cable A is \(20\mathrm{\ ft}/\mathrm{s}^2\) and the deceleration of cable B is \(2\mathrm{\ ft}/\mathrm{s}^2\), determine the tension in each cable.

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

A beam AB of mass m and of uniform cross section is suspended from two springs as shown. If spring 2 breaks, determine at that instant (a) the angular acceleration of the bar, (b) the acceleration of point A, (c) the acceleration of point B.

A beam AB of mass m and of uniform cross section is suspended from two springs as shown. If spring 2 breaks, determine at that instant (a) the angular acceleration of the bar, (b) the acceleration of point A, (c) the acceleration of point B.

A beam AB of mass m and of uniform cross section is suspended from two springs as shown. If spring 2 breaks, determine at that instant (a) the angular acceleration of the bar, (b) the acceleration of point A, (c) the acceleration of point B.

A thin plate of the shape indicated and of mass m is suspended from two springs as shown. If spring 2 breaks, determine the acceleration at that instant (a) of point A, (b) of point B. A square plate of side b

A thin plate of the shape indicated and of mass m is suspended from two springs as shown. If spring 2 breaks, determine the acceleration at that instant (a) of point A, (b) of point B. A circular plate of diameter b

A thin plate of the shape indicated and of mass m is suspended from two springs as shown. If spring 2 breaks, determine the acceleration at that instant (a) of point A, (b) of point B. A rectangular plate of height b and width a

A sphere of radius r and mass m is projected along a rough horizontal surface with the initial velocities indicated. If the final velocity of the sphere is to be zero, express, in terms of \(v_{0}\), r, and \(\mathrm{m}_{k}\), (a) the required magnitude of \(\mathrm{V}_{0}\), (b) the time \(t_1\) required for the sphere to come to rest, (c) the distance the sphere will move before coming to rest.

Solve Prob. 16.69, assuming that the sphere is replaced by a uniform thin hoop of radius r and mass m.

A bowler projects an 8-in.-diameter ball weighing 12 lb along an alley with a forward velocity \(\mathbf{v}_{0}\) of 15 ft/s and a backspin \(\mathrm{V}_{0}\) of 9 rad/s. Knowing that the coefficient of kinetic friction between the ball and the alley is 0.10, determine (a) the time \(t_{1}\) at which the ball will start rolling without sliding, (b) the speed of the ball at time \(t_{1}\), (c) the distance the ball will have traveled at time \(t_{1}\).

Solve Prob. 16.71, assuming that the bowler projects the ball with the same forward velocity but with a backspin of 18 rad/s.

A uniform sphere of radius r and mass m is placed with no initial velocity on a belt that moves to the right with a constant velocity \(\mathbf{v}_{1}\). Denoting by \(\mathrm{m}_{k}\) the coefficient of kinetic friction between the sphere and the belt, determine (a) the time \(t_1\) at which the sphere will start rolling without sliding, (b) the linear and angular velocities of the sphere at time \(t_1\).

A sphere of radius r and mass m has a linear velocity \(\mathbf{v}_{0}\) directed to the left and no angular velocity as it is placed on a belt moving to the right with a constant velocity \(\mathbf{v}_{1}\). If after first sliding on the belt the sphere is to have no linear velocity relative to the ground as it starts rolling on the belt without sliding, determine in terms of \(v_1\) and the coefficient of kinetic friction \(\mathrm{m}_{k}\) between the sphere and the belt (a) the required value of \(v_0\), (b) the time \(t_1\) at which the sphere will start rolling on the belt, (c) the distance the sphere will have moved relative to the ground at time \(t_1\).

Show that the couple \(\bar{I}\mathrm{A}\) of Fig. 16.15 can be eliminated by attaching the vectors \(m \bar{\mathbf{a}}_{t}\) and \(m \bar{\mathbf{a}}_{n}\) at a point P called the center of percussion, located on line OG at a distance \(G P=\bar{k}^{2} / \bar{r}\) from the mass center of the body.

A uniform slender rod of length L = 900 mm and mass m = 4 kg is suspended from a hinge at C. A horizontal force P of magnitude 75 N is applied at end B. Knowing that \(\bar{r}=225\mathrm{\ mm}\), determine (a) the angular acceleration of the rod, (b) the components of the reaction at C.

In Prob. 16.76, determine (a) the distance \(\bar{r}\) for which the horizontal component of the reaction at C is zero, (b) the corresponding angular acceleration of the rod.

A uniform slender rod of length L = 36 in. and weight W = 4 lb hangs freely from a hinge at A. If a force P of magnitude 1.5 lb is applied at B horizontally to the left (h = L), determine (a) the angular acceleration of the rod, (b) the components of the reaction at A.

In Prob. 16.78, determine (a) the distance h for which the horizontal component of the reaction at A is zero, (b) the corresponding angular acceleration of the rod.

The uniform slender rod AB is welded to the hub D, and the system rotates about the vertical axis DE with a constant angular velocity V. (a) Denoting by w the mass per unit length of the rod, express the tension in the rod at a distance z from end A in terms of w, l, z, and V, (b) Determine the tension in the rod for w = 0.3 kg/m, l = 400 mm, z = 250 mm, and v = 150 rpm.

The shutter shown was formed by removing one quarter of a disk of 0.75-in. radius and is used to interrupt a beam of light emanating from a lens at C. Knowing that the shutter weighs 0.125 lb and rotates at the constant rate of 24 cycles per second, determine the magnitude of the force exerted by the shutter on the shaft at A.

A 6-in.-diameter hole is cut as shown in a thin disk of 15-in. diameter. The disk rotates in a horizontal plane about its geometric center A at the constant rate of 480 rpm. Knowing that the disk has a mass of 60 lb after the hole has been cut, determine the horizontal component of the force exerted by the shaft on the disk at A.

A turbine disk of mass 26 kg rotates at a constant rate of 9600 rpm. Knowing that the mass center of the disk coincides with the center of rotation O, determine the reaction at O immediately after a single blade at A, of mass 45 g, becomes loose and is thrown off.

A uniform rod of length L and mass m is supported as shown. If the cable attached at end B suddenly breaks, determine (a) the acceleration of end B, (b) the reaction at the pin support.

A uniform rod of length L and mass m is supported as shown. If the cable attached at end B suddenly breaks, determine (a) the acceleration of end B, (b) the reaction at the pin support.

A 12-lb uniform plate rotates about A in a vertical plane under the combined effect of gravity and of the vertical force P. Knowing that at the instant shown the plate has an angular velocity of 20 rad/s and an angular acceleration of \(30\ \mathrm{rad}/\mathrm{s}^2\) both counterclockwise, determine (a) the force P, (b) the components of the reaction at A.

A 1.5-kg slender rod is welded to a 5-kg uniform disk as shown. The assembly swings freely about C in a vertical plane. Knowing that in the position shown the assembly has an angular velocity of 10 rad/s clockwise, determine (a) the angular acceleration of the assembly, (b) the components of the reaction at C.

Two uniform rods, ABC of weight 6 lb and DCE of weight 8 lb, are connected by a pin at C and by two cords BD and BE. The T-shaped assembly rotates in a vertical plane under the combined effect of gravity and of a couple M which is applied to rod ABC. Knowing that at the instant shown the tension in cord BE is 2 lb and the tension in cord BD is 0.5 lb, determine (a) the angular acceleration of the assembly, (b) the couple M.

The object ABC consists of two slender rods welded together at point B. Rod AB has a weight of 2 lb and bar BC has a weight of 4 lb. Knowing the magnitude of the angular velocity of ABC is 10 rad/s when \(\mathrm{u}=0^{\circ}\), determine the components of the reaction at point C when \(\mathrm{u}=0^{\circ}\).

A 3.5-kg slender rod AB and a 2-kg slender rod BC are connected by a pin at B and by the cord AC. The assembly can rotate in a vertical plane under the combined effect of gravity and a couple M applied to rod BC. Knowing that in the position shown the angular velocity of the assembly is zero and the tension in cord AC is equal to 25 N, determine (a) the angular acceleration of the assembly, (b) the magnitude of the couple M.

A 9-kg uniform disk is attached to the 5-kg slender rod AB by means of frictionless pins at B and C. The assembly rotates in a vertical plane under the combined effect of gravity and of a couple M which is applied to rod AB. Knowing that at the instant shown the assembly has an angular velocity of 6 rad/s and an angular acceleration of \(25\mathrm{\ rad}/\mathrm{s}^2\), both counterclockwise, determine (a) the couple M, (b) the force exerted by pin C on member AB.

Derive the equation \(\Sigma M_{C}=I_{C} \mathrm{a}\) for the rolling disk of Fig. 16.17, where \(\(\Sigma M_{C}\) represents the sum of the moments of the external forces about the instantaneous center C, and \(I_{C}\) is the moment of inertia of the disk about C.

Show that in the case of an unbalanced disk, the equation derived in Prob. 16.92 is valid only when the mass center G, the geometric center O, and the instantaneous center C happen to lie in a straight line.

A wheel of radius r and centroidal radius of gyration \(\bar{k}\) is released from rest on the incline and rolls without sliding. Derive an expression for the acceleration of the center of the wheel in terms of r, \(\bar{k}\), b, and g.

A homogeneous sphere S, a uniform cylinder C, and a thin pipe P are in contact when they are released from rest on the incline shown. Knowing that all three objects roll without slipping, determine, after 4 s of motion, the clear distance between (a) the pipe and the cylinder, (b) the cylinder and the sphere.

A 40-kg flywheel of radius R = 0.5 m is rigidly attached to a shaft of radius r = 0.05 m that can roll along parallel rails. A cord is attached as shown and pulled with a force P of magnitude 150 N. Knowing the centroidal radius of gyration is \(\bar{k}=0.4\mathrm{\ m}\), determine (a) the angular acceleration of the flywheel, (b) the velocity of the center of gravity after 5 s.

A 40-kg flywheel of radius R = 0.5 m is rigidly attached to a shaft of radius r = 0.05 m that can roll along parallel rails. A cord is attached as shown and pulled with a force P. Knowing the centroidal radius of gyration is \(\bar{k}=0.4\mathrm{\ m}\) and the coefficient of static friction is \(\mathrm{m}_{s}=0.4\), determine the largest magnitude of force P for which no slipping will occur.

A drum of 60-mm radius is attached to a disk of 120-mm radius. The disk and drum have a total mass of 6 kg and a combined radius of gyration of 90 mm. A cord is attached as shown and pulled with a force P of magnitude 20 N. Knowing that the disk rolls without sliding, determine (a) the angular acceleration of the disk and the acceleration of G, (b) the minimum value of the coefficient of static friction compatible with this motion.

A drum of 60-mm radius is attached to a disk of 120-mm radius. The disk and drum have a total mass of 6 kg and a combined radius of gyration of 90 mm. A cord is attached as shown and pulled with a force P of magnitude 20 N. Knowing that the disk rolls without sliding, determine (a) the angular acceleration of the disk and the acceleration of G, (b) the minimum value of the coefficient of static friction compatible with this motion.

A drum of 60-mm radius is attached to a disk of 120-mm radius. The disk and drum have a total mass of 6 kg and a combined radius of gyration of 90 mm. A cord is attached as shown and pulled with a force P of magnitude 20 N. Knowing that the disk rolls without sliding, determine (a) the angular acceleration of the disk and the acceleration of G, (b) the minimum value of the coefficient of static friction compatible with this motion.

A drum of 60-mm radius is attached to a disk of 120-mm radius. The disk and drum have a total mass of 6 kg and a combined radius of gyration of 90 mm. A cord is attached as shown and pulled with a force P of magnitude 20 N. Knowing that the disk rolls without sliding, determine (a) the angular acceleration of the disk and the acceleration of G, (b) the minimum value of the coefficient of static friction compatible with this motion.

A drum of 4-in. radius is attached to a disk of 8-in. radius. The disk and drum have a combined weight of 10 lb and a combined radius of gyration of 6 in. A cord is attached as shown and pulled with a force P of magnitude 5 lb. Knowing that the coefficients of static and kinetic friction are \(\mathrm{m}_{s}=0.25\) and \(\mathrm{m}_{k}=0.20\), respectively, determine (a) whether or not the disk slides, (b) the angular acceleration of the disk and the acceleration of G.

A drum of 4-in. radius is attached to a disk of 8-in. radius. The disk and drum have a combined weight of 10 lb and a combined radius of gyration of 6 in. A cord is attached as shown and pulled with a force P of magnitude 5 lb. Knowing that the coefficients of static and kinetic friction are \(\mathrm{m}_{s}=0.25\) and \(\mathrm{m}_{k}=0.20\), respectively, determine (a) whether or not the disk slides, (b) the angular acceleration of the disk and the acceleration of G.

A drum of 4-in. radius is attached to a disk of 8-in. radius. The disk and drum have a combined weight of 10 lb and a combined radius of gyration of 6 in. A cord is attached as shown and pulled with a force P of magnitude 5 lb. Knowing that the coefficients of static and kinetic friction are \(\mathrm{m}_{s}=0.25\) and \(\mathrm{m}_{k}=0.20\), respectively, determine (a) whether or not the disk slides, (b) the angular acceleration of the disk and the acceleration of G.

A drum of 4-in. radius is attached to a disk of 8-in. radius. The disk and drum have a combined weight of 10 lb and a combined radius of gyration of 6 in. A cord is attached as shown and pulled with a force P of magnitude 5 lb. Knowing that the coefficients of static and kinetic friction are \(\mathrm{m}_{s}=0.25\) and \(\mathrm{m}_{k}=0.20\), respectively, determine (a) whether or not the disk slides, (b) the angular acceleration of the disk and the acceleration of G.

A 12-in.-radius cylinder of weight 16 lb rests on a 6-lb carriage. The system is at rest when a force P of magnitude 4 lb is applied. Knowing that the cylinder rolls without sliding on the carriage and neglecting the mass of the wheels of the carriage, determine (a) the acceleration of the carriage, (b) the acceleration of point A, (c) the distance the cylinder has rolled with respect to the carriage after 0.5 s.

A 12-in.-radius cylinder of weight 16 lb rests on a 6-lb carriage. The system is at rest when a force P of magnitude 4 lb is applied. Knowing that the cylinder rolls without sliding on the carriage and neglecting the mass of the wheels of the carriage, determine (a) the acceleration of the carriage, (b) the acceleration of point A, (c) the distance the cylinder has rolled with respect to the carriage after 0.5 s.

Gear C has a mass of 5 kg and a centroidal radius of gyration of 75 mm. The uniform bar AB has a mass of 3 kg and gear D is stationary. If the system is released from rest in the position shown, determine (a) the angular acceleration of gear C, (b) the acceleration of point B.

Two uniform disks A and B, each of mass 2 kg, are connected by a 1.5-kg rod CD as shown. A counterclockwise couple M of moment 2.5 N-m is applied to disk A. Knowing that the disks roll without sliding, determine (a) the acceleration of the center of each disk, (b) the horizontal component of the force exerted on disk B by pin D.

A 10-lb cylinder of radius r = 4 in. is resting on a conveyor belt when the belt is suddenly turned on and it experiences an acceleration of magnitude \(a=6\mathrm{\ ft}/\mathrm{s}^2\). The smooth vertical bar holds the cylinder in place when the belt is not moving. Knowing the cylinder rolls without slipping and the friction between the vertical bar and the cylinder is negligible, determine (a) the angular acceleration of the cylinder, (b) the components of the force the conveyor belt applies to the cylinder.

A hemisphere of weight W and radius r is released from rest in the position shown. Determine (a) the minimum value of \(\mathrm{m}_{s}\) for which the hemisphere starts to roll without sliding, (b) the corresponding acceleration of point B [Hint: Note that \(O G=\frac{3}{5} r\) and that, by the parallel-axis theorem, \(\bar{I}=\frac{2}{5} m r^{2}-m(O G)^{2}\).]

Solve Prob. 16.111, considering a half cylinder instead of a hemisphere. [Hint: Note that OG = 4r/3p and that, by the parallel-axis theorem, \(\bar{I}=\frac{1}{2} m r^{2}-m(O G)^{2}\).]

The center of gravity G of a 1.5-kg unbalanced tracking wheel is located at a distance r = 18 mm from its geometric center B. The radius of the wheel is R = 60 mm and its centroidal radius of gyration is 44 mm. At the instant shown the center B of the wheel has a velocity of 0.35 m/s and an acceleration of \(1.2\mathrm{\ m}/\mathrm{s}^2\), both directed to the left. Knowing that the wheel rolls without sliding and neglecting the mass of the driving yoke AB, determine the horizontal force P applied to the yoke.

A small clamp of mass \(m_{B}\) is attached at B to a hoop of mass \(m_{h}\). The system is released from rest when \(u = 90^{\circ}\) and rolls without sliding. Knowing that \(m_{h} = 3m_{B}\), determine (a) the angular acceleration of the hoop, (b) the horizontal and vertical components of the acceleration of B.

A small clamp of mass \(m_{B}\) is attached at B to a hoop of mass \(m_{h}\). Knowing that the system is released from rest and rolls without sliding, derive an expression for the angular acceleration of the hoop in terms of \(m_{B}\), \(m_{h}\), r, and u.

A 4-lb bar is attached to a 10-lb uniform cylinder by a square pin, P, as shown. Knowing that r = 16 in., h = 8 in., \(u = 20^{\circ}\), L = 20 in., and v = 2 rad/s at the instant shown, determine the reactions at P at this instant assuming that the cylinder rolls without sliding down the incline.

The ends of the 20-lb uniform rod AB are attached to collars of negligible mass that slide without friction along fixed rods. If the rod is released from rest when \(u = 25^{\circ}\), determine immediately after release (a) the angular acceleration of the rod, (b) the reaction at A, (c) the reaction at B.

The ends of the 20-lb uniform rod AB are attached to collars of negligible mass that slide without friction along fixed rods. A vertical force P is applied to collar B when \(u = 25^{\circ}\), causing the collar to start from rest with an upward acceleration of \(40 \ ft/s^{2}\). Determine (a) the force P, (b) the reaction at A.

The motion of the 3-kg uniform rod AB is guided by small wheels of negligible weight that roll along without friction in the slots shown. If the rod is released from rest in the position shown, determine immediately after release (a) the angular acceleration of the rod, (b) the reaction at B.

A beam AB of length L and mass m is supported by two cables as shown. If cable BD breaks, determine at that instant the tension in the remaining cable as a function of its initial angular orientation u.

End A of a uniform 10-kg bar is attached to a horizontal rope and end B contacts a floor with negligible friction. Knowing that the bar is released from rest in the position shown, determine immediately after release (a) the angular acceleration of the bar, (b) the tension in the rope, (c) the reaction at B.

End A of the 8-kg uniform rod AB is attached to a collar that can slide without friction on a vertical rod. End B of the rod is attached to a vertical cable BC. If the rod is released from rest in the position shown, determine immediately after release (a) the angular acceleration of the rod, (b) the reaction at A.

A uniform thin plate ABCD has a mass of 8 kg and is held in position by three inextensible cords AE, BF, and CG. If cord AE is cut, determine at that instant (a) if the plate is undergoing translation or general plane motion, (b) the tension in cords BF and CG.

The 4-kg uniform rod ABD is attached to the crank BC and is fitted with a small wheel that can roll without friction along a vertical slot. Knowing that at the instant shown crank BC rotates with an angular velocity of 6 rad/s clockwise and an angular acceleration of \(15 \ rad/s^{2}\) counterclockwise, determine the reaction at A.

The 7-lb uniform rod AB is connected to crank BD and to a collar of negligible weight, which can slide freely along rod EF. Knowing that in the position shown crank BD rotates with an angular velocity of 15 rad/s and an angular acceleration of \(60 \ rad/s^{2}\) , both clockwise, determine the reaction at A.

In Prob. 16.125, determine the reaction at A, knowing that in the position shown crank BD rotates with an angular velocity of 15 rad/s clockwise and an angular acceleration of \(60 \ rad/s^{2}\) counterclockwise.

The 250-mm uniform rod BD, of mass 5 kg, is connected as shown to disk A and to a collar of negligible mass, that may slide freely along a vertical rod. Knowing that disk A rotates counterclockwise at a constant rate of 500 rpm, determine the reactions at D when u = 0.

Solve Prob. 16.127 when \(u = 90^{\circ}\).

The 4-kg uniform slender bar BD is attached to bar AB and a wheel of negligible mass that rolls on a circular surface. Knowing that at the instant shown bar AB has an angular velocity of 6 rad/s and no angular acceleration, determine the reaction at point D.

The motion of the uniform slender rod of length L = 0.5 m and mass m = 3 kg is guided by pins at A and B that slide freely in frictionless slots, circular and horizontal, cut into a vertical plate as shown. Knowing that at the instant shown the rod has an angular velocity of 3 rad/s counterclockwise and \(u = 30^{\circ}\), determine the reactions at points A and B.

At the instant shown, the 20-ft-long, uniform 100-lb pole ABC has an angular velocity of 1 rad/s counterclockwise and point C is sliding to the right. A 120-lb horizontal force P acts at B. Knowing the coefficient of kinetic friction between the pole and the ground is 0.3, determine at this instant (a) the acceleration of the center of gravity, (b) the normal force between the pole and the ground.

A driver starts his car with the door on the passenger’s side wide open (u = 0). The 80-lb door has a centroidal radius of gyration \(\bar{k}=12.5 \ \mathrm{in}.\), and its mass center is located at a distance r = 22 in. from its vertical axis of rotation. Knowing that the driver maintains a constant acceleration of \(6 \ ft/s^{2}\), determine the angular velocity of the door as it slams shut (\(u = 90^{\circ}\)).

For the car of Prob. 16.132, determine the smallest constant acceleration that the driver can maintain if the door is to close and latch, knowing that as the door hits the frame its angular velocity must be at least 2 rad/s for the latching mechanism to operate.

Two 8-lb uniform bars are connected to form the linkage shown. Neglecting the effect of friction, determine the reaction at D immediately after the linkage is released from rest in the position shown.

The 6-kg rod BC connects a 10-kg disk centered at A to a 5-kg rod CD. The motion of the system is controlled by the couple M applied to disk A. Knowing that at the instant shown disk A has an angular velocity of 36 rad/s clockwise and no angular acceleration, determine (a) the couple M, (b) the components of the force exerted at C on rod BC.

The 6-kg rod BC connects a 10-kg disk centered at A to a 5-kg rod CD. The motion of the system is controlled by the couple M applied to disk A. Knowing that at the instant shown disk A has an angular velocity of 36 rad/s clockwise and an angular acceleration of \(150 \ rad/s^{2}\) counterclockwise, determine (a) the couple M, (b) the components of the force exerted at C on rod BC.

In the engine system shown l = 250 mm and b = 100 mm. The connecting rod BD is assumed to be a 1.2-kg uniform slender rod and is attached to the 1.8-kg piston P. During a test of the system, crank AB is made to rotate with a constant angular velocity of 600 rpm clockwise with no force applied to the face of the piston. Determine the forces exerted on the connecting rod at B and D when \(u = 180^{\circ}\). (Neglect the effect of the weight of the rod.)

Solve Prob. 16.137 when \(u = 90^{\circ}\).

The 4-lb rod AB Problems and the 6-lb rod BC are connected as shown to a disk that is made to rotate in a vertical plane at a constant angular velocity of 6 rad/s clockwise. For the position shown, determine the forces exerted at A and B on rod AB.

The 4-lb rod AB and the 6-lb rod BC are connected as shown to a disk that is made to rotate in a vertical plane. Knowing that at the instant shown the disk has an angular acceleration of \(18 \ rad/s^{2}\) clockwise and no angular velocity, determine the components of the forces exerted at A and B on rod AB.

Two rotating rods in the vertical plane are connected by a slider block P of negligible mass. The rod attached at A has a weight of 1.6 lb and a length of 8 in. Rod BP weighs 2 lb and is 10 in. long and the friction between block P and AE is negligible. The motion of the system is controlled by a couple M applied to rod BP. Knowing that rod BP has a constant angular velocity of 20 rad/s clockwise, determine (a) the couple M, (b) the components of the force exerted on AE by block P.

Two rotating rods in the vertical plane are connected by a slider block P of negligible mass. The rod attached at A has a mass of 0.8 kg and a length of 160 mm. Rod BP has a mass of 1 kg and is 200 mm long and the friction between block P and AE is negligible. The motion of the system is controlled by a couple M applied to bar BP. Knowing that at the instant shown rod BP has an angular velocity of 20 rad/s clockwise and an angular acceleration of \(80 \ rad/s^{2}\) clockwise, determine (a) the couple M, (b) the components of the force exerted on AE by block P.

Draw the shear and bending-moment diagrams for the rod of Prob. 16.77 immediately after the cable at B breaks.

A uniform slender bar AB of mass m is suspended as shown from a uniform disk of the same mass m. Neglecting the effect of friction, determine the accelerations of points A and B immediately after a horizontal force P has been applied at B.

A uniform rod AB, of mass 15 kg and length 1 m, is attached to the 20-kg cart C. Neglecting friction, determine immediately after the system has been released from rest, (a) the acceleration of the cart, (b) the angular acceleration of the rod.

The 5-kg slender rod AB is pin-connected to an 8-kg uniform disk as shown. Immediately after the system is released from rest, determine the acceleration of (a) point A, (b) point B.

The 6-lb cylinder B and the 4-lb wedge A are held at rest in the position shown by cord C. Assuming that the cylinder rolls without sliding on the wedge and neglecting friction between the wedge and the ground, determine, immediately after cord C has been cut, (a) the acceleration of the wedge, (b) the angular acceleration of the cylinder.

The 6-lb cylinder B and the 4-lb wedge A are held at rest in the position shown by cord C. Assuming that the cylinder rolls without sliding on the wedge and neglecting friction between the wedge and the ground, determine, immediately after cord C has been cut, (a) the acceleration of the wedge, (b) the angular acceleration of the cylinder.

Each of the 3-kg bars AB and BC is of length L = 500 mm. A horizontal force P of magnitude 20 N is applied to bar BC as shown. Knowing that b = L (P is applied at C), determine the angular acceleration of each bar.

Each of the 3-kg bars AB and BC is of length L = 500 mm. A horizontal force P of magnitude 20 N is applied to bar BC. For the position shown, determine (a) the distance b for which the bars move as if they formed a single rigid body, (b) the corresponding angular acceleration of the bars.

(a) Determine the magnitude and the location of the maximum bending moment in the rod of Prob. 16.78. (b) Show that the answer to part a is independent of the weight of the rod.

Draw the shear and bending-moment diagrams for the rod of Prob. 16.84 immediately after the cable at B breaks.

A cyclist is riding a bicycle at a speed of 20 mph on a horizontal road. The distance between the axles is 42 in., and the mass center of the cyclist and the bicycle is located 26 in. behind the front axle and 40 in. above the ground. If the cyclist applies the brakes only on the front wheel, determine the shortest distance in which he can stop without being thrown over the front wheel.

The forklift truck shown weighs 2250 lb and is used to lift a crate of weight W = 2500 lb. The truck is moving to the left at a speed of 10 ft/s when the brakes are applied on all four wheels. Knowing that the coefficient of static friction between the crate and the fork lift is 0.30, determine the smallest distance in which the truck can be brought to a stop if the crate is not to slide and if the truck is not to tip forward.

A 5-kg uniform disk is attached to the 3-kg uniform rod BC by means of a frictionless pin AB. An elastic cord is wound around the edge of the disk and is attached to a ring at E. Both ring E and rod BC can rotate freely about the vertical shaft. Knowing that the system is released from rest when the tension in the elastic cord is 15 N, determine (a) the angular acceleration of the disk, (b) the acceleration of the center of the disk.

Identical cylinders of mass m and radius r are pushed by a series of moving arms. Assuming the coefficient of friction between all surfaces to be m < 1 and denoting by a the magnitude of the acceleration of the arms, derive an expression for (a) the maximum allowable value of a if each cylinder is to roll without sliding, (b) the minimum allowable value of a if each cylinder is to move to the right without rotating.

The uniform rod AB of weight W is released from rest when \(b = 70^{\circ}\). Assuming that the friction force between end A and the surface is large enough to prevent sliding, determine immediately after release (a) the angular acceleration of the rod, (b) the normal reaction at A, (c) the friction force at A.

The uniform rod AB of weight W is released from rest when \(b = 70^{\circ}\). Assuming that the friction force is zero between end A and the surface, determine immediately after release (a) the angular acceleration of the rod, (b) the acceleration of the mass center of the rod, (c) the reaction at A.

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

A uniform plate of mass m Review Problems is suspended in each of the ways shown. For each case determine immediately after the connection B has been released (a) the angular acceleration of the plate, (b) the acceleration of its mass center.

A cylinder with a circular hole is rolling without slipping on a fixed curved surface as shown. The cylinder would have a weight of 16 lb without the hole, but with the hole it has a weight of 15 lb. Knowing that at the instant shown the disk has an angular velocity of 5 rad/s clockwise, determine (a) the angular acceleration of the disk, (b) the components of the reaction force between the cylinder and the ground at this instant.

The motion of a square plate of side 150 mm and mass 2.5 kg is guided by pins at corners A and B that slide in slots cut in a vertical wall. Immediately after the plate is released from rest in the position shown, determine (a) the angular acceleration of the plate, (b) the reaction at corner A.

The motion of a square plate of side 150 mm and mass 2.5 kg is guided by a pin at corner A that slides in a horizontal slot cut in a vertical wall. Immediately after the plate is released from rest in the position shown, determine (a) the angular acceleration of the plate, (b) the reaction at corner A.

The Geneva mechanism shown is used to provide an intermittent rotary motion of disk S. Disk D weighs 2 lb and has a radius of gyration of 0.9 in., and disk S weighs 6 lb and has a radius of gyration of 1.5 in. The motion of the system is controlled by a couple M applied to disk D. A pin P is attached to disk D and can slide in one of the six equally spaced slots cut in disk S. It is desirable that the angular velocity of disk S be zero as the pin enters and leaves each of the six slots; this will occur if the distance between the centers of the disks and the radii of the disks are related as shown. Knowing disk D rotates with a constant counterclockwise angular velocity of 8 rad/s and the friction between the slot and pin P is negligible, determine when \(f = 150^{\circ}\) (a) the couple M, (b) the magnitude of the force pin P applies to disk S.