The system consists of 60-lb and 20-lb blocks A and B, respectively, and 5-lb pulleys C

Determine the kinetic energy of the 100-kg object.

The 80-kg wheel has a radius of gyration about its mass center O of \(k_{O} = 400 \ mm\). Determine its angular velocity after it has rotated 20 revolutions starting from rest.

The uniform 50-lb slender rod is subjected to a couple moment of \(M = 100 \ lb \cdot ft\). If the rod is at rest when \(\theta = 0^{\circ}\), determine its angular velocity when \(\theta = 90^{\circ}\).

The uniform 50-kg slender rod is at rest in the position shown when P = 600 N is applied. Determine the angular velocity of the rod when the rod reaches the vertical position.

The 50-kg wheel is subjected to a force of 50 N. If the wheel starts from rest and rolls without slipping, determine its angular velocity after it has rotated 10 revolutions. The radius of gyration of the wheel about its mass center G is \(k_{G} = 0.3 \ m\).

If the uniform 30-kg slender rod starts from rest at the position shown, determine its angular velocity after it has rotated 4 revolutions. The forces remain perpendicular to the rod.

The 20-kg wheel has a radius of gyration about its center G of \(k_{G} = 300 \ mm\). When it is subjected to a couple moment of \(M = 50 \ N \cdot m\), it rolls without slipping. Determine the angular velocity of the wheel after its mass center G has traveled through a distance of \(s_{G} = 20 \ m\), starting from rest.

If the 30-kg disk is released from rest when \(\theta = 0^{\circ}\), determine its angular velocity when \(\theta = 90^{\circ}\).

The 50-kg reel has a radius of gyration about its center O of \(k_{O} = 300 \ mm\). If it is released from rest, determine its angular velocity when its center O has traveled 6 m down the smooth inclined plane.

The 60-kg rod OA is released from rest when \(\theta = 0^{\circ}\). Determine its angular velocity when \(\theta = 45^{\circ}\). The spring remains vertical during the motion and is unstretched when \(\theta = 0^{\circ}\).

The 30-kg rod is released from rest when \(\theta = 0^{\circ}\). Determine the angular velocity of the rod when \(\theta = 90^{\circ}\). The spring is unstretched when \(\theta = 0^{\circ}\).

The 30-kg rod is released from rest when \(\theta = 45^{\circ}\). Determine the angular velocity of the rod when \(\theta = 0^{\circ}\). The spring is unstretched when \(\theta = 45^{\circ}\).

At a given instant the body of mass m has an angular velocity \(\mathbf{\omega}\) and its mass center has a velocity \(\mathbf{v}_{G}\). Show that its kinetic energy can be represented as \(T=\frac{1}{2} I_{I C} \omega^{2}\), where \(I_{I C}\) is the moment of inertia of the body determined about the instantaneous axis of zero velocity, located a distance \(r_{G / I C}\) from the mass center as shown.

The wheel is made from a 5-kg thin ring and two 2-kg slender rods. If the torsional spring attached to the wheel’s center has a stiffness \(k=2 \ \mathrm{N} \cdot \mathrm{m} / \mathrm{rad}\), and the wheel is rotated until the torque \(M=25 \ \mathrm{N} \cdot \mathrm{m}\) is developed, determine the maximum angular velocity of the wheel if it is released from rest.

The wheel is made from a 5-kg thin ring and two 2-kg slender rods. If the torsional spring attached to the wheel’s center has a stiffness \(k=2 \ \mathrm{N} \cdot \mathrm{m} / \mathrm{rad}\), so that the torque on the center of the wheel is \(M=(2 \theta) \ \mathrm{N} \cdot \mathrm{m}\), where \(\theta\) is in radians, determine the maximum angular velocity of the wheel if it is rotated two revolutions and then released from rest.

A force of P = 60 N is applied to the cable, which causes the 200-kg reel to turn since it is resting on the two rollers A and B of the dispenser. Determine the angular velocity of the reel after it has made two revolutions starting from rest. Neglect the mass of the rollers and the mass of the cable. Assume the radius of gyration of the reel about its center axis remains constant at \(k_{O} = 0.6 \ m\).

A force of P = 20 N is applied to the cable, which causes the 175-kg reel to turn since it is resting on the two rollers A and B of the dispenser. Determine the angular velocity of the reel after it has made two revolutions starting from rest. Neglect the mass of the rollers and the mass of the cable. The radius of gyration of the reel about its center axis is \(k_{G} = 0.42 \ m\).

A force of P = 20 N is applied to the cable, which causes the 175-kg reel to turn without slipping on the two rollers A and B of the dispenser. Determine the angular velocity of the reel after it has made two revolutions starting from rest. Neglect the mass of the cable. Each roller can be considered as an 18-kg cylinder, having a radius of 0.1 m. The radius of gyration of the reel about its center axis is \(k_{G} = 0.42 \ m\).

The double pulley consists of two parts that are attached to one another. It has a weight of 50 lb and a centroidal radius of gyration of \(k_{O} = 0.6 \ ft\) and is turning with an angular velocity of 20 rad/s clockwise. Determine the kinetic energy of the system. Assume that neither cable slips on the pulley.

The double pulley consists of two parts that are attached to one another. It has a weight of 50 lb and a centroidal radius of gyration of \(k_{O} = 0.6 \ ft\) and is turning with an angular velocity of 20 rad/s clockwise. Determine the angular velocity of the pulley at the instant the 20-lb weight moves 2 ft downward.

The disk, which has a mass of 20 kg, is subjected to the couple moment of \(M = (2 \theta + 4) \ N \cdot m\), where \(\theta\) is in radians. If it starts from rest, determine its angular velocity when it has made two revolutions.

The spool has a mass of 40 kg and a radius of gyration of \(k_{O} = 0.3 \ m\). If the 10-kg block is released from rest, determine the distance the block must fall in order for the spool to have an angular velocity \(\omega = 15 \ rad/s\). Also, what is the tension in the cord while the block is in motion? Neglect the mass of the cord.

The force of T = 20 N is applied to the cord of negligible mass. Determine the angular velocity of the 20-kg wheel when it has rotated 4 revolutions starting from rest. The wheel has a radius of gyration of \(k_{O} = 0.3 \ m\).

Determine the velocity of the 50-kg cylinder after it has descended a distance of 2 m. Initially, the system is at rest. The reel has a mass of 25 kg and a radius of gyration about its center of mass A of \(k_{A} = 125 \ mm\).

The 10-kg uniform slender rod is suspended at rest when the force of F = 150 N is applied to its end. Determine the angular velocity of the rod when it has rotated \(90^{\circ}\) clockwise from the position shown. The force is always perpendicular to the rod.

The 10-kg uniform slender rod is suspended at rest when the force of F = 150 N is applied to its end. Determine the angular velocity of the rod when it has rotated \(180^{\circ}\) clockwise from the position shown. The force is always perpendicular to the rod.

The pendulum consists of a 10-kg uniform disk and a 3-kg uniform slender rod. If it is released from rest in the position shown, determine its angular velocity when it rotates clockwise \(90^{\circ}\).

A motor supplies a constant torque \(M = 6 \ kN \cdot m\) to the winding drum that operates the elevator. If the elevator has a mass of 900 kg, the counterweight C has a mass of 200 kg, and the winding drum has a mass of 600 kg and radius of gyration about its axis of k = 0.6 m, determine the speed of the elevator after it rises 5 m starting from rest. Neglect the mass of the pulleys.

The center O of the thin ring of mass m is given an angular velocity of \(\omega_{0}\). If the ring rolls without slipping, determine its angular velocity after it has traveled a distance of s down the plane. Neglect its thickness.

The wheel has a mass of 100 kg and a radius of gyration of \(k_{O} = 0.2 \ m\). A motor supplies a torque \(M = (40 \theta + 900) \ N \cdot m\), where \(\theta\) is in radians, about the drive shaft at O. Determine the speed of the loading car, which has a mass of 300 kg, after it travels s = 4 m. Initially the car is at rest when s = 0 and \(\theta = 0^{\circ}\). Neglect the mass of the attached cable and the mass of the car’s wheels.

The rotary screen S is used to wash limestone. When empty it has a mass of 800 kg and a radius of gyration of \(k_{G} = 1.75 \ m\). Rotation is achieved by applying a torque of \(M = 280 \ N \cdot m\) about the drive wheel at A. If no slipping occurs at A and the supporting wheel at B is free to roll, determine the angular velocity of the screen after it has rotated 5 revolutions. Neglect the mass of A and B.

If P = 200 N and the 15-kg uniform slender rod starts from rest at \(\theta = 0^{\circ}\), determine the rod’s angular velocity at the instant just before \(\theta = 45^{\circ}\).

A yo-yo has a weight of 0.3 lb and a radius of gyration of \(k_{O} = 0.06 \ ft\). If it is released from rest, determine how far it must descend in order to attain an angular velocity \(\omega = 70 \ rad/s\). Neglect the mass of the string and assume that the string is wound around the central peg such that the mean radius at which it unravels is r = 0.02 ft.

If the 50-lb bucket, C, is released from rest, determine its velocity after it has fallen a distance of 10 ft. The windlass A can be considered as a 30-lb cylinder, while the spokes are slender rods, each having a weight of 2 lb. Neglect the pulley’s weight.

The coefficient of kinetic friction between the 100-lb disk and the surface of the conveyor belt is \(\mu_{A} = 0.2\). If the conveyor belt is moving with a speed of \(v_{C} = 6 \ ft/s\) when the disk is placed in contact with it, determine the number of revolutions the disk makes before it reaches a constant angular velocity.

The 30-kg disk is originally at rest, and the spring is unstretched. A couple moment of \(M = 80 \ N \cdot m\) is then applied to the disk as shown. Determine its angular velocity when its mass center G has moved 0.5 m along the plane. The disk rolls without slipping.

The 30-kg disk is originally at rest, and the spring is unstretched. A couple moment \(M = 80 \ N \cdot m\) is then applied to the disk as shown. Determine how far the center of mass of the disk travels along the plane before it momentarily stops. The disk rolls without slipping.

Two wheels of negligible weight are mounted at corners A and B of the rectangular 75-lb plate. If the plate is released from rest at \(\theta = 90^{\circ}\), determine its angular velocity at the instant just before \(\theta = 0^{\circ}\).

The link AB is subjected to a couple moment of \(M = 40 \ N \cdot m\). If the ring gear C is fixed, determine the angular velocity of the 15-kg inner gear when the link has made two revolutions starting from rest. Neglect the mass of the link and assume the inner gear is a disk. Motion occurs in the vertical plane.

The 10-kg rod AB is pin connected at A and subjected to a couple moment of \(M = 15 \ N \cdot m\). If the rod is released from rest when the spring is unstretched at \(\theta = 30^{\circ}\), determine the rod’s angular velocity at the instant \(\theta = 60^{\circ}\). As the rod rotates, the spring always remains horizontal, because of the roller support at C.

The 10-lb sphere starts from rest at \(\theta = 0^{\circ}\) and rolls without slipping down the cylindrical surface which has a radius of 10 ft. Determine the speed of the sphere’s center of mass at the instant \(\theta = 45^{\circ}\).

Motor M exerts a constant force of P = 750 N on the rope. If the 100-kg post is at rest when \(\theta = 0^{\circ}\), determine the angular velocity of the post at the instant \(\theta = 60^{\circ}\). Neglect the mass of the pulley and its size, and consider the post as a slender rod.

The linkage consists of two 6-kg rods AB and CD and a 20-kg bar BD. When \(\theta = 0^{\circ}\), rod AB is rotating with an angular velocity \(\omega = 2 \ rad/s\). If rod CD is subjected to a couple moment of \(M = 30 \ N \cdot m\), determine \(\omega_{AB}\) at the instant \(\theta = 90^{\circ}\).

The linkage consists of two 6-kg rods AB and CD and a 20-kg bar BD. When \(\theta = 0^{\circ}\), rod AB is rotating with an angular velocity \(\omega = 2 \ rad/s\). If rod CD is subjected to a couple moment \(M = 30 \ N \cdot m\), determine \(\omega\) at the instant \(\theta = 45^{\circ}\).

The two 2-kg gears A and B are attached to the ends of a 3-kg slender bar. The gears roll within the fixed ring gear C, which lies in the horizontal plane. If a \(10-N \cdot m\) torque is applied to the center of the bar as shown, determine the number of revolutions the bar must rotate starting from rest in order for it to have an angular velocity of \(\omega_{AB} = 20 \ rad/s\). For the calculation, assume the gears can be approximated by thin disks. What is the result if the gears lie in the vertical plane?

The linkage consists of two 8-lb rods AB and CD and a 10-lb bar AD. When \(\theta = 0^{\circ}\), rod AB is rotating with an angular velocity \(\omega_{AB} = 2 \ rad/s\). If rod CD is subjected to a couple moment \(M = 15 \ lb \cdot ft\) and bar AD is subjected to a horizontal force P = 20 lb as shown, determine \(\omega_{AB}\) at the instant \(\theta = 90^{\circ}\).

The linkage consists of two 8-lb rods AB and CD and a 10-lb bar AD. When \(\theta = 0^{\circ}\), rod AB is rotating with an angular velocity \(\omega_{AB} = 2 \ rad/s\). If rod CD is subjected to a couple moment \(M = 15 \ lb \cdot ft\) and bar AD is subjected to a horizontal force P = 20 lb as shown, determine \(\omega_{AB}\) at the instant \(\theta = 45^{\circ}\).

The assembly consists of a 3-kg pulley A and 10-kg pulley B. If a 2-kg block is suspended from the cord, determine the block’s speed after it descends 0.5 m starting from rest. Neglect the mass of the cord and treat the pulleys as thin disks. No slipping occurs.

The assembly consists of a 3-kg pulley A and 10-kg pulley B. If a 2-kg block is suspended from the cord, determine the distance the block must descend, starting from rest, in order to cause B to have an angular velocity of 6 rad/s. Neglect the mass of the cord and treat the pulleys as thin disks. No slipping occurs.

The spool has a mass of 50 kg and a radius of gyration of \(k_{O} = 0.280 \ m\). If the 20-kg block A is released from rest, determine the distance the block must fall in order for the spool to have an angular velocity \(\omega = 5 \ rad/s\). Also, what is the tension in the cord while the block is in motion? Neglect the mass of the cord.

The spool has a mass of 50 kg and a radius of gyration of \(k_{O} = 0.280 \ m\). If the 20-kg block A is released from rest, determine the velocity of the block when it descends 0.5 m.

An automobile tire has a mass of 7 kg and radius of gyration of \(k_{G} = 0.3 \ m\). If it is released from rest at A on the incline, determine its angular velocity when it reaches the horizontal plane. The tire rolls without slipping.

The spool has a mass of 20 kg and a radius of gyration of \(k_{O} = 160 \ mm\). If the 15-kg block A is released from rest, determine the distance the block must fall in order for the spool to have an angular velocity \(\omega = 8 \ rad/s\). Also, what is the tension in the cord while the block is in motion? Neglect the mass of the cord.

The spool has a mass of 20 kg and a radius of gyration of \(k_{O} = 160 \ mm\). If the 15-kg block A is released from rest, determine the velocity of the block when it descends 600 mm.

A uniform ladder having a weight of 30 lb is released from rest when it is in the vertical position. If it is allowed to fall freely, determine the angle \(\theta\) at which the bottom end A starts to slide to the right of A. For the calculation, assume the ladder to be a slender rod and neglect friction at A.

Determine the speed of the 50-kg cylinder after it has descended a distance of 2 m, starting from rest. Gear A has a mass of 10 kg and a radius of gyration of 125 mm about its center of mass. Gear B and drum C have a combined mass of 30 kg and a radius of gyration about their center of mass of 150 mm.

The 12-kg slender rod is attached to a spring, which has an unstretched length of 2 m. If the rod is released from rest when \(\theta = 30^{\circ}\), determine its angular velocity at the instant \(\theta = 90^{\circ}\).

The 12-kg slender rod is attached to a spring, which has an unstretched length of 2 m. If the rod is released from rest when \(\theta = 30^{\circ}\), determine the angular velocity of the rod the instant the spring becomes unstretched.

The 40-kg wheel has a radius of gyration about its center of gravity G of \(k_{G} = 250 \ mm\). If it rolls without slipping, determine its angular velocity when it has rotated clockwise \(90^{\circ}\) from the position shown. The spring AB has a stiffness k = 100 N/m and an unstretched length of 500 mm. The wheel is released from rest.

The assembly consists of two 10-kg bars which are pin connected. If the bars are released from rest when \(\theta = 60^{\circ}\), determine their angular velocities at the instant \(\theta = 0^{\circ}\). The 5-kg disk at C has a radius of 0.5 m and rolls without slipping.

The assembly consists of two 10-kg bars which are pin connected. If the bars are released from rest when \(\theta = 60^{\circ}\), determine their angular velocities at the instant \(\theta = 30^{\circ}\). The 5-kg disk at C has a radius of 0.5 m and rolls without slipping.

The compound disk pulley consists of a hub and attached outer rim. If it has a mass of 3 kg and a radius of gyration of \(k_{G} = 45 \ mm\), determine the speed of block A after A descends 0.2 m from rest. Blocks A and B each have a mass of 2 kg. Neglect the mass of the cords.

The uniform garage door has a mass of 150 kg and is guided along smooth tracks at its ends. Lifting is done using the two springs, each of which is attached to the anchor bracket at A and to the counterbalance shaft at B and C. As the door is raised, the springs begin to unwind from the shaft, thereby assisting the lift. If each spring provides a torsional moment of \(M = (0.7 \theta) \ N \cdot m\), where \(\theta\) is in radians, determine the angle \(\theta_{0}\) at which both the left-wound and right-wound spring should be attached so that the door is completely balanced by the springs, i.e., when the door is in the vertical position and is given a slight force upward, the springs will lift the door along the side tracks to the horizontal plane with no final angular velocity. Note: The elastic potential energy of a torsional spring is \(V_{e}=\frac{1}{2} k \theta^{2}\), where \(M=k \theta\) and in this case \(k=0.7 \ \mathrm{N} \cdot \mathrm{m} / \mathrm{rad}\).

The two 12-kg slender rods are pin connected and released from rest at the position \(\theta = 60^{\circ}\). If the spring has an unstretched length of 1.5 m, determine the angular velocity of rod BC, when the system is at the position \(\theta = 0^{\circ}\). Neglect the mass of the roller at C.

The two 12-kg slender rods are pin connected and released from rest at the position \(\theta = 60^{\circ}\). If the spring has an unstretched length of 1.5 m, determine the angular velocity of rod BC, when the system is at the position \(\theta = 30^{\circ}\).

If the 250-lb block is released from rest when the spring is unstretched, determine the velocity of the block after it has descended 5 ft. The drum has a weight of 50 lb and a radius of gyration of \(k_{O} = 0.5 \ ft\) about its center of mass O.

The slender 15-kg bar is initially at rest and standing in the vertical position when the bottom end A is displaced slightly to the right. If the track in which it moves is smooth, determine the speed at which end A strikes the corner D. The bar is constrained to move in the vertical plane. Neglect the mass of the cord BC.

If the chain is released from rest from the position shown, determine the angular velocity of the pulley after the end B has risen 2 ft. The pulley has a weight of 50 lb and a radius of gyration of 0.375 ft about its axis. The chain weighs 6 lb/ft.

If the gear is released from rest, determine its angular velocity after its center of gravity O has descended a distance of 4 ft. The gear has a weight of 100 lb and a radius of gyration about its center of gravity of k = 0.75 ft.

The slender 6-kg bar AB is horizontal and at rest and the spring is unstretched. Determine the stiffness k of the spring so that the motion of the bar is momentarily stopped when it has rotated clockwise \(90^{\circ}\) after being released.

The slender 6-kg bar AB is horizontal and at rest and the spring is unstretched. Determine the angular velocity of the bar when it has rotated clockwise \(45^{\circ}\) after being released. The spring has a stiffness of k = 12 N/m.

The pendulum consists of a 6-kg slender rod fixed to a 15-kg disk. If the spring has an unstretched length of 0.2 m, determine the angular velocity of the pendulum when it is released from rest and rotates clockwise \(90^{\circ}\) from the position shown. The roller at C allows the spring to always remain vertical.

The 500-g rod AB rests along the smooth inner surface of a hemispherical bowl. If the rod is released from rest from the position shown, determine its angular velocity at the instant it swings downward and becomes horizontal.

The 50-lb wheel has a radius of gyration about its center of gravity G of \(k_{G} = 0.7 \ ft\). If it rolls without slipping, determine its angular velocity when it has rotated clockwise \(90^{\circ}\) from the position shown. The spring AB has a stiffness k = 1.20 lb/ft and an unstretched length of 0.5 ft. The wheel is released from rest.

The system consists of 60-lb and 20-lb blocks A and B, respectively, and 5-lb pulleys C and D that can be treated as thin disks. Determine the speed of block A after block B has risen 5 ft, starting from rest. Assume that the cord does not slip on the pulleys, and neglect the mass of the cord.

The door is made from one piece, whose ends move along the horizontal and vertical tracks. If the door is in the open position, \(\theta = 0^{\circ}\), and then released, determine the speed at which its end A strikes the stop at C. Assume the door is a 180-lb thin plate having a width of 10 ft.

The door is made from one piece, whose ends move along the horizontal and vertical tracks. If the door is in the open position, \(\theta = 0^{\circ}\), and then released, determine its angular velocity at the instant \(\theta = 30^{\circ}\). Assume the door is a 180-lb thin plate having a width of 10 ft.

The end A of the garage door AB travels along the horizontal track, and the end of member BC is attached to a spring at C. If the spring is originally unstretched, determine the stiffness k so that when the door falls downward from rest in the position shown, it will have zero angular velocity the moment it closes, i.e., when it and BC become vertical. Neglect the mass of member BC and assume the door is a thin plate having a weight of 200 lb and a width and height of 12 ft. There is a similar connection and spring on the other side of the door.

The system consists of a 30-kg disk, 12-kg slender rod BA, and a 5-kg smooth collar A. If the disk rolls without slipping, determine the velocity of the collar at the instant \(\theta = 0^{\circ}\). The system is released from rest when \(\theta = 45^{\circ}\).

The system consists of a 30-kg disk A, 12-kg slender rod BA, and a 5-kg smooth collar A. If the disk rolls without slipping, determine the velocity of the collar at the instant \(\theta = 30^{\circ}\). The system is released from rest when \(\theta = 45^{\circ}\).

The bicycle and rider start from rest at the top of the hill. Show how to determine the speed of the rider when he freely coasts down the hill. Use appropriate dimensions of the wheels, and the mass of the rider, frame and wheels of the bicycle to explain your results.

Two torsional springs, \(M = k \theta\), are used to assist in opening and closing the hood of this truck. Assuming the springs are uncoiled (\(\theta = 0^{\circ}\)) when the hood is opened, determine the stiffness \(k (N \cdot m/rad)\) of each spring so that the hood can easily be lifted, i.e., practically no force applied to it, when it is closed in the unlocked position. Use appropriate numerical values to explain your result.

The operation of this garage door is assisted using two springs AB and side members BCD, which are pinned at C. Assuming the springs are unstretched when the door is in the horizontal (open) position and ABCD is vertical, determine each spring stiffness k so that when the door falls to the vertical (closed) position, it will slowly come to a stop. Use appropriate numerical values to explain your result.

Determine the counterweight of A needed to balance the weight of the bridge deck when \(\theta = 0^{\circ}\). Show that this weight will maintain equilibrium of the deck by considering the potential energy of the system when the deck is in the arbitrary position \(\theta\). Both the deck and AB are horizontal when \(\theta = 0^{\circ}\). Neglect the weights of the other members. Use appropriate numerical values to explain this result.

The pendulum of the Charpy impact machine has a mass of 50 kg and a radius of gyration of \(k_{A} = 1.75 \ m\). If it is released from rest when \(\theta = 0^{\circ}\), determine its angular velocity just before it strikes the specimen S, \(\theta = 90^{\circ}\).

The 50-kg flywheel has a radius of gyration of \(k_{0} = 200 \ mm\) about its center of mass. If it is subjected to a torque of \(M=\left(9 \theta^{1 / 2}+1\right) \ \mathrm{N} \cdot \mathrm{m}\), where \(\theta\) is in radians, determine its angular velocity when it has rotated 5 revolutions, starting from rest.

The drum has a mass of 50 kg and a radius of gyration about the pin at O of \(k_{O} = 0.23 \ m\). Starting from rest, the suspended 15-kg block B is allowed to fall 3 m without applying the brake ACD. Determine the speed of the block at this instant. If the coefficient of kinetic friction at the brake pad C is \(\mu_{k} = 0.5\), determine the force P that must be applied at the brake handle which will then stop the block after it descends another 3 m. Neglect the thickness of the handle.

The spool has a mass of 60 kg and a radius of gyration of \(k_{G} = 0.3 \ m\). If it is released from rest, determine how far its center descends down the smooth plane before it attains an angular velocity of \(\omega = 6 \ rad/s\). Neglect the mass of the cord which is wound around the central core. The coefficient of kinetic friction between the spool and plane at A is \(\mu_{k} = 0.2\).

The gear rack has a mass of 6 kg, and the gears each have a mass of 4 kg and a radius of gyration of k = 30 mm at their centers. If the rack is originally moving downward at 2 m/s, when s = 0, determine the speed of the rack when s = 600 mm. The gears are free to turn about their centers A and B.

At the instant shown, the 50-lb bar rotates clockwise at 2 rad/s. The spring attached to its end always remains vertical due to the roller guide at C. If the spring has an unstretched length of 2 ft and a stiffness of k = 6 lb/ft, determine the angular velocity of the bar the instant it has rotated \(30^{\circ}\) clockwise.

The system consists of a 20-lb disk A, 4-lb slender rod BC, and a 1-lb smooth collar C. If the disk rolls without slipping, determine the velocity of the collar at the instant the rod becomes horizontal, i.e., \(\theta = 0^{\circ}\). The system is released from rest when \(\theta = 45^{\circ}\).

At the instant the spring becomes undeformed, the center of the 40-kg disk has a speed of 4 m/s. From this point determine the distance d the disk moves down the plane before momentarily stopping. The disk rolls without slipping.