Problem 1 - Load B is connected to a double pulley by one of two inextensible cables shown. The motion of the pulley is controlled by cable C, which has a constant acceleration of 9 in/s² and an initial velocity of 12 in/s, both directed to the right. Determine (a) the number of revolutions executed by the pulley in 2 s, (b) the velocity and change in position of the load B after 2 s, and (c) the acceleration of point D on the rim of the inner pulley at t = 0.

Problem 2 - Determine the distance the load W is lifted in t = 4s using the hoist. The shaft of the motor M turns with an angular velocity = 10 t² rad/s. where t is in seconds.

Problem 3 - Link AB has angular velocity and angular acceleration as shown. Determine the velocity and acceleration of point C and the angular acceleration of link BC at this instant. Express them as Cartesian vectors.

Problem 4 - A sphere, a cylinder, and a hoop, each having the same mass and the same radius, are released from rest on an incline. Determine the velocity of each body after it has rolled through a distance corresponding to a change in elevation h.

Problem 5 - 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 rad/s² both counterclockwise, determine (a) the force P, and (b) the components of the reaction at A.

Problem 6 - A pulley weighing 12 lb and having a radius of gyration of 8 in. is connected to two cylinders. Assuming no axle friction, determine the angular acceleration of the pulley and the acceleration of each cylinder.

Problem 7 - A 2-kg sphere moving horizontally to the right with an initial velocity of 5 m/s strikes the lower end of an 10-kg rigid rod AB. The rod is suspended from a hinge at A and is initially at rest. Knowing that the coefficient of restitution between the rod and the sphere is 0.75, determine the angular velocity of the rod and the velocity of the sphere immediately after the impact.

Problem 8 - When the forward speed of the 4000 lb. truck shown (below left) was 30 ft/s, the brakes were suddenly applied, causing all four wheels to stop rotating. It was observed that the truck skidded to rest in 20 ft. Determine the magnitude of the normal reaction and the friction force at each wheel. What is the kinetic coefficient of friction between the tires and the road?

Problem 9 - A uniform sphere of mass m and radius r is projected along a rough horizontal surface with a linear velocity v and no angular velocity. Denoting by μ_k the coefficient of kinetic friction between the sphere and the surface, determine (a) the time t₂ at which the sphere will start rolling without sliding, and (b) the linear and angular velocities of the sphere at time t₂.

Problem 10 - A 30-lb slender rod AB is 5 ft long and is pivoted about a point 0 which is 1 ft from end B. The other end is pressed against a spring of constant k= 1800 lb/in until the spring is compressed 1 in. and the rod is in a horizontal position. If the rod is released from this position, determine its angular velocity and the reaction at the pivot O as the rod passes through a vertical position.

Problem 11 - A cord is wrapped around the inner drum of a wheel and pulled horizontally with a force of 200 N. the wheel has a mass of 50 kg and a radius of gyration of 70 mm. Knowing that μ_s = 0.20 and μ_k = 0.15, determine the acceleration of the wheel.

Problem 12 - Two solid spheres of radius 3 in., weighing 2 lb each, are mounted at A and B on the horizontal rod A'B', which rotates freely about the vertical with a counterclockwise angular velocity of 6 rad/s. The spheres are held in position by a cord which is suddenly cut. Knowing that the centroidal moment of inertia of the rod and pivot is I_y = 0.25 lb-ft-s², determine (a) the velocity of the spheres after they have moved to positions A' and B', (b) the energy lost due to the plastic impact of the spheres and the stops at A' and B'. Calculate (a) by treating the spheres as points with the same mass while neglecting the mass of the rod and pivot.

Problem 13 - The portion AOB of a mechanism consists of a 400-mm steel rod OB welded to a gear E of radius 120 mm which may rotate about a horizontal shaft 0. It is actuated by a gear D and, at the instant shown, has a clockwise angular velocity of 8 rad/s and a counterclockwise angular acceleration of 40 rad/s². Knowing that rod OB has a mass of 3 kg and gear E a mass of 4 kg and a radius of gyration of 85 mm, determine (a) the tangential force exerted by gear D on gear E, and (b) the components of the reaction at shaft O.

Problem 14 - Gear C rotates clockwise with the angular velocity of 2 rad/s, while the connecting link AB rotates counterclockwise at 4 rad/s. Determine the angular velocity of gear D.