R2-50. The semicircular disk has a mass of 50 kg and is

Blocks A and B weigh 50 and 10 lb, respectively. If P = 100 lb, determine the normal force exerted by block A on block B. Neglect friction and the weights of the pulleys, cord, and bars of the triangular frame.

The handcart has a mass of 200 kg and center of mass at G. Determine the normal reactions at each of the wheels at A and B if a force P = 50 N is applied to the handle. Neglect the mass and rolling resistance of the wheels.

The truck carries the 800-lb crate which has a center of gravity at \(G_{c}\). Determine the largest acceleration of the truck so that the crate will not slip or tip on the truck bed. The coefficient of static friction between the crate and the truck is \(\mu_{s}=0.6\).

*R2-4. The spool has a weight of 30 lb and a radius of gyration kQ = 0.65 ft. If a force of 40 lb is applied to the cord at -4, determine the angular velocity of the spool in / = 3 s starting from rest. Neglect the mass of the pulley and cord.

Solve Prob. R2-4 if a 40-lb block is suspended from the cord at A. rather than applying the 40-lb force.

R2-6. The uniform plate weighs 40 lb and is supported by a roller at A. If a horizontal force F = 70 lb is suddenly applied to the roller, determine the acceleration of the center of the roller at the instant the force is applied. The plate has a moment of inertia about its center of mass of lG = 0.414 slug ft2. Neglect the weight of the roller.

R2-7. The center of the pulley is being lifted vertically with an acceleration of 4 m/s2 at the instant it has a velocity of 2 m/s. If the cable does not slip on the pulleys surface, determine the accelerations of the cylinder B and point C. on the pulley.

:;R2-8. The double pendulum consists of two rods. Rod A B has a constant angular velocity of 3 rad/s, and rod BC has a constant angular velocity of 2 rad/s. Both of these absolute motions are measured counterclockwise. Determine the velocity and acceleration of point C at the instant shown.

R2-9. The link OA is pinned at O and rotates because of the sliding action of rod R along the horizontal groove. If R starts from rest when 0 = 0 and has a constant acceleration an = 60 mm/s to the right, determine the angular velocity and angular acceleration of OA when / = 2 s.

R2-10. The drive wheel A has a constant angular velocity of oja. At a particular instant, the radius of rope wound on each wheel is as shown. If the rope has a thickness T, determine the angular acceleration of wheel B.

R2-11. The dresser has a weight of 80 lb and is pushed along the floor. If the coefficient of static friction at A and B is fis = 0.3 and the coefficient of kinetic friction is /ik = 0.2, determine the smallest horizontal force P needed to cause motion. If this force is increased slightly, determine the acceleration of the dresser. Also, what are the normal reactions at A and B when it begins to move?

*R2-12. If the ball has a weight of 15 lb and is thrown onto a rough surface so that its center has a velocity of 6 ft/s parallel to the surface, determine the amount of backspin. to. the ball must be given so that it stops spinning at the same instant that its forward velocity is zero. It is not necessary to know the coefficient of kinetic friction at A for the calculation.

R2-13. The dragster has a mass of 1500 kg and a center of mass at G. If the coefficient of kinetic friction between the rear wheels and the pavement is fik = 0.6, determine if it is possible for the driver to lift the front wheels, A, off the ground while the rear wheels are slipping. If so. what acceleration is necessary to do this? Neglect the mass of the wheels and assume that the front wheels are free to roll.

R2-14. The dragster has a mass of 1500 kg and a center of mass at G. If no slipping occurs, determine the friction force Flt which must be applied to each of the rear wheels B in order to develop an acceleration a 6 m/s2. What are the normal reactions of each wheel on the ground? Neglect the mass of the wheels and assume that the front wheels are free to roll.

R2-15. If the operator initially drives the pedals at 20 rev/min. and then begins an angular acceleration of 30 rev/min2, determine the angular velocity of the flywheel F when / = 3 s. Note that the pedal arm is fixed connected to the chain wheel A. which in turn drives the sheave B using the fixed connected clutch gear D.The belt wraps around the sheave then drives the pulley E and fixed-connected flywheel.

*R2-16. If the operator initially drives the pedals at 12 rev/min. and then begins an angular acceleration of 8 rev/min2, determine the angular velocity of the flywheel F after the pedal arm has rotated 2 revolutions. Note that the pedal arm is fixed connected to the chain wheel A. which in turn drives the sheave B using the fixed-connected clutch gear D. The belt wraps around the sheave then drives the pulley E and fixed-connected flywheel.

R2-17. The drum has a mass of 50 kg and a radius of gyration about the pin at O of ka = 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 /x* = 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.

R2-18. The drum has a mass of 50 kg and a radius of gyration about the pin at O of k0 = 0.23 m. If the 15-kg block is moving downward at 3 m/s, and a force of P 100 N is applied to the brake arm, determine how far the block descends from the instant the brake is applied until it stops. Neglect the thickness of the handle. The coefficient of kinetic friction at the brake pad is jik 0.5.

R2-19. The 1.6-Mg car shown has been raked by increasing the height h = 0.2 m of its center of mass. This was done by raising the springs on the rear axle. If the coefficient of static friction between the rear wheels and the ground is fis = 0.3, show that the car can accelerate slightly faster than its counterpart for which h = 0. Neglect the mass of the wheels and driver and assume the front wheels at B are free to roll while the rear wheels slip.

*R2-20. The disk is rotating at a constant rate co = 4 rad/s. and as it falls freely, its center has an acceleration of 32.2 ft/s2. Determine the acceleration of point A on the rim of the disk at the instant shown.

The disk is rotating at a constant rate \(\omega=4 \ \mathrm{rad} / \mathrm{s}\), and as it falls freely, its center has an acceleration of \(32.2 \ \mathrm{ft} / \mathrm{s}^{2}\). Determine the acceleration of point B on the rim of the disk at the instant shown.

The board rests on the surface of two drums. At the instant shown, it has an acceleration of \(0.5\mathrm{\ m}/\mathrm{s}^2\) to the right, while at the same instant points on the outer rim of each drum have an acceleration with a magnitude of \(3 \mathrm{\ m} / \mathrm{s}^{2}\). If the board does not slip on the drums, determine its speed due to the motion.

R2-23. A 20-kg roll of paper, originally at rest, is pin-supporled at its ends to bracket A B. The roll rests against a wall for which the coefficient of kinetic friction at C is fjLC = 0.3. If a force of 40 N is applied uniformly to the end of the sheet, determine the initial angular acceleration of the roll and the tension in the bracket as the paper unwraps. For the calculation, treat the roll as a cylinder.

R2-24. At the instant shown, link A B has an angular velocity ioAS = 2 rad/s and an angular acceleration aAH = 6 rad/s2. Determine the acceleration of the pin at C and the angular acceleration of link CB at this instant, when 0 = 60.

R2-25. The truck has a weight of 8000 lb and center of gravity at Gt. It carries the 800-lb crate, which has a center of gravity at Gc. Determine the normal reaction at each of its four tires if it accelerates at a = 0.5 ft/s2. Also, what is the frictional force acting between the crate and the truck, and between each of the rear tires and the road? Assume that power is delivered only to the rear tires.The front tires are free to roll. Neglect the mass of the tires.The crate does not slip or tip on the truck.

R2-26. The 15-lb cylinder is initially at rest on a 5-lb plate. If a couple moment M = 40 lb ft is applied to the cylinder, determine the angular acceleration of the cylinder and the time needed for the end B of the plate to travel 3 ft and strike the wall. Assume the cylinder does not slip on the plate, and neglect the mass of the rollers under the plate.

R2-27. At the instant shown, two forces act on the 30-lb slender rod which is pinned at O. Determine the magnitude of force F and the initial angular acceleration of the rod so that the horizontal reaction which the pin exerts on the rod is 5 lb directed to the right.

*R2-28. The 20-lb solid ball is cast on the floor such that it has a backspin u> = 15 rad/s and its center has an initial horizontal velocity Vq = 20 ft/s. If the coefficient of kinetic friction between the floor and the ball is pA = 0.3. determine the distance it travels before it stops spinning.

R2-29. Determine the backspin to which should be given to the 20-lb ball so that when its center is given an initial horizontal velocity vG = 20 ft/s it stops spinning and translating at the same instant. The coefficient of kinetic friction is pA = 0.3.

R2-30. The wheelbarrow and its contents have a mass of 40 kg and a mass center at G, excluding the wheel. The wheel has a mass of 4 kg and a radius of gyration k() = 0.120 m. If the wheelbarrow is released from rest from the position shown, determine its speed after it travels 4 m down the incline. The coefficient of kinetic friction between the incline and A is jjla = 0.3. The wheels roll without slipping at B.

R2-31. At the given instant member AS has the angular motions shown. Determine the velocity and acceleration of the slider block C at this instant.

The spool and wire wrapped around its core have a mass of 20 kg and a centroidal radius of gyration \(k_{G}=250 \mathrm{\ mm}\). If the coefficient of kinetic friction at the ground is \(\mu_{B}=0.1\), determine the angular acceleration of the spool when the \(30-\mathrm{N} \cdot \mathrm{m}\) couple moment is applied.

R2-33. The car has a mass of 1.50 Mg and a mass center at G. Determine the maximum acceleration it can have if (a) power is supplied only to the rear wheels, (b) power is supplied only to the front wheels. Neglect the mass of the wheels in the calculation, and assume that the wheels that do not receive power are free to roll. Also, assume that slipping of the powered wheels occurs, where the coefficient of kinetic friction is = 0.3.

R2-34. The tire has a mass of 9 kg and a radius of gyration k(J = 225 mm. If it is released from rest and rolls down the plane without slipping, determine the speed of its center O when / = 3 s.

R2-35. The bar has a mass m and length /. If it is released from rest from the position 0 = 30, determine its angular acceleration and the horizontal and vertical components of reaction at the pin O.

*R2-36. The pendulum consists of a 30-lb sphere and a 10-lb slender rod. Compute the reaction at the pin O just after the cord AH is cut.

Spool B is at rest and spool A is rotating at 6 rad/s when the slack in the cord connecting them is taken up. If the cord does not stretch, determine the angular velocity of each spool immediately and the cord is jerked tight. The spools A and B have weights and radii of gyration \(W_A=30\mathrm{\ lb},\quad k_A=0.8\mathrm{\ ft}\quad\text{ and }\quad W_B=15\ \mathrm{lb},\quad k_B=0.6\ \mathrm{ft},\) respectively.

R2-38. The rod is bent into the shape of a sine curve and is forced to rotate about the y axis by connecting the spindle S to a motor. If the rod starts from rest in the position shown and a motor drives it for a short time with an angular acceleration a = (1.5ef) rad/s2, where t is in seconds, determine the magnitudes of the angular velocity and angular displacement of the rod when / = 3 s. Locate the point on the rod which has the greatest velocity and acceleration, and compute the magnitudes of the velocity and acceleration of this point when t = 3 s. The curve defining the rod is = 0.25 sin(7ry), where the argument for the sine is given in radians when y is in meters.

The scaffold S is raised by moving the roller at A toward the pin at B. If A is approaching B with a speed of 1.5 ft/s. determine the speed at which the platform rises as a function of \(\theta\). The 4-ft links are pin connected at their midpoint.

*R2-40. The pendulum of the Charpy impact machine has a mass of 50 kg and a radius of gyration of kA = 1.75 m. If it is released from rest when 0 = 0. determine its angular velocity just before it strikes the specimen S. 0 = 90.

R2-41. The gear rack has a mass of 6 kg, and the gears each have a mass of 4 kg and a radius of gyration 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.

R2-42. A 7-kg automobile tire is released from rest at A on the incline and rolls without slipping to point B. where it then travels in free flight. Determine the maximum height h the tire attains. The radius of gyration of the tire about its mass center is kG = 0.3 m.

R2-43. The two 3-lb rods Yrand HI are fixed (welded) to the link AC at E. Determine the internal axial force v. shear force Ey, and moment Mt. which the bar AC exerts on FE at E if at the instant 0 = 30 link AB has an angular velocity (o = 5 rad/s and an angular acceleration a = 8 rad/s2 as shown.

*R2-44. The uniform connecting rod BC has a mass of 3 kg and is pin-connected at its end points. Determine the vertical forces which the pins exert on the ends B and C of the rod at the instant (a) 6 = 0. and (b) 0 = 90. The crank A B is turning with a constant angular velocity = 5 rad/s.

R2-45. If bar AB has an angular velocity wAB = 6 rad/s, determine the velocity of the slider block C at the instant shown.

R2-46. The drum of mass rn, radius r. and radius of gyration ku rolls along an inclined plane for which the coefficient of static friction is fi. If the drum is released from rest, determine the maximum angle 0 for the incline so that it rolls without slipping.

R2-47. Determine the velocity and acceleration of rod R for any angle 0 of cam C if the cam rotates with a constant angular velocity oj. The pin connection at O does not cause an interference with the motion of A on C.

*R2-48. When the crank on the Chinese windlass is turning, the rope on shaft A unwinds while that on shaft B winds up. Determine the speed at which the block lowers if the crank is turning with an angular velocity oj = 4 rad/s. What is the angular velocity of the pulley at C? The rope segments on each side of the pulley are both parallel and vertical, and the rope does not slip on the pulley.

R2-49. The semicircular disk has a mass of 50 kg and is released from rest from the position shown.The coefficients of static and kinetic friction between the disk and the beam are = 0.5 and ik = 0.3, respectively. Determine the initial reactions at the pin A and roller B. used to support the beam. Neglect the mass of the beam for the calculation.

R2-50. The semicircular disk has a mass of 50 kg and is released from rest from the position shown.The coefficients of static and kinetic friction between the disk and the beam are fxs = 0.2 and nk = 0.1. respectively. Determine the initial reactions at the pin A and roller B used to support the beam. Neglect the mass of the beam for the calculation.

The hoisting gear A has an initial angular velocity of 60 rad/s and a constant deceleration of \(1\ \mathrm{rad}/\mathrm{s}^2\). Determine the velocity and deceleration of the block which is being hoisted by the hub on gear B when t = 3 s.