# Rotational Motion Part I

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3 18. Small blocks, each of mass 2.0 kg, are clamped at the ends and at the center of a light rod 1.2 m long. Compute the moment of inertia of the system about an axis passing through a point one-third of the length from one end of the rod if the moment of inertia of the light rod can be neglected kg-m Find the moment of inertia about each of the following axes for a rod that is 4.00 cm in diameter and 2.00 m long and has a mass of 8.00 kg. a) An axis perpendicular to the rod and passing through its center. b) An axis perpendicular to the rod and passing through one end. c) A longitudinal axis passing through the center of the rod kg-m kg-m kg-m Four small spheres, each of mass kg, are arranged in a square m on a side and connected by light rods of negligible mass. Find the moment of inertia of the system about an axis a) perpendicular to the plane of the square through the center kg-m 2 b) bisecting two opposite sides of the square kg-m What is the rotational inertia of a solid ball 0.50 min radius that weighs 80.0 N if it is rotated about a diameter? kg-m What is the rotational inertia of a thick ring that is rotating about an axis perpendicular to the plane of the ring passing through its center? The ring has a mass of 1.20 kg and a diameter of 45.0 cm. The hole in the ring is 15.0 cm wide kg-m A wagon wheel is constructed as shown in the figure. The radius of the wheel is m and the rim has a mass of 1.20 kg. Each of the eight spokes, which lie along a diameter and are m long has a mass of kg. What is the moment of inertia of the wheel about an axis through its center and perpendicular to the plane of the wheel? kg-m 2

6 47. A cord is wrapped around the rim of a flywheel 0.5 m in radius, and a steady pull of 50.0 N is exerted on the cord. The wheel is mounted with frictionless bearings on a horizontal shaft through its center. The moment of inertia of the wheel is 4.0 kg-m 2. Compute the angular acceleration of the wheel rad/s A grindstone in the shape of a solid disk with a diameter of 1.0 m and a mass of 50.0 kg, is rotating at 900 rev/min. A tool is pressed against the rim with a normal force of N, and the grindstone comes to rest in 10.0 s. Find the coefficient of friction between the tool and the grindstone. Neglect friction in the bearings A bucket of water of mass 20.0 kg is suspended by a rope wrapped around a windlass in the form of a solid cylinder 0.20 m in diameter, also of mass 20.0 kg. The cylinder is pivoted on a frictionless axle through its center. The bucket is released from rest at the top of a well and falls 20.0 m to the water. Neglect the weight of the rope. a) What is the tension in the rope while the bucket is falling? 65.3N b) With what velocity does the bucket strike the water? 16.2 m/s c) What was the time of fall? 2.48 sec d) While the bucket is falling, what is the force exerted on the cylinder by the axle? 261 N 50. A 60.0-kg grindstone is 1.0 m in diameter and has a moment of inertia of 3.75 kg-m 2. A tool is pressed down on the rim with a normal force of 50.0 N. The coefficient of sliding friction between the tool and the stone is 0.6, and there is a constant friction torque of 5 m-n between the axle of the stone and its bearings. a) How much force must be applied normally at the end of a crank handle 0.5 m long to bring the stone from rest to120 rev/min in 9.0 s? 50.5 N b) After attaining a speed of 120 rpm, what must the normal force at the end of the handle become to maintain a constant speed of 120 rpm? 40.0 N c) How long will it take the grindstone to come from 120 rpm to rest if it is acted on by the axle friction alone? 9.42 s 51. Dirk the Dragonslayer is exploring a castle. He is spotted by a dragon who chases him down a hallway. Dirk runs into a room and attempts to swing the heavy door shut before the dragon gets him. The door is initially perpendicular to the wall, so it must be turned through 90º to close. The door is 3.00 m tall and 1.00 m wide and weighs N. The friction at the hinges can be neglected. If Dirk applies a force of N at the edge of the door and perpendicular to it, how long will it take him to close the door? sec 52. A 5.0 kg block rests on a frictionless horizontal surface. A cord attached to the block passes over a pulley whose diameter is 0.2 m, to a hanging block also of mass 5.0 kg. The system is released from rest, and the blocks are observed to move 4.0 m in 2.0 seconds. a) What is the tension in each part of the cord? 10 N 39N b) What is the moment of inertia of the pulley? kg-m Two blocks, one of mass 4.0 kg and the other of mass 2.0 kg are connected by a light rope that passes over a pulley as shown in the figure to the right. The pulley has radius 0.10 m and moment of inertia 0.20 kg-m 2. Find the linear accelerations of Blocks A and B, the angular acceleration of wheel C, and the tension in each side of the cord a) if the surface of the wheel is frictionless; a A = a B = 3.27 m/s 2 ; a C = 0; T A = T B = 26.1 N b) if there is no slipping between the cord and the surface of the wheel. a A = a B = m/s 2 ; a C = 7.45 rad/s 2 ; T A = 36.2 N; T B = 21.1 N 54. A block of mass m = 5 kg slides down a surface inclined 37º to the horizontal, as shown in the figure to the right. The coefficient of sliding friction is A string attached to the block is wrapped around a flywheel on a fixed axis at O. The flywheel has a mass of 20.0 kg, and outer radius of 0.2 m, and a moment of inertia with respect to the axis of 0.2 kg-m 2. a) What is the acceleration of the block down the plane? 1.97 m/s 2 b) What is the tension in the string? 9.85 N 55. A flywheel 1.0 m in diameter is pivoted on a horizontal axis. A rope is wrapped around the outside of the flywheel, and a steady pull of 50.0 N is exerted on the rope. Ten meters of rope are unwound in 4.0 s. a) What is the angular acceleration of the flywheel? b) What is its final angular velocity? 2.5 rad/s 2 10 rad/s c) What is its final kinetic energy? 500 J d) What is its moment of inertia? 10 kg-m 2

7 Heavy Pulleys and Hanging Masses: 1. A 4.0 kg bicycle wheel (Mass is concentrated at the rim.) of radius 0.20 m is held on a fixed support, while a 1.1 kg mass on a string wrapped around the wheel falls as shown. What is the linear acceleration of the dropping mass? 2.11 m/s 2 2. An Atwood machine is constructed using a massive 2.0 kg hoop of 22 cm radius as shown in the diagram. A 1.5 kg mass and a 1.0 kg mass arranged as shown are released from rest. Find the linear acceleration of the falling mass m/s 2 6. An Atwood machine consists of a disk of mass M, and radius R, and two masses ml and m 2 hanging from each side as shown in the figure. Find the linear acceleration of the system. 7. A 2-disk Atwood machine with radii of 15 cm and 38 cm, has a moment of inertia of 4.0 kg-m 2 is shown in the figure below. Masses of 3.0 kg and 2.0 kg are attached to strings wrapped around the disks as shown. When released from rest, what is the linear acceleration of each mass? a 2 = m/s 2 a 1 = m/s 2 3. A bicycle wheel of radius 0.70 m and mass 3.0 kg has a small light hub of radius 0.13 m as shown in the figure. The 2.0 kg mass which is attached to a string wrapped around the hub is released from rest. What is the linear acceleration of the dropping mass? m/s 2 9. A spool (solid cylinder) of radius 27 cm is mounted to spin about its axis. A string wrapped around it is pulled with a 5.4 N force, causing the object to spin up at 14 rad/sec 2. What is the moment of inertia of the object? kg-m 2 4. An Atwood machine is constructed using two wheels (Mass concentrated at the rim.) as shown in the figure below. What is the linear acceleration of the hanging masses? 5. Find the linear acceleration of the system shown in the figure below. The mass of the pulley is concentrated at the rim. The coefficient of kinetic friction between the ramp and the 5.0 kg block is m/s 2

9 Equilibrium of Rigid Body: Recall that we said the first condition for equilibrium existed when the sum of the forces acting on the body was zero. Now we introduce the second condition for equilibrium which exists when the sum of the torques of all the forces acting on the body, with respect to any specified axis is zero. This means that the body is not accelerating and it is not rotating. If it were rotating then it would experience a centripetal acceleration. 21. A N weight is hung on the end of a horizontal pole 2.0 m long. What is the torque around the other end of the pole caused by this weight? Around the center of the pole? 400 mn 200 mn 22. Two men carry a 1500 N load by hanging it from a horizontal pole that rests on one shoulder of each man. If the men are 3.00 m apart and the load is 1.00 m from one of them, how much load does each man support? The weight of the pole is 500 N N 750 N 23. A man holds a m fishing pole horizontally with both hands, one at the end and the other m from the end. He has just caught a kg fish. The pole has a mass of kg and you can consider its weight to be concentrated m from the end near the man's hands. What is the force exerted by each hand? 93 N down 118 N up 24. A steel beam of uniform cross section weighs 2.5 x10 5 N. If it is 5.00 m long, what force is needed to lift one end of it? 1.2 x 10 5 N 25. A bar 4.0 m long weighs N. Its center of gravity is 1.5 m from one end. A weight of N is attached at the heavy end and a weight of N is attached at the light end. What are the magnitude, direction, and point of application of the force needed to achieve translational and rotational equilibrium of the bar? N up at 2.2 m from 300 N 26. A painter weighing 875 N stands on a plank 3.00 m long, which is supported at each end by a stepladder. The plank weighs 223 N. If the painter stands 1.00 m from one end of the plank, what force is exerted by each stepladder? 400 N 700 N 27. A brick layer weighing N stands 1.00 m from one end of a scaffold 3.00 m long. The scaffold weighs750 N. A pile of bricks weighing N is 1.50 m from the other end of the scaffold. What force must be exerted on each end of the scaffold in order to support it? 1070 N at end near bricklayer 800 N other end

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