AP Physics Rotational Motion Part I Introduction: Which moves with greater speed on a merry-go-round - a horse near the center or one near the outside? Your answer probably depends on whether you are considering the translational or rotational motion of the horses. Have you ever linked arms with friends at a skating rink while making a turn? If you have, you probably noticed that the person on the inside moved very little while the person on the outside had to run to keep up. The outside person traveled a greater distance per period of time and therefore had the greater translational speed. During the same period of time all skaters rotated through the same angle per period of time and had the same rotational speed. In our previous study of motion we discussed translational motion - that is the motion of bodies moving as a whole without regard to rotation. In this unit we will extend our ideas of motion to include the rotation of a rigid body about a fixed axis. If the axis is inside the body we tend to say the body rotates about its axis. If the axis is outside the body, we say the body revolves about an axis. An example of this would be the earth which daily rotates about its axis and yearly revolves around the sun. An object rotating about an axis tends to remain rotating about the same axis unless acted upon by a net external influence. This property of a body to resist changes in its rotational state is called rotational inertia. The rotational inertia of a body depends on the amount of mass the body possesses and on the distribution of that mass with respect to the axis of rotation. The greater the distance of the bulk of the mass from the axis of rotation - the greater the rotational inertia. A long pendulum has a greater rotational inertia than a short one. The period of a pendulum is directly proportional to the square root of the length of the pendulum. It takes more time to change the rotational inertia of a long pendulum as it swings back and forth. People and animals with long legs tend to walk with slower strides than those with short legs for the same reason. Have you ever tried running with your legs straight? Performance Objectives: Upon completion of the readings and activities of the unit and when asked to respond either orally or on a written test, you will be able to: state relationships between linear and angular variables. recognize that the rotational kinematics formulas are analogous to the translational ones. Use these formulas to solve problems involving rotating bodies. define rotational inertia or moment of inertia. Calculate the rotational inertia for a point mass, a system of point masses, and rigid bodies. use the parallel axis theorem to find the moment of inertia about an axis other than the center of mass. calculate the kinetic energy of a rotating body. define torque. Calculate the net torque acting on a body. state Newton s second law for rotation. Recognize that the rotational dynamics formulas are analogous to the translational ones. Use these formulas to solve problems involving rotating bodies. use the work-kinetic energy theorem for rotation to solve problems. Textbook Reference: Tipler: Chapter 9 Glencoe Physics: Chapter 8 "To every thing -- turn, turn, turn there is a season -- turn, turn, turn and a time for every purpose under heaven." -- The Byrds (with a little help from Ecclesiastes) Recall: From the definition of a radian (arc length/radius) θ = s/r, where s is the arc length, r is the radius and θ is the angle measure in radians. The following quantities are called the bridges between linear and angular measurements: s = rθ v = rω a T = rα a R = v 2 /r = rω 2 Definitions and Conversions: 1. What angle in radians is subtended by an arc 3.0 m in length, on the circumference of a circle whose radius is2.0 m? 1.5 rad 2. What angle in radians is subtended by an arc of length 78.54 cm on the circumference of a circle of diameter 100.0 cm? What is the angle in degrees? 1.57 rad 90 3. The angle between two radii of a circle of radius 2.00 m is 0.60 rad. What length of arc is intercepted on the circumference of the circle by the two radii? 1.2 m 4. What is the angular velocity in radians per second of a flywheel spinning at the rate of 7230 revolutions per minute? 757 rad/sec
5. If a wheel spins with an angular velocity of 625 rad/s, what is its frequency in revolutions per minute? 5968 rpm 6. Compute the angular velocity in rad/s, of the crankshaft of an automobile engine that is rotating at 4800 rev/min. 503 rad/sec Rotational Kinematics: Rotational motion is described with kinematic formulas just like the translational motion formulas. To get the rotational kinematic formulas, substitute the rotational variables. 7. A flywheel accelerates uniformly from rest to an angular velocity of 94 radians per second in 6.0 seconds. What is the angular acceleration of the flywheel in radians per second squared? 16 rad/s 2 8. a) Calculate the angular acceleration in radians per second squared of a wheel that starts from rest and attains an angular velocity of 545 revolutions per minute in 1.00 minutes. b) What is the angular displacement in radians of the wheel during the first 0.500 minutes? c) During the second 0.500 minutes? 0.95 rad/s 2 428 rad. 1283 rad 9. Find the angular displacement in radians during the second 20.0 second interval of a wheel that accelerates from rest to 725 revolutions per minute in 1.50 minutes? 506 radians 10. A fly wheel requires 3.0 seconds to rotate through 234 rad. Its angular velocity at the end of this time is 108 rad/s. Find a) the angular velocity at the beginning of the 3 second interval; b) the constant angular acceleration. 48 rad/s 20.0 rad/s 2 11. A playground merry-go-round is pushed by a child. The angle the merry-go-round turns through varies with time according to θ(t) = 2t + 0.05t 3, where θ is in radians and t is in seconds. a) Calculate the angular velocity of the merry-go-round as a function of time. ω = 2 + 0.15t 2 b) What is the initial value of the angular velocity? 2 rad/s c) Calculate the instantaneous velocity at t = 5.0 sec. d) Calculate the average angular velocity for the time interval t = 0 to t = 5 seconds. 5.75 rad/s 3.25 rad/s 12. A rigid object rotates with angular velocity that is given by ω = 4 + 8t 2, where ω is in rad/sec and t is in seconds. a) Calculate the angular acceleration as a function of time. b) Calculate the instantaneous angular acceleration a at t = 2 sec. c) Calculate the average angular acceleration for the time interval t = 0 to t = 2seconds. α = 16t 32 rad/s 2 16 rad/s 2 13. A bicycle wheel of radius 0.33 m turns with angular acceleration α = 1.2 0.4t, where α is in rad/s 2 and t is in seconds. It is at rest at t = 0. a) Calculate the angular velocity and angular displacement as functions of time. b) Calculate the maximum positive angular velocity and maximum positive angular displacement of the wheel. ω = 1.2t - 0.2t 2 θ = 0.6t 2-0.067t 3 1.8 rad/s 7.2 rad 14. A roller in a printing press turns through an angle θ given by θ(t) = 2.50t 2 0.400t 3. a) Calculate the angular velocity of the roller as a function of time. ω(t) = 5t 1.2t 2 b) Calculate the angular acceleration of the roller as a function of time. α(t) = 5 2.4t c) What is the maximum positive angular velocity and at what value of t does it occur? 5.21 rad/s 2.08 sec 15. a) A cylinder 0.15 m in diameter rotates in a lathe at 750 rpm. What is the tangential velocity of a point on the surface of the cylinder? b) The proper tangential velocity for machining cast iron is about 0.60 m/s. At how many rpm should a piece of stock 0.05 m in diameter be rotated in a lathe? 5.89 m/s 229 rev/min 16. Find the required angular velocity of an ultracentrifuge in rpm for the radial acceleration of a point 1.00 cm from the axis to equal 300,000g (that is 300,000 times the acceleration of gravity). 1.64 x 10 5 rev/min 17. A wheel rotates with a constant angular velocity of 10 rad/s. a) Compute the radial acceleration of a point 0.5 m from the axis using the relation, a = rω 2. 50 m/s 2 b) Find the tangential velocity of the point, and compute its radial acceleration from the relation, a c = v 2 /r. 5m/s 50 m/s 2 Rotational Inertia is the resistance of a rotating body to changes in its angular velocity. According to Newton's First Law a body tends to resist a change in its motion. The amount of inertia a body possesses is directly related to the mass. For rotational motion, an analogous situation exists. However, rotational inertia depends on the mass and on the distribution of the mass about the axis of rotation. This quantity that relates mass and position of the mass relative to the axis of rotation is called the moment of inertia and has units of kg-m 2. The symbol for moment of inertia is I. For a point mass m a distance r from the axis of rotation, the moment of inertia will be I = mr 2. For bodies made up of several small masses just add all the moments of inertia together. For bodies which are not composed of discrete point masses but are continuous distributions of matter, the methods of calculus must be used to find the moment of inertia. The moments of inertia for a few simple but important rigid bodies of uniform composition are listed on Page 295 in Tipler Physics.
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. 1.668 kg-m 2 22. 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. 2.67 kg-m 2 10.67 kg-m 2 0.0016 kg-m 2 19. Four small spheres, each of mass 0.200 kg, are arranged in a square 0.400 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. 0.0640 kg-m 2 b) bisecting two opposite sides of the square. 0.0320 kg-m 2 20. 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? 0.816 kg-m 2 21. 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. 0.0340 kg-m 2 23. A wagon wheel is constructed as shown in the figure. The radius of the wheel is 0.300 m and the rim has a mass of 1.20 kg. Each of the eight spokes, which lie along a diameter and are 0.300 m long has a mass of 0.373 kg. What is the moment of inertia of the wheel about an axis through its center and perpendicular to the plane of the wheel? 0.196 kg-m 2
Parallel Axis Theorem: The moment of inertia of any object about an axis through its center of mass is the minimum moment of inertia for an axis in that direction of space. The moment of inertia about any axis parallel to that axis through the center of mass is given by: Iparallel axis = Icom + Md 2 where I parallel axis is the moment of inertia about an new axis (parallel to the original axis of rotation), I com is the moment of inertia about the center of mass, M is the mass of the object and d is the distance between the original axis of rotation about the center of mass and the new proposed axis of rotation. 24. Use the parallel axis theorem to calculate the moment of inertia of a uniform thin rod of mass M and length l for an axis perpendicular to the rod at one end. Ml 2 /3 25. Use the parallel axis theorem to calculate the moment of inertia of a square sheet of metal of side length a and mass M for an axis perpendicular to the sheet and passing through one corner. 2Ma 2 /3 26. Find the moment of inertia of a thin-walled hollow cylinder of mass M and radius R about an axis perpendicular to its plane at an edge of the cylinder. 2MR 2 27. A thin, rectangular sheet of steel is 0.30 m by 0.40 m and has a mass of 16.0 kg. Find the moment of inertia about an axis a) in the plane of the sheet, through the center, parallel to the long sides. 0.12 kg-m 2 b) in the plane of the sheet, through the center, parallel to the short sides. 0.21 kg-m 2 c) perpendicular to the sheet and through the center. 0.33 kg-m 2 28. A uniform, thin rod is bent into a square of side length a. If the total mass is M, find the moment of inertia about an axis through the center, perpendicular to the plane of the square. Ma 2 /3 29. Which pendulum has more rotational inertia, a long or short one? 30. The four objects shown in the figure below have equal masses m. Object A is a solid cylinder of radius R. Object B is a hollow, thin cylinder of radius R. Object C is a solid square whose length of side = 2R. Object D is the same size as C, but hollow (i.e., made up of four thin sticks). The objects have axes of rotation perpendicular to the page and through the center of gravity of each object. a) Which object has the smallest moment of inertia? b) Which object has the largest moment of inertia? Kinetic Energy of Rotation: Because a rotating rigid body consists of particles in motion, it has kinetic energy. This kinetic energy is computed using the moment of inertia of the body and the angular velocity. KE = ½Iω 2 31. The rotor of an electric motor has a rotational inertia of 45 kg-m 2. What is its kinetic energy if it turns at 1500 revolutions per minute? 555 kj 32. A grinding wheel in the shape of a solid disk is 0.200 m in diameter and has a mass of 3.00 kg. The wheel is rotating at 3600 rev/min about an axis through its center. a) What is its kinetic energy? 1066 J b) How far would it have to drop in free fall to acquire the same kinetic energy? 36.3 m 33. The flywheel of a gasoline engine is required to give up 300.0 J of kinetic energy while its angular speed decreases from 660 rev/min to 540 rev/min. What moment of inertia is required for the wheel? 0.380 kg-m 2 34. A phonograph turntable has a kinetic energy of 0.0700 J when turning at 78 rpm. What is the moment of inertia of the turntable about the rotation axis? 35. Energy is to be stored in a large flywheel in the shape of a disk with radius of 1.20 m and a mass of 80.0 kg. To prevent structural failure of the flywheel, the maximum allowed radial acceleration of a point on its rim is 5000 m/s 2. What is the maximum kinetic energy that can be stored in the flywheel? 1.20 x 10 5 J 36. A light flexible rope is wrapped several times around a 160.0 N solid cylinder with a radius of 0.25 m. The cylinder which can rotate without friction about a fixed horizontal axis is initially at rest. The free end of the rope is pulled with a constant force P for a distance of 5.00 m. What must P be for the final speed of the end of the rope to be 4.00 m/s? 37. 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 0.20 m and moment of inertia 0.32 kg-m 2. The rope does not slip on the pulley rim. The larger mass is 5.0 m above the floor and released from rest. Use energy methods to calculate the velocity of the 4-kg block just before it strikes the floor. 3.74 m/s
Torque: To change the translational inertia of a body you have to apply a net external force. To change the rotational inertia of a body you have to apply a torque (rhymes with fork). If you studied torque in previous science courses it was probably defined as the product of the force and the length of the torque arm. The torque arm (sometimes called lever arm) is the perpendicular distance between the line of action of the force and the axis of rotation. In order to solve problems involving torque, you need to understand how torque is calculated and then be able to calculate the net torque acting on a body. 38. Calculate the torque (magnitude and direction) about point 0 due to the force F in each of the situations sketched in the figure. In each case the object to which the force is applied has length 4.00 m, and the force F =20.0 N. 41. A small ball of mass 0.75 kg is attached to one end of a 1.25 meter long massless rod, and the other end of the rod is hung from a pivot. When the resulting pendulum is30 from the vertical, what is the magnitude of the torque about the pivot? 4.6 m-n 42. Find the net torque on the wheel in the figure about the axle through O if a = 10 cm and b = 25 cm. 3.55 m-n cw 43. Find the mass M needed to balance the 150-kg truck on the incline shown in the figure below. The angle of inclination θ is 45º. Assume all pulleys are frictionless and massless. 17.7 kg a) 80.0 m-n ccw b) 69.3 m-n ccw c) 40.0 m-n ccw d) 34.6 m-n cw e) 0 f) 0 39. Calculate the resultant torque about point O for the two forces applied in the figure below. 28 m-n cw 40. Calculate the net torque (magnitude and direction) on the beam shown in the figure below about a) an axis through O, perpendicular to the figure. 29.5 m-n ccw b) an axis through C, perpendicular to the figure. 35.6 m-n ccw Rotational Dynamics: In studying translational dynamics we made use of Newton's Second Law, which related the acceleration of a body and the forces applied to the body. An analogous relationship exists between angular acceleration and a quantity we call a torque. Qualitatively speaking, torque is the tendency of a force to cause a rotation of the body on which it acts. Mathematically speaking, torque is defined as the cross product of the moment arm and the applied force. The moment arm is the perpendicular distance between the force applied and the axis of rotation. The unit for torque is a meter-newton. The symbol for torque is the lowercase Greek letter tau, τ. 44. A net force of 10.0 N is applied tangentially to the rim of a wheel having a 0.25 m radius. If the rotational inertia of the wheel is 0.500 kg m 2, what is its angular acceleration? 5 rad/s 2 45. A solid ball is rotated by applying a force of 4.7 N tangentially to it. The ball has a radius of 14 cm and a mass of 4.0 kg. What is the angular acceleration of the ball? 21 rad/s 2 46. A fly wheel in the shape of a thin ring has a mass of 30.0 kg and a diameter of 0.96 m. A torque of 13 m-n is applied tangentially to the wheel. How long will it take for the flywheel to attain an angular velocity of 10.0 rad/s? 5.3 sec
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. 6.25 rad/s 2 48. 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 200.0 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. 0.589 49. 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 600.0 N. The friction at the hinges can be neglected. If Dirk applies a force of 180.0 N at the edge of the door and perpendicular to it, how long will it take him to close the door? 0.597 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? 0.145 kg-m 2 53. 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 = 0.745 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 0.25. 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
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. 1.09 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 = 0.105 m/s 2 a 1 = 0.265 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? 0.220 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? 0.104 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 0.300. 376 m/s 2
Angular Momentum and Angular Impulse: The angular momentum of a rigid body about a fixed axis is defined two ways: of the remainder of her body is constant and equal to 3.0 kg-m 2. 9.26 rad/s if arms considered hollow cylinder L = Iω and For a single particle, the angular momentum relative to any point would be:...where m is the mass of the particle, r is the position vector from the point to the particle and v is the translational velocity. The product of the torque and the time interval during which it acts is called the angular impulse, J θ. The angular impulse acting on the body causes a change in the angular momentum of the body about the same axis. For a torque that varies with time, the angular impulse is defined as: Conservation of angular momentum states that when the net external torque on a system is zero, the angular momentum of the system remains constant. This principle of conservation of angular momentum ranks with the principles of conservation of linear momentum and conservation of energy as one of the most fundamental of physical laws. 19. Calculate the angular momentum of a uniform sphere of radius 0.20 m and mass 4.0 kg if it is rotating about an axis along a diameter at (a) 6.0 rad/s and (b) 5.0 rev/s. 2.0 kg-m 2 /sec 20. A solid wooden door 1.0 m wide and 2.0 m high is hinged along one side and has a total mass of 50.0 kg. Initially open and at rest, the door is struck at its center with a hammer. During the blow and average force of 2000.0 N acts for 0.01 seconds. Find the angular velocity of the door after the impact. 21. A man of mass 70.0 kg is standing on the rim of a large disk that is rotating at 0.5 rev/s about an axis through its center. The disk has mass 120.0 kg and radius 4.0 m. Calculate the total angular momentum of the manplus-disk system. 22. The outstretched arms of a figure skater preparing for a spin can be considered a slender rod pivoting about an axis through its center. When her arms are brought in and wrapped around her body to execute the spin, they can be considered a thin-walled hollow cylinder. If her original angular velocity is 6.28 rad/s, what is her final angular velocity? Her arms have a combined mass of 8.0 kg. When outstretched they span 1.8 m; when wrapped, they form a cylinder of radius 25 cm. (A disk and a cylinder rotating about an axis through the center have the same moment of inertia.) The moment of inertia
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 200.0 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. 1250 N 750 N 23. A man holds a 2.000 m fishing pole horizontally with both hands, one at the end and the other 0.300 m from the end. He has just caught a 1.500 kg fish. The pole has a mass of 1.000 kg and you can consider its weight to be concentrated 0.600 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 400.0 N. Its center of gravity is 1.5 m from one end. A weight of 300.0 N is attached at the heavy end and a weight of 500.0 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? 1200.0 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 800.0 N stands 1.00 m from one end of a scaffold 3.00 m long. The scaffold weighs750 N. A pile of bricks weighing 320.0 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
Conceptual Questions: 1. Does a record player needle ride faster or slower over the groove at the beginning or the end of the record? If fidelity increases with translational speed, what part of the record produces the highest fidelity? 2. Suppose the first and last selections on a phonograph record are 3-minutes cuts. Which, if either, of these cuts is wider on the record? (That is, which contains more grooves along a radial direction?) 3. Which moves faster on a merry-go-round, a horse near the center or one near the outside. 4. If you use large diameter tires on your car, how will your speedometer reading differ? 5. Why are the front wheels located so far out in front on the racing vehicle? 6. Which will roll down a hill faster, a cylinder or a sphere of equal radii? A hollow cylinder or a solid cylinder of equal radii? Explain. 7. Why do buses and heavy trucks have large steering wheels? 8. Which is easier for turning stubborn screws, a screwdriver with a thick handle or one with a long handle? Explain. 9. Why is the middle seating most comfortable in a bus traveling on a bumpy road? 10. Explain why a long pole is more beneficial to a tightrope walker if it droops. 11. Why do you bend forward when carrying a heavy load on your back? 12. Why is it easier to carry the same amount of water in two buckets, one in each hand, then in a single bucket? 13. Using the ideas of torque and center of gravity, explain why a ball rolls down a hill. 14. Why is it dangerous to roll open the top drawers of a fully loaded file cabinet that is not secured to the floor? 15. Why is less effort required in doing sit-ups when your arms are extended in front of you? Why is it more difficult when your arms are placed in back of your head? 16. For a rotating wheel, how do the directions of the linear velocity vector and the angular velocity vector compare at the same instant of time? Answers to conceptual questions: 1. The phonograph needle rides faster at the beginning of the record. Since fidelity is enhanced with translational speed, then fidelity would be best at the beginning of a record. 2. Both three minute selections would have the same width because they would make the same number of revolutions during a three minute time period. 3. The horse on the outer rail has a greater translational (tangential) speed, while both have the same rotational speed. 4. The circumference of a large diameter tire is greater, meaning it will move a greater distance per revolution, which results in a greater speed than that shown on the speedometer. 5. The long distance to the front wheels increases the rotational inertia of the vehicle without appreciably adding to its weight. As the back wheels are driven clockwise, the rest of the car tends to rotate counter-clockwise. This would lift the front wheels off the ground. 6. A sphere will roll faster because it has less rotational inertia than a cylinder. A solid cylinder will roll faster than a hollow cylinder for the same reason. 7. The large radius of a large steering wheel allows the driver to exert more torque for a given force. 8. More torque can be exerted by the screw driver having a thick handle. 9. A rocking bus rocks about its center of gravity which is around the center of the bus. It works something like a see-saw - the farther from the center, the more you go up and down. 10. The long drooping pole lowers the center of gravity of the pole and the tightrope walker. The pole contributes to his rotational inertia. 11. You bend forward to shift the center of gravity of you and the back pack. If you did not shift the center of gravity over the support, you would topple over. 12. There is no need to adjust your center of gravity if the water is distributed between the two buckets. 13. When a ball is on an incline its center of gravity is not above the point of support. The weight acts some distance from the point of support and produces a torque about the point of support. 14. The center of gravity could be adjusted so that it is no longer above the support. 15. When your arms are extended in front of you while doing sit-ups, not only are they not lifted as far, they are closer to the axis of rotation and give you less rotational inertia. When behind your head they are lifted farther and their farther distance from the axis of rotation increases your rotational inertia.