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Unit 6: Rotational Motion and Conservation of Angular MomentumFor this unit you must:Express the motion of an object using narrative, mathematical, and graphical representationsUse representations of the relationship between force and torqueCompare the torques on an object caused by various forcesEstimate the torque on an object caused by various forces in comparison to other situationsDesign an experiment and analyze data testing a question about torques in a balanced rigid systemCalculate torques on a two-dimensional system in static equilibrium, by examining a representation or model (such as a diagram or a physical construction)Make predictions about the change in the angular velocity about an axis for an object when forces exerted on the object cause a torque about that axisPlan data collection and analysis strategies designed to test the relationship between a torque exerted on an object and the change in angular velocity of that object about an axisPredict the behaviour of rotational collision situations by the same process that are used to analyze linear collision situations using an analogy between impulse and change of linear momentum and angular impulse and change of angular momentumJustify the selection of a mathematical routine to solve for the change in angular momentum of an object caused by torques exerted on the object in an unfamiliar context or using representations beyond equationsPlan data collection and analysis strategies designed to test the relationship between torques exerted on an object and the change in angular momentum of that objectDescribe a representation and use it to analyze a situation in which several forces exerted on a rotating system of rigidly connected objects change the angular velocity and angular momentum of the systemPlan data collection strategies designed to establish that torque, angular velocity, angular acceleration and angular momentum can be predicted accurately when the variables are treated as being clockwise or counterclockwise with respect to a well-defined axis of rotation, and refine the research question based on the examination of data.Describe a model of rotational system and use that model to analyze a situation in which angular momentum changes due to interaction with other objects or systemsPlan a data collection and analysis strategy to determine the change in angular momentum of a system and relate it to interactions with other objects and systemsUse appropriate mathematical routines to calculate values for initial or final angular momentum, or change in angular momentum of a system, or average torque or time during which the torque is exerted in analyzing a situation involving torque and angular momentumPlan a data collection strategy designed to test the relationship between change in angular momentum of a system and the product of the average torque applied to the system and the time interval during which the torque is exertedMake qualitative predictions about the angular momentum of a system for a situation in which there is no net external torqueMake calculations of quantities related to the angular momentum of a system when the net external torque on the system is zeroDescribe or calculate the angular momentum and rotational inertia of a system in terms of the locations and velocities of objects that make up the system. You are expected to do qualitative reasoning with compound objects. You are expected to do calculations with a fixed set of extended objects and point masses.Chapter 10: Rotational Kinematics and EnergySection 10-1 Angular Position, Velocity and AccelerationPosition, displacement, distance, velocity, speed and acceleration can be used by an observer to describe the motion of an object. For rotational motion, there are analogous quantities:Angular position: ____________________________________________________________________________________Sign convention = Angular displacement: _______________________________________________________________________________Arc length: ________________________________________________________________________________________Angular velocity: ____________________________________________________________________________________Sign convention = Period: ____________________________________________________________________________________________Angular Acceleration: ________________________________________________________________________________Example 1: Find the angular velocity of the object below.Example 2: (a) A wind turbine for generating electricity rotates clockwise at the rate of 17.0 rpm. What is its angular velocity? (b) A CD rotates through an angle of 106° in 0.0860 s. What is the angular speed of the CD?Example 3: The crankshaft in your car engine is turning at 3000 rpm. What is the shaft’s angular speed?Section 10-2 Rotational KinematicsLinear vs. Circular/AngularFor uniform motionFor nonuniform motionExample 4: A high-speed drill rotating counterclockwise takes 2.5 s to speed up to 2400 rpm. What is the drill’s angular acceleration?Example 5: To throw a curve ball, a pitcher gives the ball an initial angular speed of 36.0 rad/s. When the catcher gloves the ball 0.595 s later, its angular speed has decreased (due to air resistance) to 34.2 rad/s. What is the ball’s angular acceleration, assuming it to be constant?How many revolutions does the ball make before being caught?Example 6: On a game show, contestants spin a wheel when it is their turn. One contestant gives the wheel an initial angular speed of 3.40 rad/s. It then rotates through one-and-one-quarter revolutions and comes to rest on the BANKRUPT space.Find the angular acceleration of the wheel.How long does it take of the wheel to come to rest?Section 10-3 Connections Between Linear and Rotational QuantitiesAngular Speed vs. Linear SpeedFor uniform circular motion:Combining the above gives the tangential speed of a rotating object to be:Example 7: Find the angular speed a CD must have to give a linear speed of 1.25 m/s when the laser shines on the disk 2.50 cm from its center.Example 8: Two children ride on a merry-go-round, with child 1 at a greater distance from the axis of rotation than child 2. Is the angular speed of child 1 greater than, less than or the same as the angular speed of child 2? Justify your response.Tangential Acceleration vs. Centripetal AccelerationCentripetal acceleration is given byNow using our relationship between linear and rotational speedGiving us the centripetal acceleration of a rotating objectTo relate tangential acceleration to rotational acceleration: Example 9: In a microhematocrit centrifuge, small samples of blood are placed in heparinized capillary tubes. The tubes are rotated at 11,500 rpm, with the bottom of the tubes 9.07 cm from the axis of rotation.Find the linear speed of the bottom of the tubes.What is the centripetal acceleration at the bottom of the tubes?Example 10: Discus throwers often warms up by standing with both feet flat on the ground and throwing the discus with a twisting motion of their bodies. Starting from rest, the thrower accelerates the discus to a final angular velocity of +15.0 rad/s in a time of 0.270 s before releasing it. During the acceleration, the discus moves on a circular arc of radius 0.810 m. FindThe magnitude of the total acceleration of the discus just before it is released andThe angle that the total acceleration makes with the radius at this momentSection 10-4 Rolling MotionObjects that roll without slipping are instantaneously at rest at the point with which they contact the ground whereas the top of the object moves at twice the speed (rotational motion + translational motion)Section 10-5 Rotational Kinetic Energy and the Moment of InertiaFor a rotating object the kinetic energy is written asOrWhereNote: Moment of Inertias will be given (see below)Example 11: A grindstone with a radius of 0.610 m is being used to sharpen an ax. If the linear speed of the stone relative to the ax is 1.50 m/s, and the stone’s rotational kinetic energy is 13.0 J, what is its moment of inertia?If the linear speed is doubled, what is the corresponding kinetic energy of the grindstone?Example 12: If the dumbbell shape below is rotated about one end, is its moment of inertia (a) more than, (b) less than or (c) the same as the moment of inertia about its center?Section 10-6 Conservation of EnergyKinetic Energy of Rolling MotionExample 13: Bike 1 has a 10.0 kg frame and 1.00 kg wheels; bike 2 has a 9.00 kg frame and 1.50 kg wheels. Both bikes thus have the same 12.0 kg total mass. What is the kinetic energy of each bike when they are ridden at 12.0 m/s? Model each wheel as a hoop of radius 35.0 cm.Example 14: A solid sphere and a hollow sphere of the same mass and radius roll without slipping at the same speed. Is the kinetic energy of the solid sphere greater than, less than, or the same as the kinetic energy of the hollow sphere? Justify your response.Example 15: A disk and a hoop of the same mass and radius are released at the same time at the top of an inclined plane. Does the disk reach the bottom of the plane before, after, or at the same time as the hoop? Justify your response.Example 16: Yo-Yo man releases a yo-yo from rest and allows it to drop, as he keeps the top end of the string stationary. The mass of the yo-yo is 0.056 kg, its moment of inertia is 2.9 x 10-5 kg?m2, and the radius of the axle the string wraps around is 0.0064 m. What is the linear speed of the yo-yo after it has dropped through a height of 0.50 m?Example 17: A thin-walled hallow cylinder of mass 0.500 kg and radius 5.00 cm and a solid cylinder of mass 2.50 kg and radius 5.00 cm start from rest at the top of an incline. Both cylinders start at the same vertical height of 60.0 cm. Ignoring any losses due to retarding forces, determine which cylinder has the greatest translational speed upon reaching the bottom of the incline.Example 18: A block of mass m is attached to a string that is wrapped around the circumference of a wheel of radius R and moment of inertia I. The wheel rotates freely about its axis and the string wraps around its circumference without slipping. Initially the wheel rotates with an angular speed , causing the block to rise with a linear speed v. To what height does the block rise before coming to rest? Give a symbolic answer.Chapter 11: Rotational Dynamics and Static EquilibriumSection 11-1 TorqueTorqueThe lever arm is the perpendicular distance from the axis of rotation to the line of application of the forceThe magnitude of torque is the product of the magnitude of the lever arm and the magnitude of the forceOnly the force component perpendicular to the line connecting the axis of rotation and the point of application of the force results in a torque about that axisThe net torque on a balanced system is zeroTorque is a vector but the determination of the direction of torque is limited to clockwise and counterclockwise for Physics 1Example 19: Two helmsmen, in disagreement about which way to turn a ship, exert the forces shown below. The wheel has a radius of 0.74 m, and the two forces have magnitudes F1 = 72 N and F2 = 58 N. Find the net torque.Section 11-2 Torque and Angular AccelerationRotational MotionIf a net torque is applied along any axis, a rigid system or object will change its rotational motionRotational motion can be described in terms of angular displacement, angular velocity and angular accelerationThe rotational motion of a point can be related to its linear motion using the distance of the point form the axis of rotation:The angular acceleration of an object or rigid system can be calculated from the net torque and the rotational inertia:The rotational inertia of an object or system depends upon the distribution of mass within the object or systemNote: You do not need to know the equation for an object’s rotational inertia, it will be provided. You should know what factors affect rotational inertia, for instance rotational inertia is larger when the mass is farther from the axis of rotation.Problem Solving StrategyDraw a FBDLabel lever arms and pivot pointUse:Example 20: In the caber toss, a contest of strength and skill that is part of Scottish games, contestants toss a heavy uniform pole, landing it on its end. A 5.9-m-tall pole with a mass of 79 kg has just landed on its end. It is tipped by 25° from the vertical and is starting to rotate about the end that touches the ground. Estimate the angular acceleration. Example 21: The engine in a small air-plane is specified to have a torque of 500 N m. This engine drives a 2.0-m-long, 40 kg single-blade propeller. On start-up, how long does it take the propeller to reach 2000 rpm?Example 22: A person holds his outstretched arm at rest in a horizontal position. The mass of the arm is m and its length is 0.740 m. When the person releases his arm, allowing it to drop freely, it begins to rotate about the shoulder joint. Find:The initial angular acceleration of the arm,The initial linear acceleration of the man’s hand.Note: Assume that the mass of the arm is concentrated at the center and use the moment of inertia of a uniform rod of length L: I = 1/3mL2)Example 23: The rotating systems shown below differ only in that the two spherical movable masses are positioned either far from the axis of rotation (left) or near the axis of rotation (right). If the hanging blocks are released simultaneously from rest, is it observed that the block on the left lands first, the block on the right lands first, or both blocks land at the same time? Justify your response.Example 24: A fisherman is dozing when a fish takes the line and pulls it with a tension T. The spool of the fishing reel is at rest initially and rotates without friction (since the fisherman left the drag off) as the fish pulls for a time t. If the radius of the spool is R, and its moment of inertia is I, find:The angular displacement of the spool,The length of line pulled from the spool, andThe final angular speed of the spool.Section 11-3 Zero Torque and Static EquilibriumStatic equilibrium occurs when an object is neither rotating nor translating, that is, the object is at restTwo conditions must be met for an object to be in static equilibriumFor an object in static equilibrium, the net torque is zero about every point so we are free to choose any axis of rotation (pivot point)The choice of the axis of rotation is often made to simplify calculations when forces are not well understood at a particular locationExample 25: A child of mass m is supported on a light plank by her parents who exert forces F1 and F2 at opposite ends of the plank to hold her up. Find the expressions for the forces required to keep the plank in static equilibrium if the child is seated a distance of one quarter the length of the plank from the right end. Example 26: A cat walks along a uniform plank that is 4.00 m long and has a mass of 7.00 kg. The plank is supported by two sawhorses, one 0.500 m from the left end of the board and the other 1.50 m from its right end. When the cat reaches the right end, the plank just begins to tip. What is the mass of the cat?Example 27: A hiker who has broken his forearm rigs a temporary sling using a cord stretching from his shoulder to his hand. The cord holds the forearm level and makes an angle of 40.0° with the horizontal where it attaches to the hand. Considering the forearm and hand to be uniform, with a mass of 1.31 kg and a length of 0.300 m, findThe tension in the cord,The force exerted by the humerus (the bone of the upper arm) on the bones of the forearm.Example 28: A uniform ladder of length L leans against a wall at an angle of 60° with respect to the floor. What is the expression for the minimum value of μs, the coefficient of static friction with the ground, that will prevent the ladder from slipping? Assume that friction between the ladder and the wall is negligible.Section 11-4 Center of Mass and BalanceIf an object’s center of mass is directly below the suspension point, its weight creates zero torque and the object is in equilibriumIf the object’s center of mass is not directly below the suspension point, the object’s weight creates a torque that tends to rotate the object to bring the center of mass under the suspension point.Example 29: A croquet mallet balances when suspended from its center of mass, as shown in the diagram. You now cut the mallet into two pieces at its center of mass. Is the mass of piece 1 greater than, less than, or equal to the mass of piece 2? Why?Section 11-5 Dynamic Applications of Torque472226964326400Example 30: Josh has just raised a 2.5 kg bucket of water using well’s winch when he accidentally lets go of the handle. The winch consists of a rope wrapped around a 3.0 kg, 4.0 cm diameter cylinder, which rotates on an axle through the center. The bucket is released from rest 4.0 m above the water level of the well. How long does it take the bucket to reach the water?Example 30: (continued)Example 31: A crate that weights 4420 N is being lifted by the mechanism shown below. The two cables are wrapped around their respective pulleys, which have radii of 0.600 and 0.200 m. The pulleys are fastened together to form a “dual” pulley and turn as a single unit about the center axle, relative to which the combined moment of inertia is 50.0 kg?m2. If the tension of 2150 N is maintained in the cable attached to the motor, find the angular acceleration of the “dual” pulley and the tension in the cable connected to the crate.Example 32: A 0.31 kg cart on a horizontal air track is attached to a string. The string passes over a disk-shaped pulley of mass 0.080 kg and radius 0.012 m and is pulled vertically downward with a constant force of 1.1 N. Find:The tension in the string between the pulley and the cart, andThe acceleration of the cart.Section 11-6 Angular MomentumAngular momentumThe angular momentum of a system is determined by the locations and velocities of the object that make up the system.The angular momentum can be found using the following:For an extended object: multiply the rotational inertia by the angular velocity:For a point object about an axis: multiply the perpendicular distance from the axis of rotation to the line of motion by the magnitude of linear momentum:For a system: sum the angular momenta of the objects that make up the system with respect to an axis of rotationSimilarly, for a system: multiply the system’s rotational inertia by its angular velocity.For a change in angular momentum: multiply the average torque and the time the torque is exerted:When a torque is exerted on an object its angular momentum can changeIf the net torque exerted on a system is zero, the angular momentum of the system does not changeExample 33: A 0.170 kg hockey puck moves straight toward the goalie with a speed of 4.42 m/s as shown. The puck is also observed by a defender.Find the angular momentum as measured by the defender given that r = 1.45 m and θ = 72.5°Find the angular momentum as measured by the goalieAs the puck gets closer to the goalie, r from the defender increases and the angle decreases. Does the puck’s angular momentum, as observed by the defender, increase, decrease, or stay the same as it approaches the goalie? Explain.Example 34: Bicycle riders can stay upright because a torque is required to change the direction of the angular momentum of the spinning wheels. A bike with wheels with a radius of 33 cm and a mass of 1.5 kg (each) travels at a speed of 10 mph. What is the angular momentum of the bike? Treat the wheels of the bike as though all the mass is at the rim.Example 35: A ceiling fan experience a constant torque of 1.9 N·m. If the fan is initially at rest, what is its angular momentum 2.6 s later?Section 11-7 Conservation of Angular MomentumThe law of conservation of angular momentum states that the angular momentum of a rotating object subject to no net external torque is constant. That is, the initial and final angular momenta are equalChanges in the radius of a system or in the distribution of mass within the system result in changes in the system’s rotational inertia, and hence in its angular velocity and linear speed for a given angular momentum (examples include elliptical orbits)The angular momentum of a system may change due to interactions with other objects or systems (rotational collisions)Example 36: Joey, whose mass is 36 kg, stands at the center of a 200 kg merry-go-round that is rotating once every 2.5 s. While it is rotating, Joey walks out to the edge of the merry-go-round. What is the rotational period of the merry-go-round when Joey gets to the edge?Example 37: An ice skater spins around on the tips of his blades while holding a 5.0 kg weight in each hand. He begins with his arms straight out from his body and his hands 140 cm apart. While spinning at 2.0 rev/s, he pulls the weights in and holds them 50 cm apart against his shoulders. If we neglect the mass of the skater, how fast is he spinning after pulling the weights in?Example 38: An artificial satellite is placed into an elliptical orbit about Earth. Telemetry data indicate that its point of closest approach (called the perigee) is 8.37 x 106 m from the center of Earth, while its point of greatest distance (called the apogee) is 2.51 x 107 m from the center of Earth. The speed of the satellite at the perigee is 8450 m/s. Find its speed at the apogee.Example 39: A 34.0 kg child runs with a speed of 2.80 m/s tangential to the rim of a stationary merry-go-round. The merry-go-round has a moment of inertia of 512 kg·m2 and a radius of 2.31 m. When the child jumps onto the merry-go-round, the entire system begins to rotate. What is the angular speed of the system? Ignore friction and any other type of external torque.Section 11-9 The Vector Nature of Rotational MotionAngular momentum is a vector quantity whose direction is determined by a right-hand ruleGrab the rotating object around the axis or rotation with your fingers pointing in the direction of the rotational velocityYour thumb indicates the direction of the angular momentum vector ................
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