Unit 7: Rotational Motion (angular kinematics, dynamics, momentum & energy) Name: Big Idea 3: The interactions of an object with other objects can be described by forces. Essential Knowledge 3.F.1: Only 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 axis. a. The lever arm is the perpendicular distance from the axis of rotation or revolution to the line of application of the force. b. The magnitude of the torque is the product of the magnitude of the lever arm and the magnitude of the force. c. The net torque on a balanced system is zero. Essential Knowledge 3.F.2: The presence of a net torque along any axis will cause a rigid system to change its rotational motion or an object to change its rotational motion about that axis. a. Rotational motion can be described in terms of angular displacement, angular velocity, and angular acceleration about a fixed axis. b. Rotational motion of a point can be related to linear motion of the point using the distance of the point from the axis of rotation. c. The angular acceleration of an object or rigid system can be calculated from the net torque and the rotational inertia of the object or rigid system. Essential Knowledge 3.F.3: A torque exerted on an object can change the angular momentum of an object. a. Angular momentum is a vector quantity, with its direction determined by a right-hand rule. b. The magnitude of angular momentum of a point object about an axis can be calculated by multiplying the perpendicular distance from the axis of rotation to the line of motion by the magnitude of linear momentum. c. The magnitude of angular momentum of an extended object can also be found by multiplying the rotational inertia by the angular velocity. d. The change in angular momentum of an object is given by the product of the average torque and the time the torque is exerted. Learning Objective (3.F.1.1): The student is able to use representations of the relationship between force and torque. Learning Objective (3.F.1.2): The student is able to compare the torques on an object caused by various forces. Learning Objective (3.F.1.3): The student is able to estimate the torque on an object caused by various forces in comparison to other situations. Learning Objective (3.F.1.4): The student is able to design an experiment and analyze data testing a question about torques in a balanced rigid system. Learning Objective (3.F.1.5): The student is able to calculate torques on a two-dimensional system in static equilibrium, by examining a representation or model (such as a diagram or physical construction). Big Idea 4: Interactions between systems can result in changes in those systems. Essential Knowledge 4.D.1: Torque, angular velocity, angular Learning Objective (3.F.2.1): The student is able to 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 axis. Learning Objective (3.F.2.2): The student is able to plan 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 axis. Learning Objective (3.F.3.1): The student is able to predict the behavior of rotational collision situations by the same processes 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 momentum. Learning Objective (3.F.3.2): In an unfamiliar context or using representations beyond equations, the student is able to justify the selection of a mathematical routine to solve for the change in angular momentum of an object caused by torques exerted on the object. Learning Objective (3.F.3.3): The student is able to plan data collection and analysis strategies designed to test the relationship between torques exerted on an object and the change in angular momentum of that object. Learning Objective (4.D.1.1): The student is able to describe a representation and use it to analyze a Profici ent 1
Unit 7: Rotational Motion (angular kinematics, dynamics, momentum & energy) Name: acceleration, and angular momentum are vectors and can be characterized as positive or negative depending upon whether they give rise to or correspond to counterclockwise or clockwise rotation with respect to an axis. Essential Knowledge 4.D.2: The angular momentum of a system may change due to interactions with other objects or systems. a. The angular momentum of a system with respect to an axis of rotation is the sum of the angular momenta, with respect to that axis, of the objects that make up the system. b. The angular momentum of an object about a fixed axis can be found by multiplying the momentum of the particle by the perpendicular distance from the axis to the line of motion of the object. c. Alternatively, the angular momentum of a system can be found from the product of the system s rotational inertia and its angular velocity. situation in which several forces exerted on a rotating system of rigidly connected objects change the angular velocity and angular momentum of the system. Learning Objective (4.D.1.2 ): The student is able to plan 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. Learning Objective (4.D.2.1): The student is able to describe a model of a rotational system and use that model to analyze a situation in which angular momentum changes due to interaction with other objects or systems. Learning Objective (4.D.2.2): The student is able to plan 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 systems. Essential Knowledge Learning Objective (4.D.3.1): 4.D.3: The change in The student is able to use appropriate mathematical routines to calculate values for angular momentum is initial or final angular momentum, or change in angular momentum of a system, or given by the product of average torque or time during which the torque is exerted in analyzing a situation the average torque and involving torque and angular momentum. the time interval during Learning Objective (4.D.3.2): which the torque is The student is able to plan a data collection strategy designed to test the relationship exerted. between the 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 exerted Big Idea 5: Changes that occur as a result of interactions are constrained by conservation laws. Essential Knowledge 5.E.1: If the net external torque exerted on the system is zero, the angular momentum of the system does not change. Learning Objective (5.E.1.1): The student is able to make qualitative predictions about the angular momentum of a system for a situation in which there is no net external torque. Learning Objective (5.E.1.2): The student is able to make calculations of quantities related to the angular momentum of a system when the net external torque on the system is zero. Essential Knowledge 5.E.2: The angular momentum of a system is determined by the locations and velocities of the objects that make up the system. The rotational inertia of an object or system depends upon the distribution of mass within the object or system. Changes 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. Learning Objective (5.E.2.1): The student is able to describe 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. Students are expected to do qualitative reasoning with compound objects. Students are expected to do calculations with a fixed set of extended objects and point masses. 2
Rotational Motion Reading Assignment Read Chapter 7 in your book and answer the following questions: 1. Define angular position ( ), include how you calculate and the units. 2. How are the positive and negative directions defined in rotational motion? 3. What is the conversion between degrees and radians? 4. Define angular velocity ( ), include how you calculate from and the units. 5. What equation relates angular velocity to period and frequency? 6. What equation relates angular speed to linear speed when an object is rotating? 7. Describe the three types of motion of a rigid body and give an example of each: a. translational b. rotational c. combination 8. Define angular acceleration ( ), include how you calculate and the units. 9. Each linear variable we studied in 1 st semester has a rotational equivalent. For each kinematic equation in the table below write the corresponding rotational equation (the first one has been done for you) Linear Equation Rotational Equation v = (constant velocity only) = (constant angular velocity only) v f = v i + a t x f = x i + v i t + ½ a( t 2 ) v 2 f = v 2 i + 2 a( x) 3
Rotational Motion Reading Assignment 10. What is the equation that defines the relationship between tangential and angular acceleration? 11. What is the rotational equivalent of force? 12. What three factors affect the ability of a force to cause rotation? 13. What is the equation and units of torque? 14. Torque is a vector just like force. How do we define positive and negative values of torque? 15. Define center of gravity. 16. What is the equation for calculating the center of gravity of an object? 17. Answer Stop to Think 7.5. 18. How is a net torque related to angular acceleration? What is the equation that relates these variables? 19. The momentum of inertia is the rotational equivalent to mass. How is rotational inertia calculated? 20. Why is it easier to rotate a disk if the mass is concentrated around the center of the disk rather than at the outer edge? 21. When an object is rolling (translating and rotating) why is a point on the bottom of the object instantaneously at rest? How does that compare to a point on the top of the rolling object? 4
Rotational Motion Reading Assignment Read Chapter 8 sections 1 2 (pages 226-232) 22. Define static equilibrium. 23. What are the three conditions that must be met for an object to be in static equilibrium? 24. Answer Stop to Think 8.2. Explain your answer. Read Section 9.7 on angular momentum (page 270) 25. Define angular momentum (L), include its equation and units. 26. Describe the relationship between torque and angular momentum, include a mathematical equation. 27. How does this relationship parallel the impulse and momentum theorem? 28. When is angular momentum conserved? 29. Why does a skater spin faster when she brings her arms close to her body? Read Page 294 on Rotational Kinetic Energy 30. What is the equation for rotational kinetic energy? 31. When an object is translating and rotating, how do you calculate the total kinetic energy of the object? 5
Torque 1. 2. 3. 4. 5. 6
6. A beam of negligible mass is shown in the given diagram. If a force of 55 N acts at a point 0.45 m away from the hinge and making an angle at 68. What is the magnitude and direction of the torque? 7. 8. 9. 10. 7
11. A 1.7 m long barbell has a 20 kg mass on its right end and a 35 kg mass on its left end. If you ignore the weight of the bar itself, how far from the left end of the barbell is the center of gravity? Static Equilibrium: 12. Two children are sitting on a see-saw, as shown. Calculate the distance the 500-N child should sit from the fulcrum (pivot) to balance the see-saw. 13. Suppose that a meterstick is supported at the center, and a 40-N block is hung at the 80-cm mark. Another block of unknown weight just balances the system when it is hung at the 10-cm mark. What is the weight of the second block? 14. What is the mass of the uniform meterstick shown in this figure? 15. A 1.2 m long beam is free to rotate about its left end. What force F must be applied in order to achieve rotational equilibrium? 16. A beam of mass 18 kg is held by a string which can hold a maximum tension of 350N. At what distance x can this string be placed before it breaks? 8
Rotational Inertia: 17. Consider two solid uniform spheres where both have the same diameter, but one has twice the mass of the other. How much larger is the moment of inertia of the larger sphere compared to that of the smaller sphere? 18. 19. 20. A board has a mass of 3.5 kg and a total length of 5 meters. a. Calculate the rotational inertia of the board when rotated about the center. (I = 1/12 ML 2 ) b. Calculate the rotational inertia of the board when rotated about the end. (I = 1/3 ML 2 ) c. Using the values calculated above in your explanation; explain why it is easier to hold a board level from the middle then at the end. 9
Torque and Rotational Acceleration 21. 22. A wheel has a moment of inertia of 3.00 kg m 2. It is subjected to a 3.50 N m of torque. What angular acceleration does it experience? 23. A rope is wrapped around a solid cylinder of mass 10 kg with a radius of 1.5 m. What is the angular acceleration of the cylinder if a force of 5.00 N is applied parallel to the edge of the wheel? (I = ½ MR 2 ) 24. A 1.53 kg mass hangs on a rope wrapped around a disk-shaped pulley of mass 7.07 kg and radius 66 cm. (I = ½ MR 2 ) a. Draw a Free Body Diagram of all the forces on the pulley and the mass below: b. What is the linear acceleration of the hanging mass? 10
25. A child sits on a 5 meter long seesaw. The moment of inertia of the child and seesaw is 118 kg m 2. How much torque would be needed to cause an angular acceleration of 0.15 rad/s 2? Angular Kinematics: 26. Convert 1.5 revolutions to both radians and degrees. 27. A wheel rotates 3 times. What is its angular displacement? 28. 29. The figure to the right shows a steadily rotating wheel. a. Rank in order, from largest to smallest, the angular velocities 1, 2, and 3. Explain your ranking. b. Rank in order, from largest to smallest, the tangential velocities v 1, v 2, and v3. Explain your ranking. 30. As seen from the North Pole, the earth spins CCW once in 24 hours (actually it is slightly less). a. What is the θ in radians that the earth moves in 1 hour? b. What is the Earth s angular velocity (ω) in radians per second? c. If the earth s radius is about 6.4 x 10 6 meters, what tangential velocity (m/s) does an object have at the equator? 11
31. A wheel is observed to rotate 5 complete revolutions in 25 seconds. What is its angular speed? 32. In order to travel at 20 m/s (about 50 mph) a) what angular speed must your tires have if your tires have a 0.8 m radius? b) How many revolutions does the tire make in one second? 33. Imagine a ferris wheel that is rotating at the rate of 45 degrees each second. a. What is the ferris wheel s period of rotation in seconds?) b. What is the angular velocity in radians per second? c. If the tangential velocity of one of the cars is 7.85 m/s, how far (in meters) is it located from the center (axis of rotation)? 34. A particle moves in uniform circular motion with a c = 8 m/s 2. What is a c if: a. The radius is doubled without changing the angular velocity? b. The radius is doubled without chang4eing the particles speed? c. The angular velocity is doubled without changing the circle s radius? 35. The second hand of a watch rotates at a constant angular velocity (ω) of 0.105 radians/sec. a. What is the angular displacement ( θ) in radians after 90 seconds? b. How many rotations did the second hand undergo? 12
36. The figures below show the centripetal acceleration vector a c at four successive points on the trajectory of a particle moving in a counterclockwise circle. 37. In a popular game show, contestants give the wheel a spin and try to win money and prizes! The wheel is given a rotational velocity and this rotation slows down over time and the wheel eventually stops. One contestant gives the wheel an initial angular velocity (ω i ) of 1 revolution every two seconds. Because of friction, the wheel eventually comes to rest (ω f = 0 radians/sec) in 6 seconds. a. Because the wheel s angular velocity is changing, we must have an angular acceleration. Calculate, α, the angular acceleration in radians/s 2 b. Why is the angular acceleration you found in part (a) a negative value? What does the negative sign physically mean for the wheel? c. How many radians (Δθ) did the wheel sweep out as it was slowing down to a stop? d. If the mass of the wheel is 10 kg and the radius is 7m, what is the frictional torque acting on the wheel (I = ½ MR 2 )? e. Assuming that the torque is unchanged, but the mass of the wheel is halved and its radius is doubled. How will this affect the angle through which it rotates before coming to rest? (Make sure you can reason this with proportionalities.) Increase, decrease, or stay the same? 38. A race car is on a circular track with a radius of 0.30 km (300 m). The driver accelerates from rest with a constant angular acceleration (α) of 4.5 x 10-3 rad/s 2. The driver constantly accelerates as he drives one lap around the track. a. How long does it take for the driver to make one lap around the track? b. What is the driver s angular velocity (ω f ) as he finishes this first lap? 13
39. The blades of a circular fan running at low speed turn at 250 rpm. When the fan is switched to high speed, the rotation rate increases to 350 rpm. This change of the rotation rate occurs uniformly and takes 5.75 seconds. Remember to convert rpm to rad/s. a. What is the angular acceleration (α) needed to go from low to high speed? b. How many rotations do the fan blades go through while the fan is accelerating? c. If the moment of inertia of the fan blades is 0.23 kg m 2, what is the torque provided by the motor on the fan blades? 40. A top spins and slows down at an angular acceleration of -1.5 radians/sec 2 until it topples. If the top will topple at an angular speed of 150 radians per second or less, and the top toppled after spinning for 45 seconds, what was the initial angular velocity of the top? 41. A solid disk of unknown mass and known radius R is used as a pulley in a lab experiment, as shown. A small block of mass m is attached to a string, the other end of which is attached to the pulley and wrapped around it several times. The block of mass m is released from rest and takes a time t to fall the distance D to the floor. a. Write an equation that can be used to calculate the linear acceleration a of the falling block in terms of the given quantities. b. The time t is measured for various heights D and the data are recorded in the following table. ii What quantities should be graphed in order to best determine the acceleration of the block? Explain your reasoning. 14
(continued from previous page) ii. On the grid below, plot the quantities determined in (b) i., label the axes, and draw the best-fit line to the data. iii.use your graph to calculate the magnitude of the acceleration. c. Calculate the rotational inertia of the pulley in terms of m, R, a, and fundamental constants. d. The value of acceleration found in (b)iii, along with numerical values for the given quantities and your answer to (c), can be used to determine the rotational inertia of the pulley. The pulley is removed from its support and its rotational inertia is found to be greater than this value. Give one explanation for this discrepancy. Angular Momentum 32. An isolated hoop of mass M and radius R is rotating with an angular speed of 60 rpm about its axis. a. What would be its angular speed if its mass was suddenly doubled without changing its radius? Explain. b. What would be its angular speed if the radius suddenly doubled without changing its mass? Explain. 33. A solid circular disk and a circular hoop (like a bicycle wheel) both have the same mass and radius. If both are rotating with the same angular velocity, which has the larger angular momentum or will they be equal? Explain. 15
34. What is the angular momentum of a 0.25 kg mass rotating on the end of a piece of rope in a circle of radius 0.75m at an angular speed of 12.5 rad/s? 35. A bowling ball has a mass of 5.5 kg and a radius of 12.0 cm. It is released so that it rolls down the alley at a rate of 12 rev/s. Find the magnitude of its angular momentum. 36. A figure skater rotates on ice at a rate of 3.5 rad/s with her arms extended horizontally. When she lowers her arms to her side, she speeds up to 7.5 rad/s. Find the ratio of her moment of inertia in the first case to that in the second case. 37. A disk has a mass of 3.5 kg and radius of 15 cm is rotating with an angular speed of 94.2 rad/s when a second nonrotating disk of 5.0 kg, mounted on the same shaft is dropped onto it. If the second disk has a diameter of 18 cm and a mass of 5.0 kg, what is the common final angular speed of the system? 38. If the total angular momentum of a system of particles is zero, are all the particles at rest? Explain. 39. If the total angular momentum of a system is constant, does this mean that no net force acts on the system? Explain 40. A student whose mass is 60 kg is standing on the edge of a circular merry-go- round facing inward. The merry-goround has a mass of 100 kg and a radius of 2.0 m and spins at a rate of 2 rad/s. The student walks slowly from the outer edge toward the center and stops at a distance of 0.50m from the center. Calculate the magnitude of the angular velocity of the system. (treat merry go round as a solid disk/cylinder) 41. A figure skater spinning at 3.4 rad/sec with a moment of inertia of 3.2 Kgm 2 puts his arms out so that his new moment of inertia is 4.5 Kgm 2. What is his new angular speed? 16
42. An object s moment of inertia is 2.0 kg m 2. Its angular velocity is increasing at the rate of 4.0 rad/s. What is the net torque on the object? 43. A 200 g, 20 cm diameter plastic disk is spun on an axle through its center by an electric motor. What torque must the motor supply to take the disk from 0 to 1800 rpm in 4 s? 120 rpm 44. What is the magnitude of the angular momentum of the 720 g bar rotated at 120 rpm about its center with a length of 2 m? 45. A uniform bar has length 0.8 m. The bar begins to rotate from rest about the axis passing through the left end. What will the magnitude of the angular momentum be 6 seconds after the motion has begun? 46. In the figure above, a piece of clay starting from rest falls from rest a height h and sticks to a wheel that is free to rotate on a low-friction axle at location A. Several different possible locations for the collision are shown, each with the same momentum of the clay just before it hits and sticks. a. In which case will the device rotate fastest? Explain. b. In which case will the device not rotate at all? Explain c. If we increase the mass of the clay, how will that affect the rotation of the device? Explain d. If the clay is going faster just before it sticks to the ball, how will that affect the rotation of the device? Explain 17
47. A playground ride consists of a solid disk of mass 100 kg and radius 4 m mounted on a low-friction axle (Figure 9.68). A child of mass 50 kg runs at speed 3 m/s on a line tangential to the disk and jumps onto the outer edge of the disk. a. If the disk was initially at rest, now how fast is it rotating after the boy jumps on? b. If you were to calculate the kinetic energies in part (a), you would find that Kf < Ki Where has this energy gone? c. Calculate the change in linear momentum of the system consisting of the child plus the disk (but not including the axle), from just before to just after impact. What caused this change in the linear momentum? d. The child moves to a location a distance R/2 from the axle. Now what is the angular speed? Rotational Kinetic Energy 48. A solid sphere and a hollow sphere of the same mass are rolled down an inclined plane, starting from rest, through a height h. Which one gets to the bottom first? Explain. 49. Consider a thin rod of mass 5 kg uniformly distributed over a length of 2 m. If it rotates around its center of mass at an angular velocity of ω=3rad/s counter-clockwise, what is the rotational kinetic energy of the system? (I = 1/12mL²) 50. I release a solid cylinder and a hollow cylinder of equal mass so they both roll without slipping down a 30 degree incline ramp. If the vertical height of the ramp is 4 m, which cylinder will have more translational velocity at the bottom of the ramp? (You can do this with numbers but try to think about it quantitatively.) 18
51. I release a cylinder from a height of 2 m at the top of a 60 degree ramp and it rolls without slipping. If the translational kinetic energy at the bottom is 12 J and the cylinder has a mass of 1 kg and radius 25 cm, what is the moment of inertia of the cylinder? (I=1/2mR²) 52. A 34.2 gram marble rolls from rest down a ramp that loses 67.5 cm of height. a. What is the final velocity of the marble? b. What is its rotational kinetic energy at the bottom? c. What is its translational kinetic energy at the bottom? d. Assuming the ramp was linear and 3.56 m long, and the marble had a radius of.342 cm, what was the angular acceleration of the marble as it moved down the incline? 19