Name: Jenna Martinez Date: 07-19-2022 Directions: Follow the instructions to go through the simulation. Respond to the questions andprompts in the orange boxes. Student Exploration: Moment of Inertia Vocabulary: angular velocity, linear velocity, moment of inertia, rotational kinetic energy, translational kinetic energy Prior Knowledge Questions (Do these BEFORE using the Gizmo.) At the finale of her routine, a figure skater starts to spin slowly in the middle of the ice rink,her arms and legs artfully outstretched. 1. What will happen as she stands up straight and pulls her arms in? The skater will spin faster as she pulls her arms in and stands up straight. 2. Why do you think this will happen? This is due to the fact that her mass is balanced in the middle. Gizmo Warm-up The Moment of Inertia Gizmo allows you to explore the factors that affect how quickly objects spin. TheGizmo shows a weightless turntable with severalpegs. You can place the purple 1-kg masses onany of the pegs. When the Gizmo opens, there is asingle 1-kg mass located 5 m from the center. 1. Click Play ( ). What happens? The turntable spins. 2. The speed of rotation is described by angular velocity ( ω ), which is measured in radians per second. There are 2 π radians (6.28 radians) in a circle. What is the current angularvelocity? w=4 radians per sec 3. Click Reset ( ). Move the mass so that it is only 1 meter from the center ( r = 1). A. Click Play . What is the angular velocity now? w = 20 radians per sec B. How is this situation similar to a spinning skater? The closer the mass is tothe center, the faster itrotates. Reproduction for educational use only. Public sharing or posting prohibited. © 2020 ExploreLearning™ All rights reserved
Activity A: Angular velocity Get the Gizmo ready: ● Click Reset . ● Check that the Piston energy is set to 200 J. Introduction: For a spinning turntable, two types of velocity are important. As you learned, the angular velocity ( ω ) describes how quickly the whole disk is spinning in radians per second. An angular velocity of 6.28 rad/s describes a disk that makes one full revolution every second. You can also describe the linear velocity ( v ) of a point on the turntable. Linear velocity is simply the speed and direction of a point at an instant in time. Linear velocity is measured in meters per second. Question: How can you predict angular velocity based on the total energy of the turntable and how itsmass is distributed? 1. Explore: Experiment with the Gizmo to answer the following two questions: A. How does increasing the mass of the turntable affect its angular velocity? More mass would make moving it more energy-intensive. A slower turntable wouldresult from this. B. How does increasing the average distance of the masses from the center affect the angular velocity of the turntable? The angular velocity won't change. 2. Gather data: Turn on Show linear velocity . The Gizmo reports the velocity of the red dot. Place the red dot at 1 meter and a single mass at 5 meters. Click Play . For each position of the red dot, record the angular velocity of the turntable and linear velocity of the red dot. Red dot position: r = 1 m r = 2 m r = 3 m r = 4 m r = 5 m r = 6 m Angular velocity: 4 rad/s 4 rad/s 4 rad/s 4 rad/s 4 rad/s 4 rad/s Linear velocity: 4 m/s 8 m/s 12 m/s 16 m/s 20 m/s 24 m/s 3. Analyze: How does the linear velocity relate to the angular velocity and the radius? The angular velocity and radius combine to form the linear velocity. Write an equation for the linear velocity ( v ) based on angular velocity ( ω ) and position of the red dot ( r ): v= wr Reproduction for educational use only. Public sharing or posting prohibited. © 2020 ExploreLearning™ All rights reserved
4. Gather data: Place a single mass on the turntable at a radius of 6 m, and place the red dot on the mass. Click Play , and record both the angular velocity ( ω ) of the turntable and the linear velocity ( v ) of the mass. Move the mass and red dot to the next radius and repeat. Radius (m) Angular velocity (rad/s) Linear velocity (m/s) Kinetic energy (J) 6 m 3.33 rad/s 20 m/s 200 J 3 m 6.67 rad/s 20 m/s 200 J 2 m 10 rad/s 20 m/s 200 J 5. Analyze: What patterns do you notice in your data? The only velocity that changes when the radius does so is the angular velocity. 6. Calculate: The translational kinetic energy of a rotating object is equal to one half its mass multiplied by its velocity squared: TKE = ½ mv 2 . Use the linear velocity to calculate the last column of the table. (Hint: There is one mass on the turntable, so its mass is 1 kg.) A. How does the translational kinetic energy of the turntable compare to the piston energy? B. Why do you think this is so? Because energy is a finite resource, the piston's energy and the acceleration's energymust be equal. 7. Manipulate: Suppose a weightless turntable contains a single mass m that is a distance r from the center. The turntable has a kinetic energy KE . A. How would you calculate the linear velocity of the mass? v = B. How would you calculate the angular velocity of the turntable? ω = v/r C. Given the radius r, the mass m, and the kinetic energy KE, what is the angular velocity of the turntable? ω = 8. Apply: A weightless turntable has a mass of 4 kg that is located 3 m from the center. It is struck by a piston that imparts 160 J of kinetic energy to the turntable. Reproduction for educational use only. Public sharing or posting prohibited. © 2020 ExploreLearning™ All rights reserved
What is the angular velocity of theturntable? 2.98 rad/s Use the Gizmo to check your answer. (Hint: Place a mass on each of the 3-m pegs.) Activity B: Moment of inertia Get the Gizmo ready: ● Click Reset . Turn off Show linear velocity . ● Set the Piston energy to 200 J. ● Place a single mass at r = 5 m. Introduction: When studying more complex rotating objects, it is helpful to use the rotational equivalent of mass, moment of inertia ( I ). Just as mass can be defined as the resistance to changes in linear velocity, moment of inertia is a measure of an object’s resistance to changes in angular velocity. Question: How is angular velocity related to kinetic energy and moment of inertia? 1. Compare: The rotational kinetic energy of a body is given by the formula RKE = ½ Iω 2 . How is this equation similar to the equation for translational kinetic energy? The angular velocity is used in place of the linear velocity, and the moment of inertia is usedin place of the mass. 2. Calculate: Click Play . Calculate the value of I based on RKE = ½ Iω 2 , the current energy value (200 J), and the angular velocity ( ω ). The units of I are kg·m 2 . I = Turn on Show moment of inertia to check your answer. 3. Explore: Place the 1-kg mass at each radius and record the moment of inertia. Position: r = 1 m r = 2 m r = 3 m r = 4 m r = 5 m r = 6 m Moment of inertia: 4. Analyze: How is the moment of inertia related to the radius? It correlates with the radius. 5. Practice: The moment of inertia for a single object is given by the formula I = mr 2 . When there are multiple objects, the total moment of inertia is the sum of the individual moments. A. What is the moment of inertia for a system that has a 2-kg mass that is 2 meters from the center Reproduction for educational use only. Public sharing or posting prohibited. © 2020 ExploreLearning™ All rights reserved
and a 3-kg mass that is 3 meters from the center? B. Suppose the kinetic energy is 300 joules. What is the angular velocity? 4.14 rad/s Use the Gizmo to check your answer. Activity C: Continuousfigures Get the Gizmo ready: ● Click Reset . Turn off Show moment of inertia . ● Set the Piston energy to 200 J. ● In the Objects menu, choose Disk . Check that the Radius is 5.0 m and set the Mass to 1.0 kg. Introduction: The moment of inertia ( I ) for a single mass ( m ) that is rotating around a point with radius r is given by the equation I = mr 2 . However, the equation changes when the mass is not located at a single point, but spread out over an area. In this case, some of the mass is located close to the center of rotation, where itcauses less resistance to rotation. Question: How do you calculate the moment of inertia for disks, rings, and other shapes? 1. Observe: Click Play . What is the angular velocity of the disk? 5.66 rad/s 2. Calculate: Based on the rotational kinetic energy and angular velocity, what is the moment of inertia of this disk? Turn on Show moment of inertia to check 3. Explore: Click Reset . A. Double the mass of the disk to 2.0 kg. How does this affect the moment of inertia? Additionally, the moment of inertia is doubled. B. Return the mass to 1.0 kg. Record the moment of inertia for each of these radii: 4.0 m 3.0 m 2.0 m 1.0 m C. How is the moment of inertia related to the square of the radius? The square of the radius determines the moment of inertia. Reproduction for educational use only. Public sharing or posting prohibited. © 2020 ExploreLearning™ All rights reserved
4. Analyze: The general equation for moment of inertia is I = kmr 2 , where k is a constant that depends on the shape of the object. Based on what you have seen so far, what is the value of k for a disk? 1/2 5. Practice: Use the Gizmo to find the value of k for the other objects. List their values below. Ring: k = 1 Sprocket 1: k = 0.8662 Sprocket 2: k = 0.6276 Sprocket 3: k = 0.7914 6. Think and discuss: Why is the value of k lower for a disk than for a ring? (Hint: Remember that k is a constant that represents how the mass is distributed away from the center of rotation.) Because all of the mass is scattered away from the center of rotation at the edge, the ring hasa higher constant. The mass is uniformly divided between the edge and the center, resultingin a lower constant for the disk. 7. Explain: Why is the value of k the same for a ring as it is for a point mass? Because the mass is dispersed away from the center of rotation on both when the radiusaway from the axis of rotation on a point mass is equal to the radius of the ring, the value of kis the same. 8. Summarize: In general, why does the moment of inertia increase when mass is distributed farther from the center of a rotating object? Try to explain your reasoning in words rather than using equations. The object produces more rotational resistance if the mass is dispersed farther from itscenter. Consequently, it has more inertia as well. Reproduction for educational use only. Public sharing or posting prohibited. © 2020 ExploreLearning™ All rights reserved
