Moment of Inertia Rotational Kinetic Energy Angular Momentum Lecture 15 Moment of Inertia Rotational Kinetic Energy Angular Momentum 12/9/2018 Physics 103, Spring 2004, U. Wisconsin

Inertia and Acceleration Equilibrium: forces (torques) are in balance ( = zero) Dynamics: forces (torques) are non zero Cause Effect Change in motion Linear Rotary Force Torque How is torque related to angular acceleration? What is the rotational equivalent of mass? How do we express Newton’s law for rotational motion? 12/9/2018 Physics 103, Spring 2004, U. Wisconsin

Physics 103, Spring 2004, U. Wisconsin Moment of Inertia When a rigid object is subject to a net torque (≠0), it undergoes an angular acceleration Where the force is applied on the body matters Distribution of mass about the body matters The angular acceleration is directly proportional to the net torque The relationship ∑t = Ia is analogous to ∑F = ma Newton’s Second Law The angular acceleration is inversely proportional to the moment of inertia, I, of the object Mass of a piece of the object (mi) Distance from axis of rotation to that piece (ri). SI units are kg-m2 12/9/2018 Physics 103, Spring 2004, U. Wisconsin

Moment of Inertia of a Uniform Ring Image the hoop is divided into a number of small segments, m1 … These segments are equidistant from the axis 12/9/2018 Physics 103, Spring 2004, U. Wisconsin

Physics 103, Spring 2004, U. Wisconsin Preflight 14.4 The rotational inertia of a rigid body: is a measure of its resistance to changes in rotational motion. depends on the location of the axis of rotation. is large if most of the body’s mass is far from the axis of rotation. is all of the above is none of the above 12/9/2018 Physics 103, Spring 2004, U. Wisconsin

Physics 103, Spring 2004, U. Wisconsin Preflight 14.5 & 14.6 A hoop, a solid cylinder and a solid sphere all have the same mass and radius. Which of them has the largest moment of inertia when they rotate about axis shown? The hoop. The cylinder. The sphere All have the same moment of inertia 12/9/2018 Physics 103, Spring 2004, U. Wisconsin

Physics 103, Spring 2004, U. Wisconsin Preflight 14.7 & 14.8 The picture below shows two different dumbbell shaped objects. Object A has two balls of mass m separated by a distance 2L, and object B has two balls of mass 2m separated by a distance L. Which of the objects has the largest moment of inertia for rotations around x-axis? A. B. They have the same moment of inertia 12/9/2018 Physics 103, Spring 2004, U. Wisconsin

Physics 103, Spring 2004, U. Wisconsin Moments of Inertia 12/9/2018 Physics 103, Spring 2004, U. Wisconsin

Physics 103, Spring 2004, U. Wisconsin Question Imagine hitting a dumbbell with an object coming in at speed v, first at the center, then at one end. Is the center-of-mass speed of the dumbbell the same in both cases? 1. Yes. 2. No M M Case 1 Case 2 The moving object comes in with a certain momentum. If it hits the center, as in case 1, there is no rotation, and this collision is just like a one dimensional collision between object of mass m and another of mass 2m. Because of the larger mass of dumbbell, the incoming ball bounces back. In case 2, the dumbbell, starts rotating. The incoming ball encounters less “resistance” and therefore transfers less of its momentum to the dumbbell, which will therefore have a smaller center-of-mass speed than in case 1. 12/9/2018 Physics 103, Spring 2004, U. Wisconsin

Rotational Kinetic Energy Work must be done to rotate objects Force expended perpendicular to the radius Parallel to the displacement Ds q r F 12/9/2018 Physics 103, Spring 2004, U. Wisconsin

Rotation Summary (with comparison to 1-d linear motion) Angular Linear 12/9/2018 Physics 103, Spring 2004, U. Wisconsin

Physics 103, Spring 2004, U. Wisconsin Question You decide to roll two objects down a ramp to see which one gets to the bottom first. Object A is a solid cylinder and object B is a hollow cylinder. A and B have the same mass and radius. Suppose you released them from rest at the top of the ramp at same time, which one gets to the bottom of the ramp first? 1. A 2. B 3. Same CORRECT the moment of inertia for a solid cylinder is less than that of a hollow cylinder meaning that more potential energy can be converted to translational kinetic energy rather than rotational kinetic energy Iw2/2 ...this means that A will get to the bottom faster because it will cover a greater distance per unit time 12/9/2018 Physics 103, Spring 2004, U. Wisconsin

Physics 103, Spring 2004, U. Wisconsin Lecture 15, Preflight 4 & 5 Two cylinders of the same size and mass roll down an incline. Cylinder A has most of its mass concentrated at the rim, while cylinder B has most of its mass concentrated at the center. Which reaches the bottom of the incline first? 1. A 2. B 3. Both reach at the same time. CORRECT Cylinder A has higher moment of inertia than cylinder B - therefore, it takes longer to roll down. 12/9/2018 Physics 103, Spring 2004, U. Wisconsin

Physics 103, Spring 2004, U. Wisconsin Question In the rolling experiment, which object has the most kinetic energy when it gets to the bottom of the ramp? 1. A 2. B 3. Same CORRECT Gravity is responsible for motion. Conservation of energy tells us that PE will be turned into KE. Both will have the same KE at the bottom, since they had the same PE (mgh is same because they had the same mass and shape, and started from the same height) at the top of the ramp. The total kinetic energy is the sum of translational (mv2/2) and rotational (Iw2/2). How about velocity after rolling down the same height? 12/9/2018 Physics 103, Spring 2004, U. Wisconsin

Speed of Sliding Object 12/9/2018 Physics 103, Spring 2004, U. Wisconsin

Physics 103, Spring 2004, U. Wisconsin Kinetic Energy VCM w 12/9/2018 Physics 103, Spring 2004, U. Wisconsin

Kinetic Energy: Rolling without Slipping VCM w 12/9/2018 Physics 103, Spring 2004, U. Wisconsin

Application: Rolling without Slipping Down Incline VCM w h KEtotal + PEg = 0 PEg = -Mgh Solve: 12/9/2018 Physics 103, Spring 2004, U. Wisconsin

Application: Rolling without Slipping Down Incline VCM w h Larger I  smaller VCM 12/9/2018 Physics 103, Spring 2004, U. Wisconsin

Physics 103, Spring 2004, U. Wisconsin Roller Coaster Small Large CORRECT 12/9/2018 Physics 103, Spring 2004, U. Wisconsin

Physics 103, Spring 2004, U. Wisconsin Angular Momentum Similarly to the relationship between force and momentum in a linear system, we can show the relationship between torque and angular momentum Angular momentum is defined as L = Iw and 12/9/2018 Physics 103, Spring 2004, U. Wisconsin

Physics 103, Spring 2004, U. Wisconsin Lecture 15, Preflight 1 The angular momentum of a particle is independent of the specific origin of coordinates. is zero when its position and momentum vectors are parallel. is zero when its position and momentum vectors are perpendicular. CORRECT L r q w p Angular momentum, L = I w = (S mr2) (v/r) i.e., L = mv r = r p (here r and p make 90o) Angular momentum is a vector perpendicular to the position, r, and motion, p, L = r x p Right hand rule 12/9/2018 Physics 103, Spring 2004, U. Wisconsin

Physics 103, Spring 2004, U. Wisconsin Angular Momentum 12/9/2018 Physics 103, Spring 2004, U. Wisconsin

Angular Velocity, Momentum - Right Hand Rule 12/9/2018 Physics 103, Spring 2004, U. Wisconsin

Angular Momentum Conservation If the net torque is zero, the angular momentum remains constant Conservation of Linear Momentum states: The angular momentum of a system is conserved when the net external torque acting on the systems is zero. That is, when 12/9/2018 Physics 103, Spring 2004, U. Wisconsin

Angular Momentum Conservation 12/9/2018 Physics 103, Spring 2004, U. Wisconsin

Physics 103, Spring 2004, U. Wisconsin Lecture 15, Preflight 2 & 3 A figure skater stands on one spot on the ice (assumed frictionless) and spins around with her arms extended. When she pulls in her arms, she reduces her rotational inertia and her angular speed increases so that her angular momentum is conserved. Compared to her initial rotational kinetic energy, her final rotational kinetic energy after she has pulled in her arms must be: 1. Same 2. Larger because she is rotating faster 3. Smaller because her rotational inertia is smaller CORRECT Rotational kinetic energy is Iw2/2. L=Iw. Rot.K.E=Lw/2 L is constant - therefore, since w increases Rot. KE also increases. Additional energy is provided by the skater “working” to pull her arms in. 12/9/2018 Physics 103, Spring 2004, U. Wisconsin

Physics 103, Spring 2004, U. Wisconsin Lecture 15, Pre-flights You are sitting on a freely rotating bar-stool with your arms stretched out and a heavy glass mug in each hand. Your friend gives you a twist and you start rotating around a vertical axis though the center of the stool. You can assume that the bearing the stool turns on is frictionless, and that there is no net external torque present once you have started spinning. You now pull your arms and hands (and mugs) close to your body. 12/9/2018 Physics 103, Spring 2004, U. Wisconsin

Physics 103, Spring 2004, U. Wisconsin Lecture 15, Preflight 6 & 7 What happens to your angular momentum as you pull in your arms? 1. it increases 2. it decreases 3. it stays the same L1 L2 CORRECT Since there is no external torque acting on the system, the total angular momentum is conserved. 12/9/2018 Physics 103, Spring 2004, U. Wisconsin

Physics 103, Spring 2004, U. Wisconsin Lecture 15, Preflight 8 & 9 What happens to your angular velocity as you pull in your arms? 1. it increases 2. it decreases 3. it stays the same w1 w2 I2 I1 L CORRECT Your moment of inertia decreases so your angular velocity must increase to compensate for this change and keep angular momentum the same. 12/9/2018 Physics 103, Spring 2004, U. Wisconsin

Physics 103, Spring 2004, U. Wisconsin Lecture 21, Preflight 10 & 11 What happens to your kinetic energy as you pull in your arms? 1. it increases 2. it decreases 3. it stays the same w1 w2 I2 I1 L CORRECT (using L = I ) Because w increases as much as I decreases. In the equation: KErot = 1/2Iw2, w is squared so the kinetic energy increases. You are doing work by changing your moment of inertia so you increase your kinetic energy 12/9/2018 Physics 103, Spring 2004, U. Wisconsin

Physics 103, Spring 2004, U. Wisconsin Question Two different spinning disks have the same angular momentum, but disk 2 has a larger moment of inertia than disk 1. Which one has the biggest kinetic energy ? (a) disk 1 (b) disk 2 (using L = I ) w1 w2 If they have the same L, the one with the smallest I will have the biggest kinetic energy. I1 < I2 disk 1 disk 2 12/9/2018 Physics 103, Spring 2004, U. Wisconsin

Preflight 12 & 13 Turning the bike wheel A student sits on a barstool holding a bike wheel. The wheel is initially spinning CCW in the horizontal plane (as viewed from above). She now turns the bike wheel over. What happens? 1. She starts to spin CCW. 2. She starts to spin CW. 3. Nothing 12/9/2018 Physics 103, Spring 2004, U. Wisconsin

Physics 103, Spring 2004, U. Wisconsin Turning the bike wheel... Since there is no net external torque acting on the student-stool system, angular momentum is conserved. Remember, L has a direction as well as a magnitude! Initially: LINI = LW,I Finally: LFIN = LW,F + LS LS LW,I LW,I = LW,F + LS LW,F 12/9/2018 Physics 103, Spring 2004, U. Wisconsin

Angular Momentum, Angular Velocity, Torque 12/9/2018 Physics 103, Spring 2004, U. Wisconsin

Angular Momentum Addition 12/9/2018 Physics 103, Spring 2004, U. Wisconsin

Physics 103, Spring 2004, U. Wisconsin A person spins a tennis ball on a string in a horizontal circle so that the axis of rotation is vertical. At the point shown, the ball is given a sharp blow vertically downward. In which direction does the axis of rotation tilt after the blow? 1. +x direction 2. -x direction 3. +y direction 4. -y direction 5. It stays the same z y x L F v L F v L’ v’ According to the right hand rule, the torque exerted by the force is in the forward direction (+x), and so the change in the angular momentum must also be in this direction. 12/9/2018 Physics 103, Spring 2004, U. Wisconsin

Physics 103, Spring 2004, U. Wisconsin Gyroscope Spinning gyroscope has angular momentum along the axis When tilted the weight acting at its center of mass and the normal force at the pivot create a torque, and, therefore, angular momentum DL at 90o to the direction of L L and DL add to produce new angular momentum that causes the gyroscope to precess instead of falling over 12/9/2018 Physics 103, Spring 2004, U. Wisconsin

Percussion Point - Tennis racket sweet-spot Somewhere in the middle of these two extremes is a situation where the force exerted on the nail (pivot) is zero. Disk hits on one end of the bar - a force is exerted on the nail (pivot) in opposing direction. Similarly a force is exerted on the hand Disk hits close to nail - it experiences a force in the same direction. Percussion point - sweet-spot 12/9/2018 Physics 103, Spring 2004, U. Wisconsin