Energy and Angular Momentum

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Alexandrina Henderson
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1 Notes 13 Rotation Page 1 Energy and Angular Momentum The kinetic energy associate with a rotating object is simply the sum of the regular kinetic energies. Our goal is to state the rotational kinetic energy in terms of rotational quantities (recall v differs for each point on a rotating object) Similar for Angular momentum L (which is a vector). For a single point mass L is given by :

2 Notes 13 Rotation Page 2 Ball rolling down hill Consider a ball (or any round object) rolling down a hill. Previously we had considered objects slipping. If an object is rolling, energy goes to both translation and rotation. We are free to describe the c.m. as the axis, and to track the kinetic energy (translational and rotational) of and about an axis through the c.m. Solve for v

3 Notes 13 Rotation Page 3 Ball Rolling II If I=0 this reduces to "slipping" down ramp. If I is large (??) then the speed at the bottom is reduced compared to slipping. Note the terms with "1" and with "I/(mR^2) " determine how the energy splits into translation and rotation. It is typical for me to ask questions like: At the bottom of the ramp, what fraction of the total energy is rotational, translational (none of it is potential). Go back to that initial energy equation.

4 Notes 13 Rotation Page 4 Angular Momentum We have seen that net torque causes angular acceleration. I =Angular momentum If there is no net external torque applied, then there is no change in angular momentum of a system. Angular momentum is conserved!!!!!!

5 Notes 13 Rotation Page 5 Digression We have all the laws for conservation (Energy, Momentum, Angular momentum). We have Newton's 2nd law, and an equivalent torque law. We have definition of moment of inertia. WHAT PROBLEMS CAN WE DO?

6 Notes 13 Rotation Page 6 Some adding (moments of inertia) We can add moments of inertia for discrete systems and /or for continuous mass distributions.

7 Notes 13 Rotation Page 7 Hollow Sphere I II Back to Angular momentum We had If the net external torque is zero, then Angular momentum is constant If no net external torques, then Check a simple case. Ball on string (but string may wrap around finger (axis). Or, comet around Sun. SPECIAL CASE

8 Notes 13 Rotation Page 8 mvr As string wraps up, r changes. The angular momentum is still given (for this special point object case) by mvr. Since the string pulls inward toward the axis, there is no torque on the mass. If r is reduced, the object moves (tangentially) faster. Uh where did the extra kinetic energy come from. If the mass moved to 1/2 the initial radius, it has 4 times the initial kinetic energy. From where? Work is done to "pull" the mass inward (by some force).

9 Notes 13 Rotation Page 9 Comets and Bullets You are watching a comet move in orbit around the Sun. As the comet moves from initial distance, to final distance, what happens to : Speed, Angular Speed, KE, Angular momentum, momentum Angular momentum remains the same (L is conserved) Once you find v, find. KE obtain from speed. Where does the energy come from? The Sun does work (gravitational) to pull the comet closer. So it moves faster. Another way to say it, the comet "fell" closer to the Sun, has less potential energy and increased kinetic energy. Try to swing an object on a string and let it wrap around your finger. Watch what happens. Go to playground. Pull yourself in toward the center of the Merry go round. Ice skate and do a spin, pull your arms in. Sit in/on a rotating tire swing. Pull yourself in, let yourself out.

10 Notes 13 Rotation Page 10 Bullet In playing pool, it is the impact parameter that determines how the balls proceed after a collision. Angular momentum is what completes the picture. We now have three physics quantities that obey simple conservation laws. Energy, Momentum, Angular momentum.

11 Restatement of torque Given the conservation of angular momentum we can state the torque law as: In the OLD DAYS of Vinyl record players, CD Juke boxes or DVD's Drop a stationary record on top of one that is already spinning. When disk 2 (initially at rest) falls onto disk 1 initially spinning, what happens? This situation might be called a rotationally completely inelastic collision. ANGULAR MOMENTUM IS CONSERVED HERE. Notes 13 Rotation Page 11

Chapter 9-10 Test Review

Chapter 9-10 Test Review Chapter Summary 9.2. The Second Condition for Equilibrium Explain torque and the factors on which it depends. Describe the role of torque in rotational mechanics. 10.1. Angular