1. Department of Physics and Applied Physics 95.141, F2010, Lecture 19 Physics I 95.141 LECTURE 19 11/17/10

2. Department of Physics and Applied Physics 95.141, F2010, Lecture 19 Exam Prep Problem A ball (ball A) of mass m=2kg, traveling at a velocity v A =4m/s. collides with two balls at rest. Ball B (m B =4kg) leaves the collision at an angle of +45 degrees, and ball C (m C =2kg) leaves the collision at an angle of -45 degrees. Ball A is at rest after the collision. –A) (5pts) Write down the conservation of momentum expressions for this collision –B) (10pts) Solve your system of equations to determine the velocities of balls B and C after the collision –C) (10pts) Is this an elastic or inelastic collision? Explain why. If it is inelastic, how much thermal energy is generated in the collision?

3. Department of Physics and Applied Physics 95.141, F2010, Lecture 19 Exam Prep Problem m A =m C =2, v A =4m/s v B =v C =0, m B =4 θ’ B =+45°, θ’ C =-45°, v’ A =0m/s –A) (5pts) Write down the conservation of momentum expressions for this collision

4. Department of Physics and Applied Physics 95.141, F2010, Lecture 19 Exam Prep Problem m A =m C =2, v A =4m/s v B =v C =0, m B =4 θ’ B =+45°, θ’ C =-45°, v’ A =4m/s B) (10pts) Solve your system of equations to determine the velocities of balls B and C after the collision

5. Department of Physics and Applied Physics 95.141, F2010, Lecture 19 Exam Prep Problem m A =m C =2, v A =4m/s v B =v C =0, m B =4 θ’ B =+45°, θ’ C =-45°, v’ A =4m/s C) (10pts) Is this an elastic or inelastic collision? Explain why. If it is inelastic, how much thermal energy is generated in the collision?

6. Department of Physics and Applied Physics 95.141, F2010, Lecture 19 Outline Torque Rotational Inertia Moments of Inertia What do we know? –Units –Kinematic equations –Freely falling objects –Vectors –Kinematics + Vectors = Vector Kinematics –Relative motion –Projectile motion –Uniform circular motion –Newton’s Laws –Force of Gravity/Normal Force –Free Body Diagrams –Problem solving –Uniform Circular Motion –Newton’s Law of Universal Gravitation –Weightlessness –Kepler’s Laws –Work by Constant Force –Scalar Product of Vectors –Work done by varying Force –Work-Energy Theorem –Conservative, non-conservative Forces –Potential Energy –Mechanical Energy –Conservation of Energy –Dissipative Forces –Gravitational Potential Revisited –Power –Momentum and Force –Conservation of Momentum –Collisions –Impulse –Conservation of Momentum and Energy –Elastic and Inelastic Collisions2D, 3D Collisions –Center of Mass and translational motion –Angular quantities –Vector nature of angular quantities –Constant angular acceleration

7. Department of Physics and Applied Physics 95.141, F2010, Lecture 19 Review of Lecture 18 Discussed angular quantities we use to describe rotational motion –Angular Displacement: –Angular Velocity: –Angular Acceleration: Vector Nature of Angular Velocity Rotation with constant angular acceleration, and parallels with constant linear acceleration problems.

8. Department of Physics and Applied Physics 95.141, F2010, Lecture 19 Example A top is brought up to speed with α=7rad/s 2 in 1.5s. After that it slows down slowly with α=- 0.1rad/s 2 until it stops spinning. –A) What is the fastest angular velocity of the top? –B) How long does it take the top to stop spinning once it reaches its top angular velocity? –C) How many rotations does the top make in this time?

9. Department of Physics and Applied Physics 95.141, F2010, Lecture 19 Example A top is brought up to speed with α=7rad/s 2 in 1.5s. After that it slows down slowly with α=- 0.1rad/s 2 until it stops spinning. –C) How many rotations does the top make in this time?

10. Department of Physics and Applied Physics 95.141, F2010, Lecture 19 More Angular Quantities We have discussed the angular equivalents for position, velocity, and acceleration (Chapter 2). But is this where the similarities end? After we discussed linear motion, we discussed the Forces that cause this motion. Is there an equivalent to Force for rotational motion?

11. Department of Physics and Applied Physics 95.141, F2010, Lecture 19 Torque Clearly it takes a Force to make something start rotating, but is the magnitude/direction of the Force the only thing that matters? Think about opening a door. Where is it easier to push the door open, near the hinges, or near the handle? F F

12. Department of Physics and Applied Physics 95.141, F2010, Lecture 19 Torque The effect of a Force on the rotational object depends on the perpendicular distance from the axis of rotation that the Force is applied. This distance is known as the lever arm or moment arm. F

13. Department of Physics and Applied Physics 95.141, F2010, Lecture 19 Torque Angular acceleration is proportional to the product of the Force and the lever arm (torque). But its not just the total Force, of course, life’s not that easy….

14. Department of Physics and Applied Physics 95.141, F2010, Lecture 19 Torque The Torque can be defined as the product of the lever arm and the component of the Force perpendicular to the lever arm.

15. Department of Physics and Applied Physics 95.141, F2010, Lecture 19 Example Two forces are applied to compound wheel as shown below. What is the net Torque on the object? (R A =0.3m, R B =0.5m)

16. Department of Physics and Applied Physics 95.141, F2010, Lecture 19 Rotational Dynamics We know that for linear acceleration, Newton’s 2 nd Law tells us that the linear acceleration is proportional to Force. For rotational motion, we know the kinematic equations are similar to those for linear motion. So angular acceleration is proportional to the rotational equivalent of Force (Torque). What is the rotational equivalent of mass/inertia.

17. Department of Physics and Applied Physics 95.141, F2010, Lecture 19 Rotational Dynamics If we have a point mass a distance R from an axis of rotation, and we apply a Force perpendicular to R, what is acceleration? The quantity mR 2 is known as the rotational inertia of the particle: moment of inertia.

18. Department of Physics and Applied Physics 95.141, F2010, Lecture 19 Calculating Moments of Inertia Say we have two masses connected to a massless rod. 5m 2kg 3kg 4m 1m A B

19. Department of Physics and Applied Physics 95.141, F2010, Lecture 19 Rotational Dynamics What if we have, instead of a point source, a solid object? We could divide the solid object into a large number of smaller masses dm, and calculate the rotational inertia of each of these….

20. Department of Physics and Applied Physics 95.141, F2010, Lecture 19 I for a solid object Thin hoop

21. Department of Physics and Applied Physics 95.141, F2010, Lecture 19 I for a solid object Circular Plate

22. Department of Physics and Applied Physics 95.141, F2010, Lecture 19 I for a solid object Rod

23. Department of Physics and Applied Physics 95.141, F2010, Lecture 19 Common Moments of Inertia

24. Department of Physics and Applied Physics 95.141, F2010, Lecture 19 Example Problem A mass of 10kg is attached to a cylindrical pulley of radius R1 and mass m p =10kg and released from rest. What is the acceleration of the mass? R M p =10kg M b =10kg

25. Department of Physics and Applied Physics 95.141, F2010, Lecture 19 Example Problem What is the angular acceleration of the rod shown below, if it is released from rest, at the moment it is released? What is the linear acceleration of the tip?

26. Department of Physics and Applied Physics 95.141, F2010, Lecture 19 WIPEOUT

27. Department of Physics and Applied Physics 95.141, F2010, Lecture 19 Parallel Axis Theorem What is the CM for the system we looked at earlier? 5m 2kg 3kg A B

28. Department of Physics and Applied Physics 95.141, F2010, Lecture 19 Parallel Axis Theorem Says that, for rotation about an axis h from the CM What is I CM ? 5m 2kg 3kg CM h

29. Department of Physics and Applied Physics 95.141, F2010, Lecture 19 Parallel Axis Theorem Example What is the moment of inertia for a rod –Rotating about its center of mass? –Rotating about its end?

30. Department of Physics and Applied Physics 95.141, F2010, Lecture 19 Rotational Kinetic Energy We now know the rotational equivalent of mass is the moment of inertia I. If I told you there was such a thing as rotational kinetic energy, you could probably make a good guess as to what form it would take…

31. Department of Physics and Applied Physics 95.141, F2010, Lecture 19 Rotational Kinetic Energy Prove it!