1. Department of Physics and Applied Physics 95.141, F2010, Lecture 20 Physics I 95.141 LECTURE 20 11/21/10

2. Department of Physics and Applied Physics 95.141, F2010, Lecture 20 Parallel Axis Theorem Moment of inertia for a sphere, rotating about an axis through its center, is: What is moment of inertia for this sphere about an axis going through the edge of the sphere? Through an axis a distance R from the center? As R>>r o ? R

3. Department of Physics and Applied Physics 95.141, F2010, Lecture 20 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…

4. Department of Physics and Applied Physics 95.141, F2010, Lecture 20 Rotational Kinetic Energy Prove it!

5. Department of Physics and Applied Physics 95.141, F2010, Lecture 20 Rolling without slipping If the CM of the wheel is moving with speed v Place yourself at the axis, you will see road moving by at –v beneath you. So if the road is moving at –v, and the wheel isn’t slipping, you know the angular velocity of the wheel!

6. Department of Physics and Applied Physics 95.141, F2010, Lecture 20 Instantaneous Axis What is the axis of rotation for the rolling wheel? The point on the ground is fixed, the wheel is then rotating around this point.

7. Department of Physics and Applied Physics 95.141, F2010, Lecture 20 Total KE of Rolling Body The total kinetic energy of the rolling body will then be:

8. Department of Physics and Applied Physics 95.141, F2010, Lecture 20 Rolling Body Example

9. Department of Physics and Applied Physics 95.141, F2010, Lecture 20 What is speed of rock at end of incline? Use conservation of Energy Rolling Body Example

10. Department of Physics and Applied Physics 95.141, F2010, Lecture 20 Outline Angular Momentum Vector Cross Products 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 –Torque –Rotational Inertia –Moments of Inertia

11. Department of Physics and Applied Physics 95.141, F2010, Lecture 20 Review of Lecture 19 Introduced concept of Torque –“Rotational Force” Comes from Force exerted on a rotating body Depends on distance from axis of rotation Equal to radius x perpendicular component of Force Does not have same units as Force Rotational equivalent of mass is moment of inertia…how hard it is to accelerate (angularly) an object for a given Torque. –For a point mass: –For a collection of point masses: –For solid object rotating around CM: use table in book, or –use integral:

12. Department of Physics and Applied Physics 95.141, F2010, Lecture 20 Lecture 19 Review Parallel axis theorem –Can be used to determine I around an axis other than one through center of mass Discussed equivalent of Newton’s 2 nd Law for Angular Motion –Torque is what gives angular acceleration: Angular Kinetic Energy Rolling without slipping

13. Department of Physics and Applied Physics 95.141, F2010, Lecture 20 What are we missing? We have discussed angular acceleration, velocity, Force (torque) and kinetic energy All have equivalents in linear motion There is one other translational quantity we have spent a lot of time with…momentum, does this have an angular equivalent? Again, if we had to guess….

14. Department of Physics and Applied Physics 95.141, F2010, Lecture 20 Angular Momentum Angular momentum is defined as the product angular velocity of an object and the object’s moment of inertia. Can write the equivalent of Newton’s second law for angular momentum, just like we did for translational momentum

15. Department of Physics and Applied Physics 95.141, F2010, Lecture 20 Conservation of Angular Momentum Just like translational momentum, angular momentum is a conserved quantity in certain circumstances. Because a torque leads to angular acceleration (a change in angular velocity), if a torque is applied to a rotating body, angular momentum is not conserved. But if no external torques act on a body, and I is constant, then angular momentum is conserved!

16. Department of Physics and Applied Physics 95.141, F2010, Lecture 20 Conservation of Angular Momentum Law of conservation of angular momentum: The total angular momentum of a rotating object remains constant if the net external torque acting on the object is zero Import conservation law in Physics, along with conservation of linear momentum and energy!!

17. Department of Physics and Applied Physics 95.141, F2010, Lecture 20 Example Problem Lets take the rotating rod system we looked at earlier. If the rod is rotating at ω=5rad/s, what is the system’s angular momentum? If I remove block 2, what is the new angular velocity of the system? m2=3kg m1=2kg 2m

18. Department of Physics and Applied Physics 95.141, F2010, Lecture 20 Directional Nature of L For translational momentum, the momentum vector points in the same direction as the velocity vector, because mass is a scalar. In the same way, because the moment of inertia is a scalar, the angular velocity vector determines the direction of the angular momentum vector.

19. Department of Physics and Applied Physics 95.141, F2010, Lecture 20 Vector Cross Product We have discussed, so far, two types of vector products: –Multiplication by a scalar: –Dot-product of two vectors ( )

20. Department of Physics and Applied Physics 95.141, F2010, Lecture 20 Vector Cross-Product There is yet a third type of vector product, known as the vector cross product Product of two vector, that results in a vector Resulting vector has a magnitude And points in the direction perpendicular to the plane created by Direction determined by right hand rule

21. Department of Physics and Applied Physics 95.141, F2010, Lecture 20 Vector Cross Product, Mathematically

22. Department of Physics and Applied Physics 95.141, F2010, Lecture 20 Vector Cross Product What is the vector cross product of a vector and itself?

23. Department of Physics and Applied Physics 95.141, F2010, Lecture 20 Does Order Matter?

24. Department of Physics and Applied Physics 95.141, F2010, Lecture 20 Example What are the vector cross products

25. Department of Physics and Applied Physics 95.141, F2010, Lecture 20 Example What is the vector dot product of the two vectors:

26. Department of Physics and Applied Physics 95.141, F2010, Lecture 20 Torque and the Cross Product Technically, when we first introduced torque as the product of the radius and the perpendicular component of the Force, we were taking a cross product!

27. Department of Physics and Applied Physics 95.141, F2010, Lecture 20 Angular Momentum of a particle Imagine a particle of mass m moving in a circular path of radius R with angular velocity ω.

28. Department of Physics and Applied Physics 95.141, F2010, Lecture 20 Angular Momentum of Particle If we think about vectors, it becomes clear that the angular momentum must be the cross product of the linear momentum and the radius!