1. Translational vs. rotational motion  Translational Motion  What we talked about in earlier units  Motion of the center of mass  Rotational Motion  Spinning around the center of mass  Motion is often a combination of both

2. Translational vs. rotational Translational Translational  Displacement  Velocity  Force  Inertia (mass)  Momentum Rotational Rotational  Angle (or rots.)  Rotational speed  Torque  Rotational inertia  Angular Momentum

3. Forces on an object  When forces are in line with the center of mass, the result is simply translational motion  When forces don’t line up with the center of mass, object may rotate  We need to consider what happens when forces don’t “line up”

4. Torque  Torque is the result of force applied away from the center of rotation  Lever arm = perpendicular distance from center of rotation Torque = force x lever arm Units: Newton-meters (Nm) Lever arm Force Lever arm Force

5. Example: Opening a Door  Force applied: 50N  Distance from hinge to doorknob: 0.8m  Torque = 50N * 0.8m = 40 Nm Lever arm = 0.8m Force = 50N

6. Rotational Equilibrium  Remember force equilibrium?  No change in (translational) speed  Zero net force  Sum of forces in one direction = Sum of forces in opposite direction. Torque = 400 Nm  Rotational equilibrium  No change in rotational speed  Zero net torque  Sum of torques in one direction = Sum of torques in opposite direction. (clockwise/counter- clockwise)

7. Examples Torque = 400 Nm Torque = 300 Nm Torque = 100 Nm Sum of clockwise torques = Sum of counterclockwise torques

8. More specifically Force = 100 N Force = 50 N d = 1m 50N x 2m = 100N x 1m 40N x 1m + 60N x 2m = 160N x 1m Force = 160 N Force = 60 N Force = 40 N d = 1m 100 Nm = 100Nm 160 Nm = 160Nm

9. Two last rotational concepts  Rotational inertia  Angular momentum

10. Rotational Inertia  Remember inertia?  An object’s resistance to change in its state of motion  Force is required to change the state of motion  Rotational inertia  An object’s resistance to change in its state of rotational motion  Torque is required to change the state of rotational motion

11. Example: pencil  First:  Hold your pencil between two fingers, near the middle  Rotate it back and forth (you’re applying a torque)  Then:  Hold your pencil between two fingers, near the end  Rotate it back and forth.

12. Pencil, continued  Which is harder?  The further out the mass is from center of rotation, the more rotational inertia  Examples: choking up on a bat, running with your legs bent

13. Angular Momentum  Remember momentum?  Momentum = mass x velocity  Momentum is conserved  Angular momentum  Angular momentum = rotational inertia x rotational velocity  Angular momentum is conserved  Example: bike wheels, gyroscopes

14. Translational vs. Rotational, revisited  A lot of these translational concepts have rotational equivalents  Force ↔ torque  Inertia (mass) ↔ rotational inertia  Momentum ↔ angular momentum  But one key difference: You can change your rotational inertia!

15. Changing Rotational Inertia  Spread out, increase rotational inertia  Tuck in, decrease rotational inertia  Used by gymnasts, divers, skaters, freestyle skiers, falling cats