1. PH 201 Dr. Cecilia Vogel Lecture 20

2. REVIEW  Constant angular acceleration equations  Rotational Motion  torque OUTLINE  moment of inertia  angular momentum  angular kinetic energy

3. Table so Far Linear variable Angular variable Variable name x  angle (rad) v = dx/dt  d  /dt angular velocity (rad/s) a = dv/dt  d  /dt ang. acceleration (rad/s 2 ) F  torque (Nm) m Imoment of inertia (kgm 2 ) K p

4. Recall Momentum  Momentum is conserved,  if no external force  because   F=m  v CM /  t  So if LHS=0,  v CM =0  then  p=0

5. Angular Momentum   =I  /  t  So if LHS =0  then I   Define angular momentum  Angular momentum conserved  if no net external torque

6. Add to Table p Rotational Kinetic Energy (J) K rot = ½I  2 K = ½mv 2 moment of inertia (kgm 2 )I m torque (Nm)  F ang. acceleration (rad/s 2 )  d  /dt a = dv/dt angular velocity (rad/s)  d  /dt v = dx/dt angle (rad)  x Variable nameAngular variable Linear variable L=I  Angular momentum (kgm 2 /s)

7. Angular Momentum   =I  /  t  So if net torque is not zero  then L changes  angular momentum changes

8. Angular Momentum  angular momentum is a vector  direction is found by a RHR  Hold your right hand so your curved fingers point in the direction of rotation  then your thumb will point in the direction of angular momentum (out +, in -)

9. Conservation Demo  Sit on a chair, free to rotate  hold a wheel rotating so its angular momentum points to your left.  Try to tip wheel’s axis up or down.  Notice  torque required for you to change angular momentum of wheel (just direction).  You and wheel are isolated, so if you tip wheel axis down,  to conserve momentum need L ___.

10. Demo and Bikes  Sit on a bike  wheels rotate so angular momentum points to your left.  Lean the bike.  If you tip wheel axis down, (lean left)  to conserve momentum need L ___  Bike turns ___

11. Kinetic Energy of Rotation  As a rigid body rotates,  all parts are moving  but different parts are moving at different speeds,  so  If you consider  then

12. Add to Table Linear variable Angular variable Variable name x  angle (rad) v = dx/dt  d  /dt angular velocity (rad/s) a = dv/dt  d  /dt ang. acceleration (rad/s 2 ) F  torque (Nm) m Imoment of inertia (kgm 2 ) K = ½mv 2 K rot = ½I  2 Rotational Kinetic Energy p

13. Total Kinetic Energy  An object might be rotating, while also moving linearly,  like a tire on a bike that’s being ridden.  Has  and note: K rot must be rotation about CM