Rotational and Translational Motion Dynamics 8.01 W11D2

1. Rotational and Translational Motion Dynamics8.01W11D2

2. Today’s Reading Assignment: W11D2 Young and Freedman: 10.3-10.6

3. Review: Angular Momentum and Torque for a System of Particles Change in total angular momentum about a point S equals the total torque about the point S

4. Torque for Rotation and Translation The total torque about S is given by where the first term torque about S due to the total external force acting at the center-of-mass and the second term is torque about the center-of-mass is only due to the forces as seen in the laboratory frame. The total external torque produces an angular acceleration about the center-of-mass

5. Torque and Angular Momentum for Rotation and Translation The torque about a point S is the time derivative of the angular momentum about S, Once again the first and third terms vanish because So the torque about S becomes,

6. Torque and Angular Momentum for Rotation and Translation The torque about a point S is Recall that the external force is the time change of the momentum of the center-of-mass, So the first term is the torque about S due to the total external force acting at the center-of-mass

7. Torque for Rotation and Translation The time derivative that appears in the second term in the above expression, the time derivative of the momentum of a mass element in the center-of mass-frame, is equal to the force acting on that element which include both inertial and fictitious forces, The torque about the center-of-mass is then

8. Torque for Rotation and Translation When we sum the torques over all the elements in the body, the fictitious forces act at the center-of-mass, so the torque from these fictitious forces is zero, so, the torque about the center-of-mass is only due to the forces as seen in the laboratory frame.

9. Rules to Live By: Angular Momentum and Torque • About any fixed point S • Independent of the CM motion, even if and are not parallel

10. Rules to Live By: Kinetic Energy of Rotation and Translation Change in kinetic energy of rotation about center-of-mass Change in rotational and translational kinetic energy

11. Rotational and Translational

12. Concept Question A cylinder is rolling without slipping down an inclined plane. The friction at the contact point P is • Static and points up the inclined plane. • Static and points down the inclined plane. • Kinetic and points up the inclined plane. • Kinetic and points down the inclined plane. • Zero because it is rolling without slipping.

13. Table Problem: Cylinder on Inclined Plane Torque About Center of Mass A hollow cylinder of outer radius R and mass m with moment of inertia I cm about the center of mass starts from rest and moves down an incline tilted at an angle q from the horizontal. The center of mass of the cylinder has dropped a vertical distance h when it reaches the bottom of the incline. Let g denote the gravitational constant. The coefficient of static friction between the cylinder and the surface is ms. The cylinder rolls without slipping down the incline. Using the torque method about the center of mass, calculate the velocity of the center of mass of the cylinder when it reaches the bottom of the incline.

14. Concept Question: Rotation and Translation

15. Concept Question: Rotation and Translation

16. Mini-Experiment: Pulling Spool • Which way does the yo-yo roll when you pull it horizontal? • Is there some angle at which you can pull the string in which the yo-yo doesn’t roll forward or back?

17. Table Problem: Descending and Ascending Yo-Yo A Yo-Yo of mass m has an axle of radius b and a spool of radius R. It’s moment of inertia about the center of mass can be taken to be I = (1/2)mR2 and the thickness of the string can be neglected. The Yo-Yo is released from rest. What is the acceleration of the Yo-Yo as it descends.

18. Concept Question: Constants of the Motion • A bowling ball is initially thrown down an alley with an initial speed v0, and it slides without rolling but due to friction it begins to roll until it rolls without slipping. What quantities are constant before it rolls without slipping? • Energy. • Angular momentum about the center of mass. • Angular momentum about a fixed point on the ground. • Three of the above. • Two of the above. • One of the above. • None of the above.

19. Problem: Bowling Ball Conservation of Angular Momentum Method A bowling ball of mass m and radius R is initially thrown down an alley with an initial speed v0, and it slides without rolling but due to friction it begins to roll. The moment of inertia of the ball about its center of mass is I = (2/5)mR2. By cleverly choosing a point about which to calculate the angular momentum, use conservation of angular momentum to find is the velocity of the center-of-mass when the wheel rolls without slipping.

20. Next Reading Assignment: W11D3 Young and Freedman: 10.1-10.6