2. Rotational motion, Angular displacement, angular velocity, angular acceleration Rotational energy Moment of Inertia (Rotational inertia) Torque For every rotational quantity, there is a linear analog. Chapter 10:Rotation of a rigid object about a fixed axis Part 2 Reading assignment: Chapter 11.1-11.3 Homework :(due Wednesday, Oct. 12, 2005): Problems:Q4, 2, 5, 18, 21, 23, 24,

3. Black board example 11.3 HW 27 (a)What is the angular speed  about the polar axis of a point on Earth’s surface at a latitude of 40°N (b)What is the linear speed v of that point? (c)What are  and v for a point on the equator? Radius of earth: 6370 km

4. Rotational energy A rotating object (collection of i points with mass m i ) has a rotational ___________ energy of Where: Rotational inertia

5. Demo: Both sticks have the same weight. Why is it so much more difficult to rotate the blue stick?

6. Four small spheres are mounted on the corners of a frame as shown. a)What is the rotational energy of the system if it is rotated about the z-axis (out of page) with an angular velocity of 5 rad/s b)What is the rotational energy if the system is rotated about the y- axis? (M = 5 kg; m = 2 kg; a = 1.5 m; b = 1 m). Black board example 11.4 What is the rotational inertia? 1 3 2 4

7. Rotational inertia of an object depends on: - the ________ about which the object is rotated. - the __________ of the object. - the __________ between the mass(es) and the axis of rotation.

8. Calculation of Rotational inertia for ____________ ________________ objects Refer to Table11-2 Note that the moments of inertia are different for different ________ of rotation (even for the same object)

9. Rotational inertia for some objects Page 227

10. Parallel axis theorem  Rotational inertia for a rotation about an axis that is ____________ to an axis through the center of mass h What is the rotational energy of a sphere (mass m = 1 kg, radius R = 1m) that is rotating about an axis 0.5 away from the center with  = 2 rad/sec? Blackboard example 11.4

11. Conservation of energy (including rotational energy): Again: If there are no ___________________ forces: Energy is conserved. Rotational _____________ energy must be included in energy considerations!

12. Connected cylinders. Two masses m 1 (5 kg) and m 2 (10 kg) are hanging from a pulley of mass M (3 kg) and radius R (0.1 m), as shown. There is no slip between the rope and the pulleys. (a)What will happen when the masses are released? (b)Find the velocity of the masses after they have fallen a distance of 0.5 m. (c)What is the angular velocity of the pulley at that moment? Black board example 11.5

13. Torque A force F is acting at an angle  on a lever that is rotating around a pivot point. r is the ______________ between F and the pivot point. This __________________ pair results in a torque  on the lever 

14. Black board example 11.6 Two mechanics are trying to open a rusty screw on a ship with a big ol’ wrench. One pulls at the end of the wrench (r = 1 m) with a force F = 500 N at an angle   = 80 °; the other pulls at the middle of wrench with the same force and at an angle   = 90 °. What is the net torque the two mechanics are applying to the screw?

15. Particle of mass m rotating in a circle with radius r. force F r to keep particle on circular path. force F t accelerates particle along tangent. Torque  and angular acceleration  Newton’s __________ law for rotation. Torque acting on particle is ________________ to angular acceleration  :

16. Work in rotational motion: Definition of work: Work in linear motion: Component of force F along displacement s. Angle  between F and s. Torque  and angular displacement .

17. Work and Energy in rotational motion Remember work-kinetic energy theorem for linear motion: There is an equivalent work-rotational kinetic energy theorem: External work done on an object changes its __________ energy External, rotational work done on an object changes its _______________energy

18. Linear motion with constant linear acceleration, a. Rotational motion with constant rotational acceleration, 

19. Summary: Angular and linear quantities Kinetic Energy: Torque: Linear motion Rotational motion Kinetic Energy: Force: Momentum:Angular Momentum: Work: