1. 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, • 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.

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

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

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

5. Black board example 11.4 2 What is the rotational inertia? 3 1 4 • Four small spheres are mounted on the corners of a frame as shown. • 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 • 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).

6. 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.

7. 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)

8. Rotational inertia for some objects Page 227

9. Parallel axis theorem  Rotational inertia for a rotation about an axis that is ____________ to an axis through the center of mass h Blackboard example 11.4 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 w = 2 rad/sec?

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

11. Black board example 11.5 Connected cylinders. • Two masses m1 (5 kg) and m2 (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. • What will happen when the masses are released? • Find the velocity of the masses after they have fallen a distance of 0.5 m. • What is the angular velocity of the pulley at that moment?

12. Torque f A force F is acting at an angle f 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 t on the lever

13. 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 F1 = 80 °; the other pulls at the middle of wrench with the same force and at an angle F2 = 90 °. What is the net torque the two mechanics are applying to the screw?

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

15. Definition of work: Work in linear motion: Component of force F along displacement s. Angle g between F and s. Work in rotational motion: Torque t and angular displacement q.

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

17. Linear motion with constant linear acceleration, a. Rotational motion with constant rotational acceleration, a.

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