**Chapter 10:Rotation of a rigid object about a fixed axis** Reading assignment: Chapter 10.1 to10.4, 10.5 (know concept of moment of inertia, don’t worry about integral calculation), 10.6 to 10.9 Homework 10.1 (due Tuesday, Oct. 23):CQ1, CQ2, AE1, 2, 3, 6, 7, 12 Homework 10.2 (due Wednesday, Oct. 24): CQ8, QQ3, QQ4, OQ3, OQ4, OQ6, OQ8, AE3, 13, 15, 19, 26, 29 Homework 10.3 (due Friday, Oct. 26): CQ13, 35, 36, 38, 43, 49, 55, 56, 59 • Rotational motion • Angular displacement, angular velocity, angular acceleration • Rotational energy • Moment of Inertia • Torque

**Rotational motion** Look at one point P: Arc length s: Thus the angle (angular position) is: Planar, rigid object rotating about origin O. q is measured in degrees or radians (SI unit: radian) Full circle has an angle of 2p radians. Thus, one radian is 360°/2p = 57.3 Radian degrees 2p 360° p 180° p/2 90° 1 57.3°

**Define quantities for circular motion** (note analogies to linear motion!!) Angular displacement: Average angular speed: Instantaneous angular speed: Average angular acceleration: Instantaneous angular acceleration:

**Angular velocity is a vector** Right-hand rule for determining the direction of this vector. Every particle (of a rigid object): • rotates through the same angle, • has the same angular velocity, • has the same angular acceleration. q, w, a characterize rotational motion of entire object

**Rotational motion with constant rotational acceleration, a.** Linear motion with constant linear acceleration, a. Exactly the same equations, just different symbols!!

**Black board example 10.1** A wheel starts from rest and rotates with constant angular acceleration and reaches an angular speed of 12.0 rad/s in 3.00 s. 1. What is the magnitude of the angular acceleration of the wheel (in rad/s2)? A. 0 B. 1 C. 2 D. 3 E. 4 2. Through what angle does the wheel rotate in these 3 sec (in rad)? A. 18 B. 24 C. 30 D. 36 E. 48 3. Through what angle does the wheel rotate between 2 and 3 sec (in rad)? A. 5 B. 10 C. 15 D. 20 E. 25

**Relation between angular and linear quantities** Arc length s: Tangential speed of a point P: Tangential acceleration of a point P: Note: This is not the centripetal acceleration ar This is the tangential acceleration at

**Black board example 10.2** vt • A fly is sitting at the end of a ceiling fan blade. The length of the blade is 0.50 m and it spins with 40.0 rev/min. • Calculate the (tangential) speed of the fly. • What are the tangential and angular speeds of another fly sitting half way in? • Starting from rest it takes the motor 20 seconds to reach this speed. What is the angular acceleration? • At the final speed, with what force does the fly (m = 0.01 kg, r = 0.50 m) need to hold on, so that it won’t fall off? • (Note difference between angular and centripetal acceleration).

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

**Rotational kinetic energy** A rotating object (collection of i points with mass mi) has a rotational kinetic energy of Where: Moment of inertia or rotational inertia

**Black board example 10.3** i-clicker 2 • Four small spheres are mounted on the corners of a weightless frame as shown. • M = 5 kg; m = 2 kg; • a = 1.5 m; b = 1 m 1 3 4 • 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? • i-clicker for question b): • A) 281 J B) 291 J C) 331 J D) 491 J E) 582 J

**Moment of inertia (rotational inertia) of an object depends** on: • the axis about which the object is rotated. • the mass of the object. • the distance between the mass(es) and the axis of rotation.

**Calculation of Moments of inertia for continuous extended** objects Refer to Table10.2 Note that the moments of inertia are different for different axes of rotation (even for the same object)

**Moment of inertia for some objects** Page 287

**Black board example 10.4** • Rotational energy earth. • The earth has a mass M = 6.0×1024 kg and a radius of R = 6.4×106 m. Its distance from the sun is d = 1.5×1011 m What is the rotational kinetic energy of • its motion around the sun? • its rotation about its own axis?

**Parallel axis theorem** Rotational inertia for a rotation about an axis that is parallel to an axis through the center of mass h Blackboard example 10.5 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?

**Conservation of energy (including rotational energy):** Again: If there are no non-conservative forces energy is conserved. Rotational kinetic energy must be included in energy considerations!

**Black board example 10.6** Connected cylinders. • Two masses m1 (5.0 kg) and m2 (10 kg) are hanging from a pulley of mass M (3.0 kg) and radius R (0.10 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?

**Torque** f A force F is acting at an angle f on a lever that is rotating around a pivot point. r is the distance between the pivot point and F. This force-lever pair results in a torque t on the lever

**Black board example 10.7** i-clicker 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? A. 742 Nm B. 750 Nm C. 900 Nm D. 1040 Nm E. 1051 Nm

**Torque tand ** angular acceleration a. Particle of mass m rotating in a circle with radius r. Radial force Fr to keep particle on circular path. Tangential force Ft accelerates particle along tangent. Torque acting on particle is proportional to angular acceleration a:

**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.

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

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

**Rolling motion** Superposition principle: Rolling motion = Pure translation + Pure rotation Kinetic energy of rolling motion:

**Black board example 10.8** Demo A ring, a disk and a sphere (equalmass and diameter) are rolling down an incline. All three start at the same position; which one will be the fastest at the end of the incline? • All the same • The disk • The ring • The sphere