Chapter 9 Rotation of Rigid Bodies Power Point

Chapter 9 Rotation of Rigid Bodies Power. Point® Lectures for University Physics, Thirteenth Edition – Hugh D. Young and Roger A. Freedman Lectures by Wayne Anderson Copyright © 2012 Pearson Education Inc. Modifications by Mike Brotherton

Goals for Chapter 9 • To describe rotation in terms of angular coordinate, angular velocity, and angular acceleration • To analyze rotation with constant angular acceleration • To relate rotation to the linear velocity and linear acceleration of a point on a body • To understand moment of inertia and how it relates to rotational kinetic energy • To calculate moment of inertia Copyright © 2012 Pearson Education Inc.

Uniform circular motion—Figure 3. 27 • For uniform circular motion, the speed is constant

Acceleration for uniform circular motion • For uniform circular motion, the instantaneous acceleration always points toward the center of the circle and is called the centripetal acceleration. • The magnitude of the acceleration is arad = v 2/R. • The period T is the time for one revolution, and arad = 4π2 R/T 2. Copyright © 2012 Pearson Education Inc.

Nonuniform circular motion—Figure 3. 30 • If the speed varies, the motion is nonuniform circular motion. • The radial acceleration component is still arad = v 2/R, but there is also a tangential acceleration component atan that is parallel to the instantaneous velocity. Copyright © 2012 Pearson Education Inc.

Introduction – Rigid Rotating Bodies • A wind turbine, a CD, a ceiling fan, and a Ferris wheel all involve rotating rigid objects. • Real-world rotations can be very complicated because of stretching and twisting of the rotating body. But for now we’ll assume that the rotating body is perfectly rigid. Copyright © 2012 Pearson Education Inc.

Angular coordinate • A car’s speedometer needle rotates about a fixed axis, as shown at the right. • The angle that the needle makes with the +x-axis is a coordinate for rotation. Copyright © 2012 Pearson Education Inc.

Units of angles • An angle in radians is = s/r, as shown in

Angular velocity • The angular displacement of a body is = 2 – 1. • The average angular velocity of a body is av-z = / t. • The subscript z means that the rotation is about the z-axis. • The instantaneous angular velocity is z = d /dt. • A counterclockwise rotation is positive; a clockwise rotation is negative. Copyright © 2012 Pearson Education Inc.

Calculating angular velocity • We first investigate a flywheel. • Follow Example 9. 1.

Angular velocity is a vector • Angular velocity is defined as a vector whose

Angular acceleration • The average angular acceleration is av-z = z/ t. • The instantaneous angular acceleration is z = d z/dt = d 2 /dt 2. • Follow Example 9. 2. Copyright © 2012 Pearson Education Inc.

Angular acceleration as a vector • For a fixed rotation axis, the angular acceleration

Rotation with constant angular acceleration • The rotational formulas have the same form as

Relating linear and angular kinematics • For a point a distance r from the axis of rotation: its linear speed is v = r its tangential acceleration is atan = r its centripetal (radial) acceleration is arad = v 2/r = r Copyright © 2012 Pearson Education Inc.

An athlete throwing a discus • Follow Example 9. 4 and Figure 9. 12.

Rotational kinetic energy • The moment of inertia of a set of particles is I = m 1 r 12 + m 2 r 22 + … = miri 2 • The rotational kinetic energy of a rigid body having a moment of inertia I is K = 1/2 I 2. • Follow Example 9. 6 using Figure 9. 15 below. Copyright © 2012 Pearson Education Inc.

Moments of inertia of some common bodies • Table 9. 2 gives the moments

An unwinding cable • Follow Example 9. 7. Copyright © 2012 Pearson Education Inc.

More on an unwinding cable • Follow Example 9. 8 using Figure 9. 17

Gravitational potential energy of an extended body • The gravitational potential energy of an extended body is the same as if all the mass were concentrated at its center of mass: Ugrav = Mgycm. Copyright © 2012 Pearson Education Inc.