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Chapter 9. Rotation of Rigid Bodies. Goals for Chapter 9. To describe rotation in terms of angular coordinate, angular velocity, and angular acceleration To analyze rotation with constant angular acceleration

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## Chapter 9

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**Chapter 9**Rotation of Rigid Bodies**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**Introduction**• 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.**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.**Units of angles**• An angle in radians is = s/r, as shown in the figure. • One complete revolution is 360° = 2π radians.**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.**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 direction is given by the right-hand rule shown in Figure 9.5 below.**Angular acceleration**• The average angular acceleration is av-z = z/t. • The instantaneous angular acceleration is z = dz/dt = d2/dt2. • Follow Example 9.2.**Angular acceleration as a vector**• For a fixed rotation axis, the angular acceleration and angular velocity vectors both lie along that axis.**Rotation with constant angular acceleration**• The rotational formulas have the same form as the straight-line formulas, as shown in Table 9.1 below.**Rotation of a Blu-ray disc**• A Blu-ray disc is coming to rest after being played. • Follow Example 9.3 using Figure 9.8 as shown at the right.**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 = v2/r = r**An athlete throwing a discus**• Follow Example 9.4 and Figure 9.12.**Designing a propeller**• Follow Example 9.5 and Figure 9.13.**Rotational kinetic energy**• The moment of inertia of a set of particles is • I = m1r12 + m2r22 + … = miri2 • The rotational kinetic energy of a rigid body having a moment of inertia I is K = 1/2 I2. • Follow Example 9.6 using Figure 9.15 below.**Moments of inertia of some common bodies**• Table 9.2 gives the moments of inertia of various bodies.**An unwinding cable**• Follow Problem-Solving Strategy 9.1. • Follow Example 9.7.**More on an unwinding cable**• Follow Example 9.8 using Figure 9.17 below.**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.**The parallel-axis theorem**• The parallel-axis theorem is: IP = Icm + Md2. • Follow Example 9.9 using Figure 9.20 below.**Moment of inertia of a hollow or solid cylinder**• Follow Example 9.10 using Figure 9.22.**Moment of inertia of a uniform solid sphere**• Follow Example 9.11 using Figure 9.23.

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