Chapter 10
Copyright © 2010 Pearson Education, Inc. ROTATIONAL KINEMATICS AND ENERGY Chapter 10 Copyright © 2010 Pearson Education, Inc. Units of Chapter 10 • Angular Position, Velocity, and Acceleration • Rotational Kinematics • Connections Between Linear and Rotational Quantities • Rolling Motion • Rotational Kinetic Energy and the Moment of Inertia • Conservation of Energy Copyright © 2010 Pearson Education, Inc. Angular Position, Velocity, and Acceleration 10-1 Copyright © 2010 Pearson Education, Inc. Angular Position, Velocity, and Acceleration 10-1 Degrees and revolutions: Copyright © 2010 Pearson Education, Inc. Angular Position, Velocity, and Acceleration 10-1 • Arc length s, measured in radians: • Radian: angle for which the arc length on a circle of radius r is equal to the radius of the circle Copyright © 2010 Pearson Education, Inc. Angular Position, Velocity, and Acceleration 10-1 Copyright © 2010 Pearson Education, Inc. Angular Position, Velocity, and Acceleration 10-1 Copyright © 2010 Pearson Education, Inc. Angular Position, Velocity, and Acceleration 10-1 Copyright © 2010 Pearson Education, Inc. Angular Position, Velocity, and Acceleration 10-1 Copyright © 2010 Pearson Education, Inc. 10-2 Rotational Kinematics If the angular acceleration is constant: Copyright © 2010 Pearson Education, Inc. 10-2 Rotational Kinematics Analogies between linear and rotational kinematics: Copyright © 2010 Pearson Education, Inc. Example A high speed dental drill is rotating at 3.14×104 rads/sec. Through how many degrees does the drill rotate in 1.00 sec? Copyright © 2010 Pearson Education, Inc. Connections Between Linear and Rotational Quantities 10-3 Copyright © 2010 Pearson Education, Inc. Connections Between Linear and Rotational Quantities 10-3 Copyright © 2010 Pearson Education, Inc. Connections Between Linear and Rotational Quantities 10-3 Copyright © 2010 Pearson Education, Inc. Connections Between Linear and Rotational Quantities 10-3 • This merry-go-round has both tangential and centripetal acceleration. Copyright © 2010 Pearson Education, Inc. 10-4 Rolling motion • If a round object rolls without slipping, there is a fixed relationship between the translational and rotational speeds: Copyright © 2010 Pearson Education, Inc. 10-4 Rolling motion • We may also consider rolling motion to be a combination of pure rotational and pure translational motion: Copyright © 2010 Pearson Education, Inc. Rotational Kinetic Energy and the Moment of Inertia 10-5 • For this mass, Copyright © 2010 Pearson Education, Inc. Rotational Kinetic Energy and the Moment of Inertia 10-5 • We can also write the kinetic energy as • Where I, the moment of inertia, is given by Copyright © 2010 Pearson Education, Inc. Example (a) Find the moment of inertia of the system below. The masses are m1 and m2 and they are separated by a distance r. Assume the rod connecting the masses is massless. Copyright © 2010 Pearson Education, Inc. Example continued Copyright © 2010 Pearson Education, Inc. Rotational Kinetic Energy and the Moment of Inertia 10-5 • Moments of inertia of various regular objects can be calculated: Copyright © 2010 Pearson Education, Inc. Example • What is the rotational inertia of a solid iron disk of mass 49.0 kg with a thickness of 5.00 cm and a radius of 20.0 cm, about an axis through its center and perpendicular to it? Copyright © 2010 Pearson Education, Inc. 10-6 Conservation of energy • The total kinetic energy of a rolling object is the sum of its linear and rotational kinetic energies: • The second equation makes it clear that the kinetic energy of a rolling object is a multiple of the kinetic energy of translation. Copyright © 2010 Pearson Education, Inc. 10-6 Conservation of energy • If these two objects, of the same mass and radius, are released simultaneously, the disk will reach the bottom first – more of its gravitational potential energy becomes translational kinetic energy, and less rotational. Copyright © 2010 Pearson Education, Inc. Example • Two objects (a solid disk and a solid sphere) are rolling down a ramp. Both objects start from rest and from the same height. Which object reaches the bottom of the ramp first? Copyright © 2010 Pearson Education, Inc. Example continued Copyright © 2010 Pearson Education, Inc. Example continued Copyright © 2010 Pearson Education, Inc. Summary of Chapter 10 • Describing rotational motion requires analogs to position, velocity, and acceleration • Average and instantaneous angular velocity: • Average and instantaneous angular acceleration: Copyright © 2010 Pearson Education, Inc. Summary of Chapter 10 • Period: • Counterclockwise rotations are positive, clockwise negative • Linear and angular quantities: Copyright © 2010 Pearson Education, Inc. Summary of Chapter 10 • Linear and angular equations of motion: • Tangential speed: • Centripetal acceleration: • Tangential acceleration: Copyright © 2010 Pearson Education, Inc. Summary of Chapter 10 • Rolling motion: • Kinetic energy of rotation: • Moment of inertia: • Kinetic energy of an object rolling without slipping: • When solving problems involving conservation of energy, both the rotational and linear kinetic energy must be taken into account.
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