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Rotational Kinematics and Energy

Rotational Kinematics and Energy
Chapter 10 Rotational Kinematics and EnergyUnits 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 (Not required)101 Angular Position, Velocity, and Acceleration101 Angular Position, Velocity, and Acceleration Degrees and revolutions:101 Angular Position, Velocity, and Acceleration Arc length s, measured in radians:101 Angular Position, Velocity, and Acceleration101 Angular Position, Velocity, and Acceleration101 Angular Position, Velocity, and Acceleration101 Angular Position, Velocity, and Acceleration102 Rotational Kinematics If the angular acceleration is constant:102 Rotational Kinematics Analogies between linear and rotational kinematics:103 Connections Between Linear and Rotational Quantities103 Connections Between Linear and Rotational Quantities103 Connections Between Linear and Rotational Quantities103 Connections Between Linear and Rotational Quantities This merrygoround has both tangential and centripetal acceleration.104 Rolling Motion If a round object rolls without slipping, there is a fixed relationship between the translational and rotational speeds:104 Rolling Motion We may also consider rolling motion to be a combination of pure rotational and pure translational motion:105 Rotational Kinetic Energy and the Moment of Inertia For this mass,105 Rotational Kinetic Energy and the Moment of Inertia We can also write the kinetic energy as Where I, the moment of inertia, is given by Use this equation to calculate I, for a few point masses. 105 Rotational Kinetic Energy and the Moment of Inertia Moments of inertia of various regular shapes can be calculated using these known results:Summary of Chapter 10 • Describing rotational motion requires analogs to position, velocity, and acceleration • Average and instantaneous angular velocity: • Average and instantaneous angular acceleration:Summary of Chapter 10 • Period: • Counterclockwise rotations are positive, clockwise negative • Linear and angular quantities:Summary of Chapter 10 • Linear and angular equations of motion: Tangential speed: Centripetal acceleration: Tangential acceleration:Summary of Chapter 10 • Rolling motion: • Kinetic energy of rotation: • Moment of inertia for a few points: • Moment of inertia for different shapes…… • Kinetic energy of an object rolling without slipping: • Optional: When solving problems involving conservation of energy, both the rotational and linear kinetic energy must be taken into account.106 Conservation of Energy Optional Not Required 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.106 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.