Angular Position, Velocity, and Acceleration Degrees and revolutions:
Angular Position, Velocity, and Acceleration Arc length s, measured in radians:
Rotational Kinematics If the angular acceleration is constant:
Rotational Kinematics Analogies between linear and rotational kinematics:
Connections Between Linear and Rotational Quantities This merry-go-round has both tangential and centripetal acceleration.
10-4 Rolling Motion If a round object rolls without slipping, there is a fixed relationship between the translational and rotational speeds:
10-4 Rolling Motion We may also consider rolling motion to be a combination of pure rotational and pure translational motion:
Torque From experience, we know that the same force will be much more effective at rotating an object such as a nut or a door if our hand is not too close to the axis. This is why we have long-handled wrenches, and why doorknobs are not next to hinges.
Torque We define a quantity called torque: The torque increases as the force increases, and also as the distance increases. Note:has the same unit (N . M) as work but it is a very different thing!
Torque Only the tangential component of force causes a torque:
Torque This leads to a more general definition of torque:
Torque If the torque causes a counterclockwise angular acceleration, it is positive; if it causes a clockwise angular acceleration, it is negative.
Rotational Kinetic Energy and the Moment of Inertia For this mass,
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
Rotational Kinetic Energy and the Moment of Inertia Moments of inertia of various regular objects can be calculated:
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.
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.
Summary • Describing rotational motion requires analogs to position, velocity, and acceleration • Average and instantaneous angular velocity: • Average and instantaneous angular acceleration:
Summary • Period: • Counterclockwise rotations are positive, clockwise negative • Linear and angular quantities:
Summary • Linear and angular equations of motion: Tangential speed: Centripetal acceleration: Tangential acceleration:
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.