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## Rotational Kinematics

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**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.