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Rotational Kinematics

Rotational Kinematics. Angular Position, Velocity, and Acceleration

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Rotational Kinematics

Angular Position, Velocity, and Acceleration

Angular Position, Velocity, and Acceleration

Degrees and revolutions:

Angular Position, Velocity, and Acceleration

Arc length s, measured in radians:

10-1 Angular Position, Velocity, and Acceleration

Angular Position, Velocity, and Acceleration

Angular Position, Velocity, and Acceleration

Angular Position, Velocity, and Acceleration

Rotational Kinematics

If the angular acceleration is constant:

Rotational Kinematics

Analogies between linear and rotational kinematics:

Connections Between Linear and Rotational Quantities

Connections Between Linear and Rotational Quantities

Connections Between Linear and Rotational Quantities

Connections Between Linear and Rotational Quantities

This merry-go-round has both tangential and centripetal acceleration.

10-4 Rolling MotionIf 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.