Angular Motion Notes

Angular Motion

• Measured in radians, not degrees.

• 2\pi radians in one rotation (360^\circ).

• Counterclockwise is typically positive.

• Angular displacement represented by \theta.

Angular Displacement

• Conversion to linear displacement: d = r\theta, where r is the radius and \theta is the angle.

Angular Velocity

• Rate of change of angular position.

• Represented by \omega (omega).

• \omega = \frac{\Delta\theta}{\Delta t}, measured in rad/s.

• Relationship to linear velocity: v = r\omega

Angular Acceleration

• Rate of change of angular velocity.

• Represented by \alpha.

• \alpha = \frac{\Delta\omega}{\Delta t}, measured in rad/s\textsuperscript{2}.

Centripetal Acceleration and Force

• Centripetal acceleration: a_c = \frac{v^2}{r}, where v is linear velocity.

• Centripetal force: Fc = mac

Tangential Acceleration

• Component of acceleration tangent to the circular path.

• a_t = r\alpha

Moment of Inertia

• Resistance to rotational motion.

• Depends on mass and distribution of mass around the axis of rotation.

• For a single particle: I = mr^2

• For multiple particles: I = \sum mr^2

Torque

• A force's tendency to cause rotation.

• \tau = rFsin\theta , where r is the distance from the axis of rotation to the point where the force is applied, F is the magnitude of the force, and \theta is the angle between the force vector and the lever arm.

• Also, \tau = I\alpha , where I is the moment of inertia and \alpha is the angular acceleration.

Rotational Kinetic Energy

• Energy due to rotational motion.

• KE_{rot} = \frac{1}{2}I\omega^2 , where I is the moment of inertia and \omega is the angular velocity.

Work and Power in Rotational Motion

• Work: W = \tau \theta , where \tau is torque and \theta is angular displacement.

• Power: P = \tau \omega , where \tau is torque and \omega is angular velocity.

Angular Momentum

• Measure of the extent to which an object will continue to rotate.

• For a single particle: L = r \times p = r m v sin\theta , where r is the position vector, p is the momentum, m is the mass, v is the velocity, and \theta is the angle between r and p.

• For a rigid object: L = I \omega , where I is the moment of inertia and \omega is the angular velocity.

Conservation of Angular Momentum

• If the net external torque on a system is zero, the total angular momentum of the system remains constant.

• Ii \omegai = If \omegaf , where Ii and \omegai are the initial moment of inertia and angular velocity, and If and \omegaf are the final moment of inertia and angular velocity.