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: $F<em>c = ma</em>c$

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.

• $I<em>i \omega</em>i = I<em>f \omega</em>f$, where $I<em>i$ and $\omega</em>i$ are the initial moment of inertia and angular velocity, and $I<em>f$ and $\omega</em>f$ are the final moment of inertia and angular velocity.