Angular Variables and Their Applications

Angular Variables
Rotational Motion

  • Restrictions: Considers only rigid objects with no radial motion, focusing solely on rotation.

  • Arc Length: S = r \theta , where S is arc length, r is radius, and \theta is angular position in radians.

  • Angular Position: All points on a rotating body sweep the same angle \theta, though their arc lengths (S) differ based on radius (r).

  • Conversions: 1 \text{ [rev]} = 2\pi \text{ [rad]} and 360\text{ [deg]} = 2\pi\text{ [rad]} .

Angular Velocity

  • Average: \omega = \frac{\Delta \theta}{\Delta t} (units: [rad/s]).

  • Instantaneous: \omega = \lim_{\Delta t \to 0} \frac{\Delta \theta}{\Delta t}. All points in a rigid body share the same angular velocity.

Angular Acceleration

  • Average: \alpha = \frac{\Delta \omega}{\Delta t} (where \Delta \omega = \omegaf - \omegai).

  • Instantaneous: limΔt→0Δt/Δω​

  • Coordinate Convention: Counterclockwise (ccw) rotation is considered positive.

Relating Linear and Angular Quantities

  • Summary of Relations for rigid bodies:

    1. Displacement: s = r\theta

    2. Velocity (tangential): v = r\omega

    3. Acceleration (tangential): a_{tan} = r\alpha

Polar Coordinates & Acceleration

  • Centripetal Acceleration (aR): Directed inward, computed as aR =(−v^2/r)​R^ (radial component).

  • Total Acceleration (\mathbf{a}): Combination of radial and tangential components: a=aRR^+atanθ^

Angular Frequency

  • Frequency (f): Number of revolutions per unit time.

  • Period (T): Time for one complete revolution; T = \frac{1}{f} .

Rolling Motion (Without Sliding)

  • The distance traveled by the center of mass equals the arc length touched by the wheel on the ground.

Torque

  • Definition: \tau = rF\sin(\theta). It's a force applied at a distance from an axis, causing a rotational effect.

  • How Torques Add: For a point mass, the sum of torques leads to rotational acceleration: \Sigma \tau = I\alpha.

Moment of Inertia (I)

  • Represents an object's resistance to changes in its angular velocity: I=Σmiri²

  • Examples: Solid disk (I = \frac{1}{2} m r^2), thin spherical shell (I = \frac{2}{3} m r^2).

  • Parallel Axis Theorem: I = I_{cm} + mh^2 (relates moment of inertia about any axis to that about the center of mass).

Conservation of Angular Momentum

  • In isolated systems where the net external torque is zero, angular momentum is conserved: L{initial} = L{final}.