PHYSICS: Rotational Kinematics

ANGULAR POSITION

By convention, we measure angles in a circle in a counterclockwise direction from the positive x-axis. The angular position of a particle is the angle, Φ, made between the lines connecting that particle to the origin, O, and the positive x-axis, measured counterclockwise.

In this figure, point P has an angular position of Φ. Note that every point on the line OP has the same angular position: the angular position of a point does not depend on how far that point is from the origin, O.

We can relate the angular position of P to the length of the arc of the circle between P and the x-axis by means of an easy equation: Φ = L/r; where L = length of the arc, and r = radius of the circle.

ANGULAR DISPLACEMENT

Imagine that the wheel is rotated so that every point on the line OP moves from an initial angular position of Φ_i to a final angular position of Φ_f. The angular displacement, θ, of line OP is: θ = Φ_f – Φ_i

For example, if you rotate a wheel counterclockwise such that the angular position of line OP changes from Φ_i = 45^o = π/4 to Φ_f = 135^o = 3π/4, as illustrated below, then the angular displacement of line OP is 90^o or π/2radians.

For line OP, to move in the way described above, every point along the line must rotate 90^o counterclockwise. By definition, the particles that make up a rigid body must stay in the same relative position to one another. As a result, the angular displacement is the same froe every point in a rotating rigid body.

Note: The angular distance a point has rotated may or may not equal that point’s angular displacement.

ANGULAR VELOCITY, is defined as the change in the angular displacement over time. Average angular velocity, is defined by:

Angular velocity is typically given in units of rad/s. As with angular displacement, the angular velocity of every point on a rotating object is identical.

ANGULAR ACCELERATION, a, is defined as the rate of change of angular velocity over time. Average angular acceleration, ā, is defined by:

Angular acceleration is typically given in units of rad/s².

Sources: www.sparknotes.com/