The angular position of this line is the angle of the line, relative to the reference line, which is fixed in the body, perpendicular to the rotation axis and rotating with the body. This is taken as the zero angular position.
The angular position θ is measured relative to the positive direction of the x-axis. From geometry θ = s/r (radians) rev or degree no dimensions is the length of a circular arc that extends from the x-axis (the zero angular position) to the reference line, and is the radius of the circle.
The circumference of a circle of radius r is 2πr, there are 2πradians in a complete circle: 1 rev = 360° = 2πr/r = 2πrad. 1 rad = 57.3= 0.159 rev.
Note:
Θ is never reset to zero with each complete rotation of the reference line about the rotation axis. If the reference line completes two revolutions from the zero angular position, then the angular position θ of the line is θ = 4πrad.
For a translation along x-axis, we can know all that we need to know if we know x(t),
Similarly, for pure rotation, we can know all there is to know about a rotating body if we know θ(t), the angular position of the body’s reference line as a function of time.
Angular Displacement
If a body rotates about the rotation axis as indicated, changing the angular position of the reference line from θ1 to θ2, the body undergoes an angular displacement Δθ given by Δθ = θ2–θ1
This definition of angular displacement holds not only for the rigid body as a whole but also for every particle within that body.
The angular displacement u of a rotating body is either +veor -ve, according to the following rule:
An angular displacement in the counterclockwise direction is positive, and one in the clockwise direction is negative.
Angular Velocity
Suppose that our rotating body is at angular position θ1 at time t1 and at angular position θ2 at time t2, the average angular velocity of the body in the time interval Δt from t1 to t2 is ωavg = (θ2–θ1)/(t2 -t1) = Δθ/Δt
where Δθ is the angular displacement during Δt.
We shall be most concerned with the (instantaneous) angular displacement, ω, which is the limit of ω = limΔθ/Δt = dθ/dt(rad/s or rev/s) as Δt approaches zero.