Revolutions Per Minute (RPM) is a term that, as the name suggests, denotes the number of full rotations an object makes in one minute. This metric is widespread in engineering, especially with machinery like car engines.

To correlate RPM with frequency (number of rotations per second), you simply divide the RPM by 60, because there are 60 seconds in a minute. Thus, frequency = RPM/60.

Angular velocity, on the other hand, tells us how much angle (in radians) an object sweeps out in one second. Since there are 2π radians in one full rotation, angular velocity, ω, is calculated as 2π times the frequency. Therefore, to determine the angular velocity using RPM: angular velocity, ω = 2π(RPM/60).

The choice of radians over degrees is deeply rooted in the geometry and calculus of circles. A radian is defined by the arc length that is equal to the radius of the circle. Using radians streamlines many mathematical processes.

For instance, when you're working with trigonometric functions in calculus, having angle measurements in radians simplifies the equations and derivations. Using radians ensures that the relationship between arc length, angle in radians, and radius (arc length = angle x radius) holds true. Because of this intrinsic relationship between radians and circle geometry, radians per second becomes a more natural choice for measuring angular velocity in physics.

Absolutely, an object in circular motion can experience changes in its angular velocity. When an object's angular velocity isn't consistent, it's undergoing what's known as angular acceleration. Just as linear acceleration is a change in linear velocity over time, angular acceleration represents the rate of change of angular velocity.

For example, when you spin a top, it doesn't instantly reach its full speed. It starts from rest and its spin or angular velocity increases until external forces like friction slow it down. The change in its spin rate over time is due to angular acceleration.

The radius is a crucial element in the relationship between linear and angular velocities in circular motion. Angular velocity signifies how swiftly an object revolves around a central axis. In contrast, linear velocity quantifies the actual speed of a particular point on that object as it moves along its circular path.

The connection between these two types of velocities is given by the formula: v = ω * r. Here, v represents the linear velocity, ω stands for the angular velocity, and r is the radius. If you keep the angular velocity constant but change the radius, the linear velocity will change. A point on a larger radius will cover a greater distance in the same amount of time, thereby resulting in a greater linear velocity. This underlines the pivotal role the radius plays in determining linear velocity when the angular velocity is known.

In a circular motion scenario, angular velocity refers to how quickly an object rotates around a central point. Imagine a spinning CD or record; every point on the CD (from the centre to the edge) completes one rotation in the same amount of time. Thus, all these points have the same angular velocity.

However, linear velocity describes the speed at which a point travels along a circular path. The further out from the centre you go, the longer the path or circumference becomes. Hence, for points to complete their respective paths in the same amount of time, those further from the centre must travel faster. This is why they have a higher linear velocity. The relationship can be described with the equation: v = ω * r, where v is linear velocity, ω is angular velocity, and r is the radius.