Proof of Absolute Directions
John David Best
Angular velocity, ω, is defined as the rate of change of angular displacement, where angular displacement is the angle formed by a reference direction, the axis of rotation, and a line between the axis of rotation and a rotating point.
The relationship between angular velocity and the amount of centripetal force it generates at radius R is Fc = mRω2 (eq 1). This is a physical law that is thought to be valid anywhere in the universe. This means that for a given mass, all rotations with the same angular velocity will produce the same centripetal force at the same radius, anywhere in the universe. This requires that the reference direction for any rotation be rotationally stationary with respect to the reference direction for any other rotation with a parallel axis of rotation. Otherwise, angular velocities that are measured as being the same would differ when compared to each other, as would the centripetal force they produce (see addendum below for additional discussion of this point). In fact, the reference direction must be the same for all rotations with parallel axes of rotation. If the reference direction were to vary for rotations with parallel axes separated by a lineal distance, then translational motion of an axis of rotation would cause its reference direction to rotate, which it cannot, as previously discussed.
The reference direction that all rotations with parallel axes share, cannot be represented by a single point. The direction to a point varies, depending on the location of the rotating object, so translational motion would cause the reference direction to rotate, which, as discussed above, cannot happen. This can be seen in the diagram below:
In order for the relationship between angular velocity and centripetal force (eq 1) to be true anywhere, the reference directions for all rotations with parallel axes, must be parallel to each other, as shown in the next diagram:
This situation can be accurately represented by the three-dimensional Cartesian coordinate system, with the reference directions always being parallel to one of the three coordinate axes, for rotations whose planes are parallel to said coordinate axis. By placing a coordinate axis so that it is parallel to a reference direction, the three perpendicular axes of the Cartesian coordinate system can represent the reference directions for three perpendicular planes of rotation, with each coordinate axis representing the reference direction for all rotations whose axes are perpendicular to it.
Addendum: An examination of the dependence of centripetal force on velocity:
First, consider an illustrative scenario –
A string, maybe a meter long, has one end attached to a small rock, and the other end attached to the shaft of a motor standing so the shaft points upwards. When the motor is turned off, the string hangs limp. When the motor runs, the rock and string swing around in a circle like the blades of a helicopter. The motor is mounted inside a windowless room that sits atop a giant turntable similar to a merry-go-round, so that the entire room can be made to spin. If the motor is turned off so that the motor shaft is motionless with respect to its housing; but the turntable supporting the room is made to rotate at the same velocity as the motor does when it runs, there will be the same tension in the string, and the rock will swing around in the same manner as it did when being spun by the motor, but now the string and rock are motionless relative to everything inside the room including the motor housing, since the entire room is spinning along with the string and rock on the same axis. The angular velocity of the motor shaft that causes the tension in the string, cannot be measured with reference to the motor housing or anything inside the room. Nor necessarily can it be relative to the base of the turntable supporting the room, because that turntable could be mounted on a still larger turntable. Whatever is used as the reference direction would also have to serve for the rotation of objects in any location in space, if the relation between centripetal force and angular velocity holds true equally everywhere.