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Other thinkers suggest that pure logic implies only relative rotation makes sense. For example, Bishop Berkeley and Ernst Mach (among others) suggested that it is relative rotation with respect to the fixed stars that matters, and rotation of the fixed stars relative to an object has the same effect as rotation of the object with respect to the ...
An everyday example of a rotating reference frame is the surface of the Earth. (This article considers only frames rotating about a fixed axis. For more general rotations, see Euler angles.) In the inertial frame of reference (upper part of the picture), the black ball moves in a straight line.
Examples of fictitious forces are the centrifugal force and the Coriolis force in rotating reference frames. To apply the Newtonian definition of an inertial frame, the understanding of separation between "fictitious" forces and "real" forces must be made clear. For example, consider a stationary object in an inertial frame.
In his Philosophiae Naturalis Principia Mathematica, Newton tried to demonstrate that one can always decide if one is rotating with respect to the absolute space, measuring the apparent forces that arise only when an absolute rotation is performed. If a bucket is filled with water, and made to rotate, initially the water remains still, but then ...
Rotation formalisms are focused on proper (orientation-preserving) motions of the Euclidean space with one fixed point, that a rotation refers to.Although physical motions with a fixed point are an important case (such as ones described in the center-of-mass frame, or motions of a joint), this approach creates a knowledge about all motions.
To support his views, Newton provided some empirical examples: according to Newton, a solitary rotating sphere can be inferred to rotate about its axis relative to absolute space by observing the bulging of its equator, and a solitary pair of spheres tied by a rope can be inferred to be in absolute rotation about their center of gravity by ...
Finally, the top can wobble up and down; the inclination angle is the nutation angle. The same example can be seen with the movements of the earth. Though all three movements can be represented by a rotation operator with constant coefficients in some frame, they cannot be represented by these operators all at the same time.
A supplementary thought experiment with the same objective of determining the occurrence of absolute rotation also was proposed by Newton: the example of observing two identical spheres in rotation about their center of gravity and tied together by a string. Occurrence of tension in the string is indicative of absolute rotation; see Rotating ...