Rotating spheres

Newton was concerned to address the problem of how it is that we can experimentally determine the true motions of bodies in light of the fact that absolute space is not something that can be perceived. Such determination, he says, can be accomplished by observing the causes of motion (that is, forces) and not simply the apparent motions of bodies relative to one another (as in the bucket argument). As an example where causes can be observed, if two globes, floating in space, are connected by a cord, measuring the amount of tension in the cord, with no other clues to assess the situation, alone suffices to indicate how fast the two objects are revolving around the common center of mass. (This experiment involves observation of a force, the tension). Also, the sense of the rotation—whether it is in the clockwise or the counter-clockwise direction—can be discovered by applying forces to opposite faces of the globes and ascertaining whether this leads to an increase or a decrease in the tension of the cord (again involving a force). Alternatively, the sense of the rotation can be determined by measuring the apparent motion of the globes with respect to a background system of bodies that, according to the preceding methods, have been established already as not in a state of rotation, as an example from Newton's time, the fixed stars. In the 1846 Andrew Motte translation of Newton's words: Mach took some issue with the argument, pointing out that the rotating sphere experiment could never be done in an empty universe, where possibly Newton's laws do not apply, so the experiment really only shows what happens when the spheres rotate in our universe, and therefore, for example, may indicate only rotation relative to the entire mass of the universe. An interpretation that avoids this conflict is to say that the rotating spheres experiment does not really define rotation relative to anything in particular (for example, absolute space or fixed stars); rather the experiment is an operational definition of what is meant by the motion called absolute rotation. ==Formulation of the argument==

This sphere example was used by Newton himself to discuss the detection of rotation relative to absolute space. Checking the fictitious force needed to account for the tension in the string is one way for an observer to decide whether or not they are rotating – if the fictitious force is zero, they are not rotating. (Of course, in an extreme case like the gravitron amusement ride, you do not need much convincing that you are rotating, but standing on the Earth's surface, the matter is more subtle.) Below, the mathematical details behind this observation are presented. Figure 1 shows two identical spheres rotating about the center of the string joining them. The axis of rotation is shown as a vector Ω with direction given by the right-hand rule and magnitude equal to the rate of rotation: |Ω| = ω. The angular rate of rotation ω is assumed independent of time (uniform circular motion). Because of the rotation, the string is under tension. (See reactive centrifugal force.) The description of this system next is presented from the viewpoint of an inertial frame and from a rotating frame of reference. Inertial frame Adopt an inertial frame centered at the midpoint of the string. The balls move in a circle about the origin of our coordinate system. Look first at one of the two balls. To travel in a circular path, which is not uniform motion with constant velocity, but circular motion at constant speed, requires a force to act on the ball so as to continuously change the direction of its velocity. This force is directed inward, along the direction of the string, and is called a centripetal force. The other ball has the same requirement, but being on the opposite end of the string, requires a centripetal force of the same size, but opposite in direction. See Figure 2. These two forces are provided by the string, putting the string under tension, also shown in Figure 2. Rotating frame Adopt a rotating frame at the midpoint of the string. Suppose the frame rotates at the same angular rate as the balls, so the balls appear stationary in this rotating frame. Because the balls are not moving, observers say they are at rest. If they now apply Newton's law of inertia, they would say no force acts on the balls, so the string should be relaxed. However, they clearly see the string is under tension. (For example, they could split the string and put a spring in its center, which would stretch.) To account for this tension, they propose that in their frame a centrifugal force acts on the two balls, pulling them apart. This force originates from nowhere – it is just a "fact of life" in this rotating world, and acts on everything they observe, not just these spheres. In resisting this ubiquitous centrifugal force, the string is placed under tension, accounting for their observation, despite the fact that the spheres are at rest. Coriolis force What if the spheres are not rotating in the inertial frame (string tension is zero)? Then string tension in the rotating frame also is zero. But how can that be? The spheres in the rotating frame now appear to be rotating and should require an inward force to do that. According to the analysis of uniform circular motion: : \mathbf{F}_{\mathrm{Fict}} = - 2 m \boldsymbol\Omega \times \mathbf{v}_{B} - m \boldsymbol\Omega \times (\boldsymbol\Omega \times \mathbf{x}_B ) \ - m \frac{d \boldsymbol\Omega }{dt} \times \mathbf{x}_B \ . The subscript B refers to quantities referred to the non-inertial coordinate system. Full notational details are in Fictitious force. For constant angular rate of rotation the last term is zero. To evaluate the other terms we need the position of one of the spheres: : \mathbf{x}_B = R\mathbf{u}_R \ , and the velocity of this sphere as seen in the rotating frame: :\mathbf{v}_B = \omega_SR \mathbf{u}_{\theta} \ , where uθ is a unit vector perpendicular to uR pointing in the direction of motion. The frame rotates at a rate ωR, so the vector of rotation is Ω = ωR uz (uz a unit vector in the z-direction), and Ω × uR = ωR (uz × uR) = ωR uθ ; Ω × uθ = −ωR uR. The centrifugal force is then: :\mathbf{F}_\mathrm{Cfgl} = - m \boldsymbol\Omega \times (\boldsymbol\Omega \times \mathbf{x}_B ) =m\omega_R^2 R \mathbf{u}_R\ , which naturally depends only on the rate of rotation of the frame and is always outward. The Coriolis force is :\mathbf{F}_\mathrm{Cor} = - 2 m \boldsymbol\Omega \times \mathbf{v}_{B} = 2m\omega_S \omega_R R \mathbf{u}_R and has the ability to change sign, being outward when the spheres move faster than the frame ( ωS > 0 ) and being inward when the spheres move slower than the frame ( ωS \mathbf{F}_{\mathrm{Fict}} = \mathbf{F}_\mathrm{Cfgl} + \mathbf{F}_\mathrm{Cor} =\left( m\omega_R^2 R + 2m\omega_S \omega_R R\right) \mathbf{u}_R = m\omega_R \left( \omega_R + 2\omega_S \right) R \mathbf{u}_R ::=m(\omega_I-\omega_S)(\omega_I+\omega_S)\ R \mathbf{u}_R = -m \left(\omega_S^2-\omega_I^2\right)\ R \mathbf{u}_R . Consequently, the fictitious force found above for this problem of rotating spheres is consistent with the general result and is not an ad hoc solution just "cooked up" to bring about agreement for this single example. Moreover, it is the Coriolis force that makes it possible for the fictitious force to change sign depending upon which of ωI, ωS is the greater, inasmuch as the centrifugal force contribution always is outward. ==Rotation and cosmic background radiation==

The isotropy of the cosmic background radiation is another indicator that the universe does not rotate. ==See also==