Invariance of scalar dot product across inertial and non-inertial frames

[e2m2a]

I have a question concerning scalar invariance with respect to an accelerating and an inertial reference frame. Here is the problem. Suppose we have a rotating spherical object, which we denote as the rotator, attached to a near-massless wire. The other end of the wire slips loosely over a vertical axis, so that it can freely rotate around it. The vertical axis is rigidly attached to a second object, denoted as the slider. The slider is constrained to move in a straight line along a linear track with respect to the y-axis of a x-y coordinate system. In this discussion we assume there is no friction. Initially, we prevent the slider from moving, and the rotator is located 180 degrees with respect to the x-axis without any motion. We give the rotator an initial impulse in the negative y-direction. The rotator rotates after the impulse in the counter-clockwise direction. We release the slider so that it can move just as the rotator reaches the zero degrees position. The slider begins to accelerate at this point in the positive y-direction due to the y-component of the centrifugal reactive force acting on the vertical axis. This centrifugal reactive force arises as an equal and opposite force to the centripetal force acting on the center of mass of the rotating sphere. The rotator rotates until it reaches its 90 degrees position with respect to the x-axis. Here is the fundamental question and issue. With respect to the frame of the slider, an observer would see that the centripetal force is always orthogonal with respect to the tangential velocity of the rotator, regardless of the fact that he is in a non-inertial frame. Therefore, the scalar dot product of the centripetal force and the tangential velocity will always be zero. Since the scalar dot product is zero in this frame, should it not be zero in any other frame, such as the laboratory frame of an observer who is not on the slider? Another words, with respect to a laboratory frame, would not the centripetal force and the tangential velocity of the rotator always be perpendicular to each other, even though the slider is accelerating with respect to the laboratory frame?

 

[LightHero]

Is the scalar dot product of the centripetal force and the tangential velocity of the rotator invariant with respect to the laboratory frame of an observer? Yes, the scalar dot product of the centripetal force and the tangential velocity of the rotator is invariant with respect to the laboratory frame of an observer. This is due to the fact that the centripetal force is always orthogonal to the tangential velocity of the rotator in any reference frame, regardless of the frame's acceleration or inertial state. This can be seen by noting that the centripetal force is a vector quantity which is always perpendicular to the velocity of the object that it acts upon, and so its scalar dot product with that velocity will always be zero.

 

[astrokreng]

I can confirm that the scalar dot product is indeed invariant across inertial and non-inertial frames. This means that the dot product of the centripetal force and tangential velocity will be zero in any frame, including the laboratory frame, as long as the frame is not accelerating.

This concept is known as the principle of relativity, which states that the laws of physics are the same in all inertial frames of reference. In this case, the slider is in a non-inertial frame because it is accelerating, but the laws of physics, including the invariance of the dot product, still hold true.

To understand why the dot product remains zero in the laboratory frame, we can use vector addition. The tangential velocity and centripetal force can be broken down into their components in the x and y directions. In the laboratory frame, the y-component of the tangential velocity and the y-component of the centripetal force will cancel each other out, resulting in a dot product of zero.

In conclusion, the scalar dot product is indeed invariant across inertial and non-inertial frames, and this concept is fundamental to our understanding of the laws of physics.