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e2m2a

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Suppose we have an object, denoted as a slider with mass M, that has one degree of freedom of movement along a linear track. The slider can move in the positive y-direction along this track with respect to an x-y coordinate system that is fixed to an inertial laboratory reference frame. The slider is initially at rest. On the surface of the slider is another track which has a section that is aligned perpendicular to the y-axis. On one end of this track section is a smaller object, denoted as the mover with mass m. (We choose the ratio of M and m such that M >> m.) The mover can slide along this track in the positive x-direction until the track curves it around through a quarter turn, such that the mover emerges from the curved section in the positive y-direction. Also at the end of the track is a near-massless spring with a force constant k. We push the mover against the spring a distance ∆x, compressing it, and giving the mover an initial elastic potential energy equal to:

Elastic Energy = 1/2 k ∆x² (1).

At some point in time we release the mover. The velocity of the mover in the positive x-direction with respect to our laboratory frame as it emerges from the spring is easily derived as:

Vmover = ∆x√(k/m) (2).

Hence, the kinetic energy would be:

KEmover = 1/2 m Vmover² (3).

As the mover travels at a constant speed in the positive x-direction, we push the slider in the positive y-direction, giving the slider a velocity Vslider. Thus, the total initial kinetic energy of the system with respect to our laboratory frame would be:

Total Initial Energy = 1/2 M Vslider² + 1/2 m Vslider ² + 1/2 m Vmover² (4).

At some later point in time the mover travels counter-clockwise around the quarter section of the track and emerges traveling in the positive y-direction. With respect to the frame of the slider because we assume there is no friction, the magnitude of the velocity of the mover in the positive y-direction would be the same as the magnitude of the initial velocity in the positive x-direction expressed in equation 2. (Remember, we also assume M >> m, such that the effect the mover has on the slider as it rounds the curved section of the track due to the centrifugal reactive force acting on the track is inconsequential.) The velocity of the mover in the positive y-direction with respect to the laboratory frame is easily derived from the Galilean velocity addition theorem as:

Vmover lab = Vslider + Vmover (5).

Hence, with respect to the laboratory frame, the mover's kinetic energy would now be:

KEmover lab = 1/2 m (Vslider + Vmover)² (6).

Expanding, we have:

KEmover lab = 1/2 m Vslider² + m Vslider Vmover + 1/2 m Vmover² (7).

Combining with the kinetic energy of the slider, we have the total final energy of

the system as:

Total Final Energy = 1/2 M Vslider² +1/2 m Vslider² + m Vslider Vmover + 1/2 m Vmover² (8).

Subtracting the initial total energy, equation 4, from the above, we arrive at:

∆Energy = m Vslider Vmover (9).

The above expression represents a gain in energy of the system. The initial energy of the system can be accounted for by the work we did in compressing the spring and the work we did in moving the system in the positive y-direction. We define the system as the mass of the slider and the mass of the mover. We include the mass of the track on the slider as part of the mass of the slider. The total initial kinetic energy of the system expressed in equation 4 must by the conservation of energy be accounted for by an equal decrease in the internal energy of our body so that the net change in energy at this initial time is zero. However, when the mover emerges from the curved section of the track, the system has

*gained*energy by the amount expressed in equation 9.

Would anyone like to venture an explanation where the source of this gain in energy

comes from? Would it be in the domain of quantum physics? Could we link it to the vacuum energy?