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A Mexican jumping bean is a hollow shell with a live larva inside. To get away from heat, the larva will propel itself along the inside of the shell and that most often causes the shell to rotate but it can sometimes cause the shell to "jump", hence the name. The motion inside the shell cause the center of mass of the shell/larva system to shift and that causes the shell to move. In this animation we will simplify the bean to just a cylindrical shell with a spherical larva inside where the bean is either floating in space or is on a frictonless table. We imagine that the larva takes a leap from the left end of the shell and this causes the larva to proceed to the right end (where it sticks) while, by Newton's laws, the shell moves toward the left.

Exactly the same sort of thing occurs when a very short light pulse is emitted from the left end and gets absorbed at the right end. In this animation, we will track the center of mass of a system that has a hollow cylinder of mass M that contains a (smaller) mass m that can "jump" from the left end to the right end at speed `v`. Of course, since the cylinder is on a frictionless table, it will react by moving backward at speed `V` which is a function of the ratio of masses m/M. When m reaches the right end, it invokes a force impulse on that end that is opposite to the force that it provided to the left end when it jumped away from that end. Since the two forces provided to the cylinder are equal and opposite, the leftward motion of the cylinder stops at that time. However, if the length of the cylinder is `L`, then, to first order, the time to travel to the right end is `L/v` and the leftward distance traveled by the cylinder is then `(LV)/v`. So the final result is that the cylinder that was originally centered at `X=0` is now centered at

`X_("final")=-LV/v`

and the mass which was originally at `x=-L/2` is now at `x=L/2`. We would like to know what is difference between the final and initial centers of mass. The initial center of mass is`chi_("initial")=-m/(M+m)L/2`

and the final center of mass is`chi_("final")=(MX_("final")+mL/2)/(m+M)`

Taking the difference we get:`deltachi_(fi)=(mL+Mchi_("final"))/(m+M)=(mL-MLV/v)/(m+M)`

. The velocity ratio is obviously V/v=m/M. Then we have`deltachi_(fi)=(mL-mL)/(m+M)=0 `

as expected. There is a slight problem with this derivation. The time interval we computed is not quite right because the cylinder was moving while the small mass was progressing to its right hand end. The time interval is really given by`deltat=L/(v-V)`

so`deltat=(L/v)/(1+m/M)`

Then the leftward distance that the cylinder moves is modified to:`X_("final")=-L(V/v)/(1+m/M)=-Lm/(m+M)`

. Then, since m is now at the right hand side of the cylinder the center of mass becomes`chi_("final")=[m(L/2+xCf)+M(xCf)]/(m+M)=[(m+M)(xCf)+mL/2]/(m+M)= X_("final")+mL/(2(m+M))`

`chi_("initial")` was at `-mL/2/(M+m)`. Difference in `chi` is then`deltachi=-mL/(m+M)+mL/2(m+M))+mL/2/(m+M)=0 ` as expected.

This completes the math for the material (matter) object. For Einstein's derivation, the larva (mass m) becomes a short light pulse of energy E. The light pulse moves at speed c relative to either an obserever moving with the sliding cylinder or to an observer in the lab. Of course the cylinder is shorter by 1/gamma for the observer in the lab so the duration of the traverse between ends becomes the same for both, t=L/c. Since the pulse exerts a force impulse of value E on the emitting surface, the cylinder's speed during the traverse is`V=-E/Mc`.

Then the cylinder moves to the left by distance`(E/M)(L/c^2)`.

If the light pulse had no mass, this would result in a displacement of the center of mass of the system to the left by distance`E/M(L/c^2)`

with no outside force applied. However, if the light pulse has mass `E/c^2`, then the initial center of mass (relative to the center of the cylinder) is`chi_("initial")=-(E/c^2)(L/2)/(M+E/c^2)`.

After being absorbed at the right hand end of the cylinder the final center of mass becomes (taking into account the movement of the center of the cylinder)`chi_("final")=[-EL/c^2+(E/c^2)L/2]/(M+E/c^2)=chi_("initial")`

as expected from ordinary laws of mechanics. An important point to note here is that, at least to first order in `V/c`, this derivation of the result that `m=E/c^2` has NO dependence on relative velocity between frames of reference. The force due to the light pulse is a result of Maxwell's equations and this is all that is required for the proof that energy has mass which increases inertia. This begs the question of whether `E=mc^2` relates to relativity at all.**
In the animation below, please place your finger on the Center of Mass legend to confirm that
that the center of mass is not movingeven though both shell and Larva move.
**

Mexican Jumping Bean Science

Einstein's E=mc^2 Derivation (Doc 35)