# The relativistic mechanics of a battery that is being charged and accelerated at the same time

This might be an interesting question:

Let's attach a battery into one end of an electric cable. Then we rotate the battery around, with accelerating speed, using 100 Watts of power, while the battery rotates around frictionlessly. (The battery moves in a circle whose radius is the length of the cable)

At the same time we apply 200 Watts of electric power into the cable, this energy travels through the cable into the battery.

We apply 100 Watts of mechanical energy into our end of the cable, and we apply 200 Watts of electric energy into our end of the cable, then the mechanical energy and the electric energy travel into the battery.

The question: What is the speed of the battery after a very long time?

Is this question too easy or too difficult?

Naively I would think that after 1 Giga seconds the battery has 200 Giga Joules of electric energy, and 100 Giga Joules of kinetic energy.

If an object has rest energy of 200 Giga Joules and kinetic energy of 100 Giga Joules, then the object can be said to be a non-relativistic object.

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BTW, you don't require increasing angular velocity to have the battery under acceleration, circular motion imposes a centripetal acceleration even at steady angular velocity. –  dmckee Mar 29 '12 at 0:06

The speed of the battery will be independent of the electrical energy you've supplied to it. If you look at $E=mc^2$, you need a hell of a lot of energy to change the mass of the battery, and as mass is defined as the measurement of the resistance of a body to acceleration, you'd need a hell of a lot of electrical energy to affect the acceleration of the battery.

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Essentially you're asking: what speed does a battery on the end of a rigid cable, undergoing rotational acceleration, get to as it's supplied electrical energy to increase its rest energy?

There are three main effects to consider

• the rotating current will radiate energy at a certain rate which will set a limit on the electrical energy that can be supplied to the battery.