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I am wondering if any space agency has ever considered using a grinding machine as a propulsion system for an interplanetary spacecraft. This system would not be used to lift the spacecraft off of a planet's surface, but rather used solely as an interplanetary/interstellar propulsion system.

The working principle is a simple one (see picture below). This grinding machine would be located at the stern of the spacecraft. Metal shavings flying off of the grinding wheel of this grinding machine would produce a propelling force for the spacecraft via Newton's Third Law of Motion. The amount of propulsion produced at any one time would be achieved by increasing/decreasing the rotational speed of the grinding wheel.

The electrical source for the electric motor of the grinding machine could be a small nuclear power plant on board the spacecraft. The 'fuel' source could be long steel rods or large rocks. A source of rocks could be obtained by mining an asteroid field or a small moon. Lastly, I think that diamond grinding wheels would probably be the most ideal to use due to their durability and longevity.

Would a grinding machine be a simple and workable propulsion system for an interplanetary spacecraft?

I don't know if it has ever been considered by anyone.

In my view, this is not a good idea for at least the following reasons:

1. It is equivalent to mechanically throwing things retrograde. See this video for an overly simple example. This is obviously not a good way for propulsion, as the specific impulse is very low. Let's talk just about the impulse $$p=mv$$ here, where $m$ is the "reaction mass", i.e. the mass of the material that's being ground, the object being throw backwards, or chemical propellant. $v$ is the velocity of the reaction mass relative to the spacecraft. The velocity $v$ of the sparks is in the order of a few m/s (same velocity as the edge of the grinding wheel. With chemical propellants, it is a few km/s. So, for the same amount $m$ of reaction mass that you carry, classical propulsion gives you a factor of about 1000 more impulse than grinding.


2. The produced momentum is kind of stochastic. As can be seen in your graphic, the sparks form a cone instead of a straight line. While the upward and the downward motions statistically cancel each other, their vertical components are a waste. While admittedly this also applies to chemical rocket engines (and ion thrusters?), just throwing some stuff overboard would be more efficient in this respect.

Still, I like this question for thinking out of the box. On a side note, reading the title reminded me of this passage of J.D. Clark's book "Ignition!":

F.A. Tsander in Moscow [...] had suggested that an astronaut might stretch his fuel supply by imitating Phileas Fogg. When a fuel tank was emptied, the astronaut could simply grind it up and add the powdered aluminum thus obtaining to the remaining fuel, whose heating value would be correspondingly enhanced!

I think this was actually tried, but found not to work well because the Aluminium particles take too long to combust, i.e. they continue to burn after they have left the combustion chamber. (Some?) Solid rocket propellants are based on Aluminium, though, but that's different.

Good for you, for thinking outside the box! Fearlessly pursuing new ideas is where any new breakthrough comes from.

But rocket exhaust moves at thousands of meters per second -- supersonic speeds. Recalling the formula relating acceleration to velocity for circular motion, a=v^2/r. So, given a velocity of 3,000 meters per second and a wheel radius of, say, 1 meter, the acceleration at the rim would be roughly 9,000,000 m/s^2, or 900,000 times Earth gravity. And angular velocity, v=Rw, or 9,000,000 radians per second. I think you would have trouble spinning the wheel up to that kind of speed, and I think you would have trouble finding a wheel material that wouldn't fly apart.