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Force that changes mass?

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I was thinking about Newton's law. The case of changing mass is often ignored outside the textbook rocket problems. Are there and examples of the effect of an external force is to change the rest mass of an object while leaving velocity constant?

I am familiar with something like in a particle accelerator in which particle are moving near the speed of light, but I do know that in relativistic mechanics, the measurement of "mass" is different from the rest mass. I'm specifically inquiring about phenomena in the classical regime, although I guest that classically, mass doesn't change.

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