Variable Mass in Newtonian Mechanics [duplicate]

Question

In the context of Newtonian mechanics, is it possible to have an isolated system which does not lose/gain particles, but whose mass changes over time? I have been trying to think of such an example for a while, but have been unable to come up with one.

I understand that this phenomenon is common in special relativity, but, as I have been told, Newton did not postulate mass invariance for Newtonian mechanics either. The question is also somewhat related to Newton's formulation of the second principle,

$$\overrightarrow{F}=\frac{\overrightarrow{dp}}{dt}$$

which, given that the system referred to does not lose/gain particles, could also be written as

$$\overrightarrow{F}=m\frac{\overrightarrow{dv}}{dt} + \frac{dm}{dt}\overrightarrow{v}$$

However, I have never seen an example outside of relativity where this formula is used. Is there any such example in Newtonian mechanics?

Answer 1

In Newtonian mechanics mass cannot disappear, it is assumed to be constant. To imagine why it is not a good idea to have variable mass, think that if mass could be created or destroyed, momentum would no longer be conserved, because in absence of a force, it will be accelerated: $\frac{dv}{dt}=-\frac{v}{m}\frac{dm}{dt}$

thus a moving object would slow down as if a friction force proportional to $v$ was acting. This violates the first law.

In addition, the law is not invariant across reference frames. Imagine you are in a reference frame in which the particle is momentarily at rest. Then by the equation above, the particle should remain at rest. However as seen from a moving frame, the particle should be decelerating.

Answer 2

Yeah of course there are many problems you can find in the text on Newtonian mechanics. Here's the one that is familiar to most of us.

Sand falls from a stationary hopper onto a freight car that moves with uniform velocity $v$. The sand falls at a rate of $dm/dt$. What force is needed to keep the freight car moving at the speed of $v$?

Using the force equation for variable mass, we have $$F=\frac{dp}{dt}=m\frac{dv}{dt}+v\frac{dm}{dt}$$ As car is moving with uniform velocity so that time derivative of velocity is zero. $$F=v\frac{dm}{dt}$$ this is the force required to move the freight car at constant speed $v$.