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Frequently I see the expression $$F = \frac{dp}{dt} = \frac{d}{dt}(mv) = \frac{dm}{dt}v + ma,$$ which can be applied to variable mass systems.

But I'm wondering if this derivation is correct, because the second law of Newton is formulated for point masses.

Furthermore if i change the inertial frame of reference, only $v$ on the right side of the formula $F = \frac{dm}{dt}v+ma$ will change, meaning that $F$ would be dependent of the frame of reference, which (according to me) can't be true.

I realize there exists a formula for varying mass systems, that looks quite familiar to this one, but isn't exactly the same, because the $v$ on the right side is there the relative velocity of mass expulsed/accreted. The derivation of that formula is also rather different from this one.

So my question is: is this formula, that I frequently encounter in syllabi and books, correct? Where lies my mistake.

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3 Answers

You'll probably find the wikipedia article useful: http://en.wikipedia.org/wiki/Variable-mass_system

It is formulated so that $F+v_{rel} \frac{dm}{dt} = ma$, where $v_{rel}$ is the relative velocity of the mass being ejected to the center of mass of the body. This takes care of your question about reference frames, because $v$ will be the same in all frames. The term gets moved to the left side of the equation because $-v$ describes the velocity of the center of mass relative to the ejected matter.

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  1. Whether a mass can be considered a point or not depends on the scale at which it is studied. The earth can be considered to be a point mass when we are studying its motion around the sun but not so when we are studying its own rotation. Newton's laws are applied to systems of many particles.

  2. Newton's second law says that the rate of change of momentum of a system is proportional to the applied force. We choose units in such a manner that the constant of proportionality is 1. With this definition, the equation $md\vec{v}/dt + \vec{v} dm/dt = \vec{F}$ makes sense with $\vec{v}$ being the velocity of the body in the same frame of reference in which other vectors are measured.

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You can only apply Newton's second law to closed systems. But, since you are applying second law to a open system, you are getting contradictory results. The correct procedure for solving variable mass system, is by calculating the change in momentum and then equating it to

    Force = (change in momentum)/small time interval in which change occurred.

Here is a article on it that you can find useful, apart from the wikipedia article. See, this website. http://www.thestudentroom.co.uk/wiki/Revision:Motion_With_Variable_Mass

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