Is it correct to say that $F= ma$ is valid for cases where mass remains constant? Suppose a snow ball is rolling down a hill and gaining mass continously so can't we say net force on snow ball at any instant is equal to mass of ball at that instant multiplied by acceleration of ball at that instant? What I have learnt is that first law is a differential principle whereas Impulse-momentum theorem is an integral principle so we can apply $F=ma$ to any body at any instant.
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2Does this answer your question? What is the fundamental definition of force? – tryst with freedom May 14 '23 at 05:45
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Newton's 2nd law applies to a single definite collection of particles, the mass of which cannot change. However, we can use momentum conservation to derive an equation describing the dynamics of a system with variable mass. See my answer here: https://physics.stackexchange.com/a/732876/37496 – d_b May 14 '23 at 06:41
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$F=dp/dt=d/dt(m\cdot v)=dm/dt\cdot v + m\cdot dv/dt$
where the latter term is $m\cdot a$
So your snowball has something in common with rocket science, where the rocket reduces mass at whatever rate.
The principle is only relevant for deriving the concept of force. No matter, which route you take, differential, integral, virtual forces, minimal principles … the results must be the same. Nature doesn‘t care about wording, while we tend to overlook things at first glance, i.e. oversimplify.
So for your snowball, like often in physics, you need some special requirements for the simple approach:
- it must have a high mass already
- or picks up only a faint quantity of snow while rolling
- it rolls very slowly
- so you can neglect $dm/dt$.
MS-SPO
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2$F=dp/dt=d/dt(m\cdot v)=dm/dt\cdot v + m\cdot dv/dt$ is correct Mathematically but is not correct in the context of Newton's second law. The correct relationship is $F=dp/dt= d(mv)/dt$ after the system under consideration has been defined. – Farcher May 14 '23 at 07:44