The generalization of Newton's 2nd Law to apply to variable mass systems

  • Thread starter sedaw
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  • #1
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F = dp/dt = d(mv)/dt = m(dv/dt) + v(dm/dt) = a + v(dm/dt)

i dont understand why d(mv)/dt = m(dv/dt) + v(dm/dt)
can somone help ?

TNX !:smile:
 

Answers and Replies

  • #2
ehild
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How do you calculate the derivative of a product?

ehild
 
  • #3
This is simply obtained by using the product rule.

dp/dt = d/dt (mv) = m(dv/dt) + v(dm/dt) (using product rule).
 
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  • #4
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I thought that Newton's second law was true only for systems with constant mass.
 
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  • #5
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Well, written in the form of
[tex] F=m \frac{dv}{dt} [/tex]
it is valid inly for constant mass system since you assume that the mass is constant :P.

But assuming, that mass is dependant on velocity, which is true (according to SR) you get what sedaw wrote.
Thus, the best, most general form of Newton's 2nd law is
[tex] F=\frac{dp}{dt} [/tex]
as it doesn't imply anything being constant ;P
 
  • #6
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Well, written in the form of
[tex] F=m \frac{dv}{dt} [/tex]
it is valid inly for constant mass system since you assume that the mass is constant :P.

But assuming, that mass is dependant on velocity, which is true (according to SR) you get what sedaw wrote.
Thus, the best, most general form of Newton's 2nd law is
[tex] F=\frac{dp}{dt} [/tex]
as it doesn't imply anything being constant ;P

I know that [tex]\bold{F}=\frac{d\bold{p}}{dt}[/tex] is Newton's second law, but I read that this was true only with constant masses. Also, I don't know much about special relativity (I just finished my freshman year) but I thought it proved that Galilean transformation was flawed. Since Newton's laws are based on Galilean relativity, I don't think SR can show that N2 holds even for variable masses. Can it?
 
  • #7
diazona
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Well, what you read was wrong. In nonrelativistic physics, [itex]\mathbf{F} = \mathrm{d}\mathbf{p}/\mathrm{d}t[/itex] is valid regardless of whether mass changes.

If I remember correctly, in special relativity, Newton's second law (in the above form) is used to define the relativistic generalization of force. So that equation holds true in all cases. (Caveat: there are actually a couple of ways to define the relativistic generalization of force that are not quite compatible with each other... it turns out that force is not as useful a concept in special relativity as it is in non-relativistic, Newtonian physics.)
 

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