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1D (net) work done by (net) force on a variable mass system.

  1. Oct 3, 2011 #1
    So I was sitting on the train last weekend, reading through my physics book on mechanical work and its relation to kinetic energy. One example would be that a box on a frictionless table being pushed and they would conclude that W = ΔK = ½mΔv2.

    Looking at this equation got me thinking, obviously this applies to a system of constant mass, but what if the mass varied? I tried looking it up on the Internet, but either I'm wording it wrong or there isn't much info out there freely available. So I figured I should try to derive an expression on my own.

    Now then, consider a box of some mass on a frictionless table in vacuum. Then let's say a piston pushes the box from rest with a force that varies through time i.e. [tex]\overrightarrow{F} = \overrightarrow{F}(t)[/tex]

    Now since the mass isn't constant, we'll use a more general equation for Newton's second law of motion: [tex]\overrightarrow{F}(t) = \frac{d(m\overrightarrow{v})}{dt} = \frac{dm}{dt}\overrightarrow{v} + \frac{d\overrightarrow{v}}{dt}m[/tex] where mass and velocity are functions of time.

    Then, consider that the work done by a force is [tex]W = \int \overrightarrow{F}\cdot d\overrightarrow{s}[/tex] or in this case: [tex]W_{F}(t) = \int \overrightarrow{F}(t)\cdot d\overrightarrow{s}[/tex]

    In this 1D problem, the force acts in the same direction as the displacement, so the angle between the vectors are zero and so we can write the equation thusly:

    [tex]W_{F}(t) = \int F(t)ds[/tex] Expanding F(t):

    [tex]W_{F}(t) = \int v\frac{dm}{dt}ds + \int m\frac{dv}{dt}ds[/tex] Using the chain rule and cancelling ds:

    [tex]W_{F}(t) = \int v\frac{dm}{ds}\frac{ds}{dt}ds + \int m\frac{dv}{ds}\frac{ds}{dt}ds = \int v^{2}dm + \int mvdv = mv^{2} + \frac{1}{2}mv^{2} + C = \frac{3}{2}mv^{2} + C[/tex] Since the box was at rest in the beginning, the constant C = 0.

    [tex]W_{F}(t) = \frac{3}{2}mv^{2} \left( = \frac{3}{2}m(t)v(t)^{2}\right)[/tex]

    So my question is, is this equation valid classically? Have I made any mistakes in my reasoning?

    Thanks in advance.
  2. jcsd
  3. Oct 3, 2011 #2

    Ken G

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    Gold Member

    The integrals are not valid-- you did the integral over dm as if v was constant, and the integral over vdv as if m was constant. Where it gets interesting is if you imagine an object moving with constant velocity that starts out with negligible mass an gradually accumulates to mass m, maintained at v by a force. You then can take just your first integral to show that the work done by the force is mv2, even though the final kinetic energy is 1/2 that. The remaining 1/2 is thermalized internally when you stick on mass, i.e., it goes into heat.
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