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Need some help with differential equations for mechanics.

  1. Jan 30, 2013 #1
    1. The problem statement, all variables and given/known data

    I'm having problems with some differential equations, just need to know where I'm going wrong.


    2. Relevant equations



    3. The attempt at a solution

    a) mv[itex]\stackrel{dv}{dx}[/itex]=F(x)
    mvdv=F(x)dx
    m∫vdv=∫F(x)dx
    v^2=v[itex]v^{2}_{0}[/itex]+[itex]\stackrel{2}{m}[/itex]∫F(x)dx

    Setting F(x)=-kx
    v^2=[itex]v^{2}_{0}[/itex]-[itex]\stackrel{k}{m}[/itex](x^2-[itex]x^{2}_{0}[/itex])

    I then have to find the position as a function of time..
    [itex]\stackrel{dx}{dt}[/itex]=[itex]\sqrt{v^{2}_{0}-\stackrel{k}{m}(x^2-x^{2}_{0}}[/itex]
    dx/[itex]\sqrt{v^{2}_{0}-\stackrel{k}{m}(x^2-x^{2}_{0}}[/itex]=dt

    I'm not sure how to do that integral, though, or if that's even right.

    b) This problem involves a disc moving along a rough surface, so it has friction (F) and linear air resistance (-bv) acting on it.

    ma=-bv+F
    mdv/dt=-bv+F
    -[itex]\stackrel{m}{b}[/itex]∫dv/v=∫Fdt
    -[itex]\stackrel{m}{b}[/itex]ln([itex]\stackrel{v}{v_0}[/itex]=Ft
    e^(-[itex]\stackrel{m}{b}[/itex])v/v_o=e^(Ft)
    v=v_0e^(-Fbt/m)

    Thanks for any help!
     
  2. jcsd
  3. Jan 30, 2013 #2
    For a), transform the integrand to this form: ## \displaystyle \frac {a} {\sqrt {1 - (cx)^2 } } ##, then use the substitution ## u = cx ##.

    For b), you are not doing it correctly. You should have gotten ## \displaystyle \int \frac {dv} {F - bv}##.
     
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