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Homework Help: Frictional force on a sphere

  1. Nov 3, 2005 #1
    Ok, I need to find the frictional force on sphere of radius [itex]a[/itex] and mass [itex]M[/itex] moving with velocity [itex]v[/itex] in an ideal gas at temperature [itex]T[/itex].

    If I put myself in the sphere frame, then diffrential cross-section is
    [tex]
    \frac{d\sigma}{d\Omega} = \frac{a^2}{4}
    [/tex]
    and the total cross-section is [itex]\sigma_{\textrm{tot}}=\pi a^2[/itex]. How do I find the frictional force from this? Ellastic collisions between the sphere and the gas particules are assumed.

    Any help greatly appreciated.
     
    Last edited: Nov 3, 2005
  2. jcsd
  3. Nov 6, 2005 #2
    Ok, here's what I tought:

    If the gas has density [itex]\rho[/itex] than the number of molecules in a volume [itex]\sigma_{\textrm{tot}}dx[/itex] is [itex]dN=\pi \rho a^2 dx[/itex]. If collisions are ellastic, then
    [tex]
    \textbf{p}_s + \textbf{p}_i = \textbf{p}_s' + \textbf{p}_i'
    [/tex]
    where the [itex]s[/itex] and the [itex]i[/itex] denote respectively the momentum of the sphere and the [itex]i[/itex]th molecule. The prime denotes the momentum after collision. (I assumed that molecules do not collide simultaneously.)

    The change in speed of the sphere is
    [tex]
    dv = \frac{|\textbf{p}_s' - \textbf{p}_s|}{M}
    [/tex]
    From accelaration [itex]dv/dx[/itex] if the [itex]x[/itex] direction is chosen along the movement of the sphere we can find the resistance force
    [tex]
    F = M\frac{dv}{dx}
    [/tex]
    Now I need to evaluate either [itex]\textbf{p}_s' - \textbf{p}_s[/itex] or [itex]\textbf{p}_i - \textbf{p}_i'[/itex].

    Anyone can help?
     
    Last edited: Nov 6, 2005
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