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Using Newtonian field equation to find the gravity inside a sphere

  1. Mar 21, 2012 #1
    1. The problem statement, all variables and given/known data

    work out g field at a distance r=R/2 from the centre of a spherically symmetric body of radius R.

    2. Relevant equations

    [itex]\nabla^{2}\psi=4\pi G \rho[/itex]

    [itex]\psi=G\int_{V'}\rho\frac{1}{|\vec{r}-\vec{r^{'}}|}d^{3}\vec{r^{'}}[/itex]

    [itex]-\int_{V}\nabla\cdot g dV = \int_{V}\rho dV[/itex]

    V=4/3∏r3

    3. The attempt at a solution

    V=4/3∏r3

    V1=4/3∏(R/2)3 = 1/6∏R3
    V2=4/3∏R3
    [itex]\int_{1/6\pi R^{3}}^{4/3\pi R^{3}}\rho dV[/itex]

    [itex]=\rho(4/3\pi R^{3}-1/6\pi R^{3})[/itex]

    [itex]=\frac{7}{6}\rho \pi r^{3}[/itex]
     
  2. jcsd
  3. Mar 22, 2012 #2

    should be a G*2*pi in the front of one of those integrals
     
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