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Math Help - Question regarding mass density?

  1. #1
    Super Member fardeen_gen's Avatar
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    Question regarding mass density?

    Mass density(mass per unit volume) of a sphere is varying as ρ = (1 + 2r) kg/m^3 where r is the distance of the position of the point from the centre of sphere(in metre). Calculate mass of the sphere if R is the radius of the sphere?

    Ans: 4π[(R^3/3) + (R^4/2)]

    How do we find the answer?
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  2. #2
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    Quote Originally Posted by fardeen_gen View Post
    Mass density(mass per unit volume) of a sphere is varying as ρ = (1 + 2r) kg/m^3 where r is the distance of the position of the point from the centre of sphere(in metre). Calculate mass of the sphere if R is the radius of the sphere?

    Ans: 4π[(R^3/3) + (R^4/2)]

    How do we find the answer?
    The volume dV of a shell of radius r and “infinitesimal” thickness dr is

    \mathrm{d}V=4\pi r^2\mathrm{d}r

    Hence

    M=\int_{r=0}^{r=R}{\rho}\,\mathrm{d}V=\int_0^R{(1+  2r)\cdot4\pi r^2}\,\mathrm{d}r
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  3. #3
    Super Member fardeen_gen's Avatar
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    I did not get the same answer. Can anybody help?
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  4. #4
    Senior Member nikhil's Avatar
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    Hi!fardeen_gen
    Mass=density*volume
    let us take a small sphere at a distance x from centre of thickness dx
    mass of this small sphere is dm.volume of this sphere
    =(4/3)pi[(x+dx)^3-x^3]
    =4pix^2dx(neglecting higher power of dx)
    so dm=(1+2x)(4pi(x^2)dx)
    dm=4pi(x^2+2x^3)
    [int x^n=[x^(n+1)]/n+1 where int means integration]
    now integrate taking limit of RHS 0 to R.
    Finally you will get
    M=4pi[(R^3)/3+(R^4)/2]
    Last edited by nikhil; June 30th 2008 at 11:10 PM.
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