1. 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?

2. Originally Posted by fardeen_gen
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

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

Hence

$\displaystyle M=\int_{r=0}^{r=R}{\rho}\,\mathrm{d}V=\int_0^R{(1+ 2r)\cdot4\pi r^2}\,\mathrm{d}r$

3. I did not get the same answer. Can anybody help?

4. 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]