# Using Mass and Density of a Sphere

Question: A sphere of radius $$R$$ has total mass $$M$$ and density function given by $$ρ = kr$$, where $$r$$ is the distance a point lies from the centre of the sphere. Give an expression for the constant $$k$$ in terms of $$M$$ and $$R$$.

My Attempt: $$ρ$$ is defined as density, meaning $$ρ=\frac{M}{Volume}$$. Volume of a sphere $$= \frac{4}{3}\pi R^3$$. Therefore, $$ρ=\frac{M}{\frac{4}{3}\pi R^3}=\frac{3M}{4\pi R^3}$$. Substituting in $$ρ = kr$$, I am left with $$kr=\frac{3M}{4\pi R^3}$$ and therefore need to find an expression for $$r$$ in terms of $$k$$, $$M$$ and/or $$R$$, but am unsure how to continue from here.

• density = mass/volume is only true if density is a constant, which it is not. You are told it is a function. May 2, 2020 at 8:45
• The sphere in the question is solid, but think of it as being made up of many hollow concentric spherical shells, each shell being $\delta r$ thick. Each shell has uniform density and for the shell with radius $r, 0 \leq r \leq R$ the mass is $\rho(r) \cdot 4\pi r^2 \delta r$. Use calculus to calculate the total mass in terms of $\rho$ and equate to $M$. May 2, 2020 at 8:50

Take a shell of infinitesimal thickness $$dr$$ at a distance $$r$$ from the centre. The mass of this shell would be $$dM = 4\pi r^2dr \times \rho(r) = 4 \pi r^2 kr dr$$ Then integrate both sides to find the mass as a function of $$k$$ $$M = \int_0^R 4 \pi r^2 \cdot kr dr = 4\pi k \frac{R^4}{4}$$ Therefore, the value of k is $$\boxed{k = \frac{M}{\pi R^4}}$$

• I'm a bit confused as to how you got that integral? May 2, 2020 at 9:03
• I'll edit to add some verbosity. May 2, 2020 at 9:03

If we consider a spherical shell with radius $$r$$ and thickness $$dr$$, then

Now, to find the mass of this spherical shell

\begin{align} \text{Volume of spherical shell } &= \text{Surface area} \times \text{Thickness} \\ &= 4\pi r^{2} \cdot dr \end{align}

Since $$\rho = kr$$,

\begin{align} \text{Mass } &= \text{Volume} \times \text{Density} \\ &= 4 \pi k r^{3}dr \end{align}

Now, to find the mass of the sphere, we integrate the above from $$r=0$$ to $$r=R$$

\begin{align} M &= \int_{0}^{R} 4 \pi k r^{3} dr \\ &= 4 \pi k \int_{0}^{R} r^{3} dr \\ &= 4 \pi k \left[\frac{r^4}{4} \right]_{0}^{R} \\ &= \pi k R^{4} \end{align}

Thus, an expression for the constant $$k$$ in terms of $$M$$ and $$R$$ is

$$k = \frac{M}{\pi R^{4}}$$