Review question

# What is the average density of this spherical planet? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R8152

## Suggestion

A spherical planet of radius $a$ has a variable density $f(r)$ which depends only on the distance $r$ from the planet’s centre. Show that the average density of the planet is

$3 \int_0^1 t^2 f(at) \:dt.$

The tricky part is going to be finding the total mass of the planet. Since the density depends only on $r$ we should think about the mass of a thin hollow spherical shell.

Find the average density correct to two significant figures in each of the three cases:

1. $f(r) = \exp\left[\left(-\dfrac{r}{a}\right)^3\right]$, where $\exp(x)$ denotes $e^x$,

We will need to use the result from the first part.

1. $f(r) = \exp\left(-\dfrac{r}{a}\right)$,

We will need to find a tool to help us evaluate the integral we get this time.

1. $f(r) = \dfrac{a^2r}{(a+r)^3}$.

This time the integral involves an awkward looking fraction. Again, we’ll need to find an appropriate tool to help.