Question

Find the density of a planet with a radius of 8000 m if the gravitational acceleration for the planet, gp, has the same magnitude as the gravitational constant, G (keep the right units for both), where G = 6.67 x 10-11 m3/(kg s2) Hint: Use the expression for the gravitational force and Newton's second law.

Answer 1

Density = 3 x 10⁻⁵ kg/m³

Step-by-step explanation:

First, we will find the volume of the planet:

`V = (4)/(3)\pi r^3\ (radius\ of\ sphere)\\\\V = (4)/(3)\pi (8000\ m)^3\\\\V = 2.14\ x\ 10^(12)\ m^3`

Now, we will use the expression for gravitational force to find the mass of the planet:

`g = (Gm)/(r^2)\\\\m = (gr^2)/(G)`

where,

m = mass = ?

g = acceleration due to gravity = 6.67 x 10⁻¹¹ m/s²

G = Universal Gravitational Constant = 6.67 x 10⁻¹¹ Nm²/kg²

r = radius = 8000 m

Therefore,

`m = ((6.67\ x\ 10^(-11)\ m/s^2)(8000\ m)^2)/(6.67\ x\ 10^(-11)\ Nm^/kg^2)\\\\m = 6.4\ x\ 10^7\ kg`

Therefore, the density will be:

`Density = (m)/(V) = (6.4\ x\ 10^7\ kg)/(2.14\ x\ 10^(12)\ m^3)`

Density = 3 x 10⁻⁵ kg/m³