A planet has twice the mass of Earth. How much larger would the radius of the planet have to be for the gravitational field strength, at the planet's surface to be the same as on Earth's surface?

Question

How to solve.

Answer 1

There are two ways of approaching this kind of a problem -- formal and informal.

Let me first use the formal approach.

Let

M be the mass of the Earth,

m be the mass of the planet,

R be the radius of the Earth,

r be the radius of the planet,

A be the gravitational acceleration on the surface of the Earth,

a be the gravitational acceleration on the surface of the planet.

We are told

m/M = 2 (1)

A = a (2)

We are to compute r/R

Using the formula for gravitational acceleration

A = GM/R²

a = Gm/r²

Using (2)

GM/R² = Gm/r²

Simplify into

r²/R² = m/M

Using (1)

(r/R)² = 2

r/R = √2

Now the informal solution:

Without changing radius, the larger mass of the planet would make the the acceleration twice as big as A.

The change in radius must compensate.

To reduce the acceleration we need the increase the radius.

We need to reduce the acceleration by a factor of 2;

but acceleration is not inversely proportional to radius, but rather its square.

Therefore we need to increase the radius by factor of √2.