# Need help in understanding accelaration due to gravity

I've been having some trouble in understanding acceleration due to gravity.

On earth, the acceleration due to gravity is an average of $9.80$ $m/s^2$. The mass of the earth is approximately $5.972 \times 10^{24} kg$. The acceleration due to gravity on the surface of the sun is $273.7$ $m/s^2$ and its mass is about $1.989 \times 10^{30}$.

So

$$\frac{\textrm{Mass of Sun}}{\textrm{Accelaration at Sun surface}} = \frac{\textrm{Mass of Earth}}{\textrm{Accelaration at Earth surface}}$$

Why don't the above numbers equal each other? Is it because I am doing mass divided by acceleration?

• Why would you expect the ratio $m/a$ to be constant? – probably_someone Jan 1 '17 at 5:17

## 1 Answer

The gravitational force is given by $F_g=\frac{GMm} {r^2}$ where symbols have their usual meaning. And the acceleration due to gravity at the surface of a body (uniform and spherical) is $a_g=\frac{GM} {R^2}$, where R is the radius of the body.

What you are doing is just dividing M by $a_g$ which gives you $\frac{MR^2} {GM} = \frac{R^2} {G}$. So do you see the problem? Saying that $\frac{M} {a_g}$ is same for Sun and Earth would imply that their radii are equal, which is clearly not true.