Why does Earth accelerate every object toward itself with acceleration of $9.8\, m/s^2$

Question

Why Earth accelerates objects due to gravity with $9.8\,\mathrm{m/s}^2$. How to demonstrate that?

Answer 1

This is an experimental fact. A measurement.

Physics is about making mathematical models, called theories, that describe the measurements and predict new phenomena with success. The physics model describing (and predicting ) measurements/gravity is called Newton's gravitational model. It is very successful.

Keep in mind if you go on to study physics that why questions in physics end up on the statement "because that is what we have observed". We accept Newton's theory of gravity because it agrees with all we have ever observed in every day life. Physics answers "how" from postulates and mathematical formulae an observation can be explained. The "why" ends up on the postulates which were chosen so that the observations can be explained, a circular logic.

For very high energies and cosmological distances new theories are necessary, because like every theory the region of validity is limited. But that is another story.

Answer 2

The gravitational field of a spherical body of mass $M$ at a distance $r$ from its center is $$\frac{GM}{r^2}$$.

This is a Newtonian Mechanics answer, but it is an extremely good approximation when dealing with objects of small mass like the Earth, the Moon. etc. For higher mass and energy scales we use the Theory of General Relativity, which describes the same phenomenon in terms of spacetime curvature.

$G$, in that equation , is the gravitational constant, which is the proportionality constant for the Gravitational Force. Its value can be determined by various precision experiments, the most well known of which was the Cavendish Experiment.

The value of $G$ is approximately $6.67*10^{-11}$ in appropriate units. The mass of the Earth is $5.97*10^{24}$ kg. The radius of the Earth is $6371$ km. With a simple enough calculation, you can see that the acceleration (equal to the field at that point) of any body close to the surface of the Earth would be reasonably close to $9.8$ $ m/s^2$.

This is true because it's an experimental fact. As @anna v mentioned in her answer, Physics is about mathematical models that fit reality, and the models we have come up with are mind-blowingly accurate. The thing about experimental facts is that they can be verified by anyone, if they are persistent and accurate enough. You could jump, throw a ball up, stand on a weighing machine, watch how the moon and the planets travel through space, or even measure something as simple as the time period of a pendulum! All of that would show that the $\frac{GM}{r^2}$ variation is true on this scale, and that is the proof you ask for.