Why is gravity, on earth, approximatley 9.81 m/s[SUP]2[/SUP].What

• jsmith613
In summary, the acceleration due to gravity on Earth is approximately 9.81 m/s^2 and is caused by the mass of the Earth and the universal law of gravitation discovered by Newton. The acceleration only depends on the distance from the center of the Earth and is not influenced by the mass of the objects. This is because the mass term cancels out in the equation, making the acceleration the same for all masses. This is a unique property of gravity and was a crucial factor in the development of Einstein's General Theory of Relativity.
jsmith613
why is gravity, on earth, approximatley 9.81 m/s2.

What causes it to be so?

thanks

Well, its an observational property of all matter in the universe that it attracts other matter by an equation which Newton found (http://en.wikipedia.org/wiki/Newton's_law_of_universal_gravitation)
$$F_G = G \frac{m_1 m_2}{r^2}$$
For two masses (m1 and m2), separated by a distance (r); given 'Newton's gravitational constant' (G).

Now, according to one of Newton's other laws
$$F = ma$$
If you combine these equations you find out that the acceleration due to gravity only depends on the mass of the Earth and your distance away from its center... if you plug in the values you get ~9.81 m/s^2

jsmith613 said:
why is gravity, on earth, approximatley 9.81 m/s2.

What causes it to be so?

thanks

Mass of the Earth and the Newton law that F=m a=-G m M/R^2. In classical physics the acceleration does depend only on the distance to the center of the earth. (This means that the acceleration is on a mountain smaller.)
That m does cancel out in this equation is very important, it is unique in physics and let Einstein think about the General Theory of Relativity.

Jens

zhermes said:
Well, its an observational property of all matter in the universe that it attracts other matter by an equation which Newton found (http://en.wikipedia.org/wiki/Newton's_law_of_universal_gravitation)
$$F_G = G \frac{m_1 m_2}{r^2}$$
For two masses (m1 and m2), separated by a distance (r); given 'Newton's gravitational constant' (G).

Now, according to one of Newton's other laws
$$F = ma$$
If you combine these equations you find out that the acceleration due to gravity only depends on the mass of the Earth and your distance away from its center... if you plug in the values you get ~9.81 m/s^2

So essetisally, because the Earth's mass is so large, the mass of the other objects is next to insignificant. Because the value of r is also large, changing the value by an amount will make little differernce.

Therefore will the value of gravity be in the range 9 - 10

(M1 = earth, M2 = object)

correct?

jsmith613 said:
So essetisally, because the Earth's mass is so large, the mass of the other objects is next to insignificant.
Not quite; it doesn't matter how large one body's mass is---because it cancels out in the equations, it has no influence on the acceleration.

jsmith613 said:
Because the value of r is also large, changing the value by an amount will make little differernce.
That's exactly right. The Earth's radius is something like 6000km, while the highest mountain is only about 10km... so the change in gravity will be about 0.3% (or something like that, so very small).

jsmith613 said:
Therefore will the value of gravity be in the range 9 - 10
Yeah, if you plug in the usual values, that's what you get.

Not quite; it doesn't matter how large one body's mass is---because it cancels out in the equations, it has no influence on the acceleration.

I don't quite understand what you mean. Please explain

Question 2: Using the equation G = M1 * M2 / r^2 does that mean I cause the Earth to accelerate upwards at a rate of 9.81 m/s^2. If this is the case, why does the Earth not move upwards dramatically due to the upward accealration of every landmass on earth.

No. That equation gives you the force. Acceleration is a=f/m so the larger the mass, the smaller the acceleration for the same force.

russ_watters said:
No. That equation gives you the force. Acceleration is a=f/m so the larger the mass, the smaller the acceleration for the same force.

Does that not mean, therefore that Gravity should change based on force and mass for each object, if we assume a = f/m

jsmith613 said:
Does that not mean, therefore that Gravity should change based on force and mass for each object, if we assume a = f/m

Simply put, the only variables in the calculation are the masses of the 2 objects and the distance between them. (Of course, this assumes an isolated system or one in which other objects are too distant to influence it.)

Other forces may act to overcome some or all of the influence of gravity, say a velocity of one object greater than the escape velocity of the larger object, but otherwise, that's really and truly all there is to it.

russ_watters said:
No. That equation gives you the force. Acceleration is a=f/m so the larger the mass, the smaller the acceleration for the same force.
jsmith613 said:
Does that not mean, therefore that Gravity should change based on force and mass for each object, if we assume a = f/m
Note that Russ said 'for the same force'. The force of gravity does change based on the mass. In fact the force is proportional to the mass, which makes the acceleration of a free falling object due to gravity the same for all masses, since the mass drops out of the equation.

The acceleration gravitational field depends only on the mass of the "source".

Given Earth with mass M and body on its surface with mass m, the force between them is:
$$F = G\frac{Mm}{r^2}$$

Now to calculate the acceleration of Earth you divide the force by Earth's mass
$$a = \frac{F}{M}$$
which is
$$a = G\frac{m}{r^2}$$

The body feels acceleration
$$a = \frac{F}{m}$$
$$a = G\frac{M}{r^2}$$

Now you see that acceleration of bodies on surface of Earth is in fact the same because it does not depends on body's mass.

Gravity is really nice force!

jsmith613 said:
Does that not mean, therefore that Gravity should change based on force and mass for each object, if we assume a = f/m
gravitational force, yes: that's what you measure with a bathroom scale.

Now to calculate the acceleration of Earth you divide the force by Earth's mass
$$a = \frac{F}{M}$$
which is
$$a = G\frac{m}{r^2}$$

Why is this so,

Does all this mean that no matter the mass of the smaller object, it will always have the same acceleration?

jsmith613 said:
Why is this so,

I understand it know

THanks everyone

Drakkith said:
Does all this mean that no matter the mass of the smaller object, it will always have the same acceleration?

Yes it does.

zhermes said:
Yes it does.

Is this because as the mass increases, the attraction due to gravity increases as well, but so does the amount of force required to move it? I think that's right, just wanting to make sure.

Drakkith said:
Is this because as the mass increases, the attraction due to gravity increases as well, but so does the amount of force required to move it? I think that's right, just wanting to make sure.
Yes! That's exactly it.

And as someone said above, this is a very interesting feature of the laws of physics, in Newton's force equation there is an inertial mass
$$F_{net} = m_{inertial} a$$
and in his gravitational equation, there is a gravitational mass
$$F_g = G\frac{M m_{grav}}{r^2}$$

And it turns out, they seem to be the same.
$$m_{inertial} = m_{grav}$$

That's what einstein called the "Equivalence principle," which led to his formulation of 'general relativity.' These days we take it for granted that those two masses are equal, but there is no fundamental physical law that says it has to be so, its an observational fact.

Awesome.

Why is gravity, on earth, approximately 9.81 m/s2?

The value of gravity on earth, or the acceleration due to gravity, is approximately 9.81 m/s2 because of the mass and size of the earth. Gravity is a force that exists between any two objects with mass, and the strength of this force is directly proportional to the masses of the objects and inversely proportional to the square of the distance between them. Since the earth is a very massive object, it has a strong gravitational pull on objects near its surface.

How was the value of 9.81 m/s2 for gravity on earth determined?

The value of 9.81 m/s2 for gravity on earth was determined through experiments and observations conducted by scientists over many centuries. In the 17th century, Sir Isaac Newton first proposed the law of universal gravitation, which explained the relationship between the masses of objects and the force of gravity between them. Through further experiments and advancements in technology, scientists were able to accurately measure the acceleration due to gravity on earth and determine its value to be approximately 9.81 m/s2.

Does the value of gravity on earth vary in different locations?

Yes, the value of gravity on earth can vary slightly in different locations. This is because the earth is not a perfect sphere and has irregularities in its shape and mass distribution. Additionally, factors such as altitude and the density of the materials below the surface can also affect the strength of gravity in a particular location. However, these variations are very small and do not significantly impact the overall value of 9.81 m/s2 for gravity on earth.

Is the value of gravity on earth constant?

No, the value of gravity on earth is not constant. It can vary slightly depending on the factors mentioned above, as well as other external influences such as the gravitational pull of other celestial bodies. However, for most practical purposes, the value of 9.81 m/s2 is considered to be constant and is used in calculations and scientific experiments.

Can the value of gravity on earth change over time?

Yes, the value of gravity on earth can change over time, but these changes are very small and occur over long periods. This is because the earth is constantly undergoing changes in its mass and shape, such as tectonic plate movements and changes in sea level. However, these changes are not significant enough to affect the overall value of 9.81 m/s2 for gravity on earth in our daily lives.

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