# Q & A: Where does the 'g' in the formulae 'F=mg' come from?

Q:
I was wondering how can I show that "F=mg" is actually an approximation of "F=Gm1m2/r^2" using Taylor Series... I'm not sure if mass is the dependent variable or radius?
- Farahnaz (age 18)
Toronto
A:
Hello Farahnaz,

To answer your question, let's just place the formula F = G * m1  m2 / R^2 into a Taylor series expansion (as distance from the center of he earth is changed)  and compare it with the approximation F = mg.

So, at Earth's surface
F = G m1 m2 / R^2
But if we to go just a little higher (say a height of h) than Earth's surface, we would have
F = G m1 m2 / (R+h)2
F = G m1 m2 / (R2 (1+h/R)2)
F = G m1 m2 / R2    *  (1+h/R)-2
Now, take note that since F = mg is an approximation for the above formula at h << R, we can take a Taylor series expansion of (1+h/R)-2.
This expansion would be 1 - 2h/R + 3(h/R)2 - 4(h/R)3 +-... and so on.

Since h/R is close to 0 when h << R, we can just take the first term "1" of the Taylor series expansion to approximate F = G m1 m2 / R^2. From this expansion, we can also see that unless h is big enough to cause the next few terms "-2h/R" and "3(h/R)2" to be significant, F = mg is a pretty good approximation for the gravitational force at heights close to the Earth's radius.

I'd also like to show that the two formulas are numerically identical.
Formula 1: F = m * (g)
where F is the amount of force felt by an object of mass m, and g is the acceleration due to Earth's gravity near its surface.
Formula 2: F = m1 * (G * m2 / r2)
where F is the amount of force felt by an object of mass m1 due to an object with mass m2 at a distance of r apart from each other. G is the gravitational constant.

So, the numerical value of g is commonly used as 10 ms-2, or more accurately 9.81 ms-2.
G is gravitational constant, and its value is 6.67300 * 10-11 m3kg-1s-2.
m2 is actually the mass of the Earth, and its value is 5.9742 * 1024 kg.
r is actually the radius of the Earth, and its value is 6.3781 * 106 m.
Now, if you calculate the value of the term (G*m2/r^2), we get a value that is 9.7998 ms-2-- which is extremely close to the value of g (9.8ms-2)!

Hope that clears things up!

Maurice

(published on 04/14/2012)