Calculating g using the Law of Universal Gravitation
Recall that g around the surface of the Earth is a constant with a calculated value of 9.8 m/s^{2} or 9.8 N/Kg
Another way to calculate the acceleration due to gravity g is by using Newton's Law of Universal Gravitation
Recall that the Force of gravity around the surface of the Earth is calculated with the formula F_{g }= m x g
But we can also use Newton's Law of Universal Gravitation to calculate the value of g.
We start with the Law of Universal Gravitation
Recall that In the above equation F is the force of gravity, You can also call it F_{g}
Also recall that F_{g} = m g
Thus the two expressions can be equated to one another' and we can say that
where:
- m is the mass of any object at the surface of the Earth.
- M is the mass of the Earth
- G is the constant of Universal Gravitation (6.67 x 10^{-11} Nm^{2}/kg^{2}).
Note that the mass m can be dropped out from each side of the equation and we can simplify the above as:
The value of g on Earth can be calculated if we know the mass of the Earth and its radius R.
This fact should not surprise us after all we've always stated that the acceleration due to gravity on an object is independent of the object's mass.
Example:
Use the information that the mass of the Earth is 5.98 x 10^{24} kg and that its radius is 6.38 x 10^{3} km (6.38 x 10^{6 } m) to confirm that g on Earth is approximately 9.8 N/kg
Solution:
Using the formula above we have:
Do a unit analysis check on your own to verify that the answer is in N/kg
It's always a good practice to do this at the end of every problem to verify that the units are correct!