PMNIMG said:
you are right. But just for once follow my equation and calculate. It will be Earth's gravitational constant
When you say "my equation", which of the many equations you have posted do you mean?
At a guess, you mean:
PMNIMG said:
Here I expect that ##G## is Newton's universal gravitational constant, ##M## will be the mass of the Earth and ##k## will be the constant in Kepler's third law as it applies to the orbital period and orbital diameter of the moon.
Note that to some extent, this calculation is backward. We calculate ##GM_{\text{earth}}## based on the observed gravitational acceleration ##g## at the surface of the earth and the known radius of the earth. We measure ##G## with something like the Cavendish experiment and we infer ##M_{\text{earth}}## from there. But you are asking that I compute ##G## based on the mass of the Earth when the truth is that we compute the mass of the Earth based on ##G##.
In any case, you asked for the calculation. I will
show my work. We want to evaluate ##G## in ##G = \frac{4 \pi^2}{Mk}## where ##k## is the constant of proportionality in Kepler's third law for the orbit of Earth's moon.
We have ##M_\text{earth} = 5.972 \times 10^{24}\text{ kg}##
We need ##k## from Kepler's third law: ##T^2 = ka^3##
We solve for ##k## yielding ##k = \frac{T^2}{a^3}##.
We substitute in ##T## which is about 2.4 million seconds (one sidereal month).
We substitute in ##a## which is about 384 million meters. (384 thousand km - orbital radius)
So ##k = \frac{(2.4 \times 10^6)^2}{(3.84 \times 10^8)^3} = 1.02 \times 10^{-13} \text{ s}^2/\text{m}^3##
We now evaluate ##G = \frac{4 \pi^2}{Mk} = \frac{4 \pi^2}{(5.972 \times 10^{24})(1.02 \times 10^{-13})} = 6.5 \times 10^{-11} \text{ m}^3/\text{kg}/\text{s}^2##
Which is roughly correct.
(I cheated and simply assumed that the units came out right without carefully checking. But on review, it looks like the units do indeed come out properly).
Note that because I showed my work, you can easily see whether I solved the wrong problem, used the wrong input data, fumbled the algebra, fumbled the calculation, used an equation incorrectly, or used the wrong units. You can point to a specific difficulty. You are not forced to say only "that does not match what I got".
