How to find the gravitational acceleration constant?

Question

If mass of earth is 5.98e24 kg and its diameter is 15000 km.
How can I get constant (a)?

Answer 1

It wasn't and isn't simple to measure either G or M (such as Earth's mass) with precision. What is relatively easy to measure is the product, GM. This can be done with accurate knowledge of R(earth) and g at earth's surface.
Cavendish made the first attempt to measure G using suspended weights configured as a torsion balance pendulum. His measurements have stood up pretty well, but now there are several conflicting authoritative values for G, generally differing in the 5th significant figure. Wiki (ref.) has interesting history and data.
Getting back to the question, if you're asking how to find the value of earth surface gravity g "from scratch", the classic experiment would be to drop a weight from a known height h and measure its fall time t. Then h = gt^2/2 so g = 2h/t^2. For the experiment, you'd want to minimize air resistance (use a heavy, streamlined weight) and not use too high a tower (since g decreases with height). You'd also need an accurate, preferably photoelectric, time measurement and a synchronized release mechanism.
Or if you want to calculate g or G from assumed values for M and R, just use the formulas
g = GM/R^2
G = gR^2/M

Answer 2

No, it works the other way around.

First people agreed that 40,000 km constitute the length of meridian of the Earth, and its diameter became
D = 40,000 km / π = 12,730 km

Then they established how long 1 meter is.
Then they measured gravitational acceleration
g = 9.8 m/s².
Then they independently measured universal constant of gravitation
G = 6.67e-11 m³/s² kg

And only after all this activities they used the formula
g = GM/R²
and derived the mass of earth:
M = 5.87e+24 kg.


How else? You do not think someone ever weighted the planet on scales, do you?

Answer 3

Once Newt came up with F = GmM/R^2 = mg = W; where m is your mass, W is your weight, M is Earth's mass, and R = 1/2 diameter is the distance between your center of mass and Earth's, it was simple to solve for G = gR^2/M, given experiments to show that g ~ 9.81 m/sec^2 (we usually use g, not a, to show the acceleration is due to gravity).

The experiments to find g are straightforward. Just drop objects of mass m and measure the accelerations in a vacuum. Another way to find g is to calculate your mass (using buoyancy forces) and simply divide your weight W/m = g.