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**Physics, Astronomy & Cosmology / Re: How did scientists measure the mass of the Earth and other planets?**

« **on:**14/10/2018 09:37:33 »

Quotefrom: alancalverd on 12/10/2018 09:49:23Using F = GmM/r2 you can calculate the force on a falling object of mass m in terms of M, the mass of the earth, and G, which we assume to be a universal constant.That doesn’t work. We’re trying to compute at least a rough G and M here. We don’t know either of them yet. We do know force F is 9.8 newtons for a 1KG mass. We can assume we know r. We therefore know the product of G and M, but not either separately.

We don't "know" F = 9.8 N/kg. We define the newton as1 kg.m.s

^{-2}, and have a

*measured*value of

*g*= F/m

We don't "assume" r: it is a

*measured*value - see below.

As I said, Cavendish (1797)

*measured*G directly.

Maskeleyne estimated the density of the planet from an experiment in 1774 at Schiehallion, measuring the deflection of a plumb bob on two sides of the mountain. In fact, that was a measure of G rather than ρ

_{earth}since the assumption that the mountain was homogeneous was more defensible. Newton had considered the possibility but rejected the experiment as too difficult.

Quote

QuoteAs F = ma, we can measure the acceleration of a falling object or the period of a pendulum to get a value for F/m

F=ma works (F 9.8 = 1 (mass) * 9.8 m/sec acceleration), but that doesn’t yield either mass of Earth M nor G, which are the two things we’re trying to determine here.

The pendulum thing is a function of acceleration (9., not of the mass of Earth. Put a pendulum in a rocket accelerating at that rate and it will have the same period as here on Earth. It tells you nothing about the mass of the Earth under you.

You have missed the point. If we know F/m we know GM/r

^{2}, and we have known r since 200 BC (Eratosthenes), so all that was required for an accurate estimate of M was Cavendish's determination of G. The reason for using a pendulum is that we know the period T = 2π√(Lm/F) for small oscillations, which is a simpler experiment than measuring the acceleration of an object in free fall.

The most amazing fact in all this, hundreds of years and thousands of experiments later, is that the gravitational masses of all bodies are identical to their inertial masses. Why?

Interestingly, Hipparchus estimated the radius of the moon's orbit around 130 BC, though his assumptions regarding the earth's orbit of the sun were somewhat wild.

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