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AskUs: Why can’t I weigh the earth by putting a scale upside-down?

So one of our readers asked us this question, Why can’t I weigh the earth by putting a scale upside-down?

PLEASE READ THIS BEFORE ANSWERING

This is a theoretical question about gravity not just a stupid question to be funny. Gravity pulls two objects with mass together. The force of gravity is equal to a mass of the object multiplied by an acceleration of a body (in this case, the acceleration of gravity). Both earth and the scale experience the same gravity acceleration because they are both on earth. The force of the scale on the earth should be its mass multiplied by the acceleration. Conversely, the force the earth exerts on the scale should be its mass multiplied by gravity acceleration.

But Newton’s second law states there are equal and opposite forces so the force the scale exerts on the earth should be equal to the force exerted by the earth on the scale. It seems that this case is true because the scale doesn’t rocket off into space when you turn it upside down but stays in place.

So is force really mass x acceleration? Where is this discontinuity coming from?

ANSWER

This is a brilliant question because in some sense you are measuring the earth’s mass.

By Newton’s third law, the force exerted on the scale by the earth is the same as the force exerted on the earth by the scale – you know, the ‘equal and opposite reaction’ law.

In this case that force is the force of gravity. The force between two objects of masses M and m separated by a distance R is equal to

F = G M m / R^2

The key point is that scale just happens to be calibrated to measure the mass for an object experiencing earth surface gravitational acceleration – i.e. it assumes GM/R2 is a constant value (which is equal to g=9.81 m/s2), and then returns the value for m that when scaled by this constant is equal to the force the scale measures.

If you had a scale of a known mass and you turned it upside down you could then calculate the value of this constant – GM/R2. Then, with known values of G and R, you could calibrate your scale to measure the mass of the earth rather than the mass of the scale.

And to clarify an important point – the earth and the scale don’t experience the same accelerations. Use Newton’s second law:

m a_1 = G M m / R^2

and find the acceleration of the scale is

a_1 = GM/R^2

Conversely, the force of the earth is

M a_2 = G M m / R^2

So the acceleration of the earth is

a_2 = G m / R^2

Since M is the mass of the earth and m is the mass of the scale, a_2 is much much much smaller than a_1.

The earth has a mass of 5.972×1024 kilograms. It has a variable weight because it isn’t on a planet. You could say that the force of attraction between it and the sun kind of gives it weight in the same way the force of attraction between you and the earth gives you weight, but that’s not a useful scale of measurement, and is really just faffing about changing the definitions of things.

For funsies: If the earth were on the earth, and you used one earth as a reference, the earth would weigh something like 13,000,000,000,000,000,000,000,000,000 lbs. By unit analysis (convert kg to lbs., mass->force = mass • acceleration (gravitational pull of earth) aka weight).

So in short, the scale is designed to give the weight of the thing on it. It uses the mass of the earth as a constant. If you flipped it upside down, (and knowing how much the scale weighs) you could use that to guesstimate the mass of the earth using Newton’s Laws, and some fancy math.

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