Monday, May 29, 2023

How to determine the mass of the earth with balls and string

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It’s fun to think about how we know it. For example, the sun has a mass of about 2 x 10 30 kilograms. It is such a huge mass that it is difficult to understand. And if it is so difficult for us to imagine such large numbers, how could we find these values? Well, the original method was to use little weights, a stick and a string. Yes, this is one of the important steps in determining the masses of the sun and of all the planets in our solar system. This is called the Cavendish experience –first played by Henry Cavendish in 1798. It’s really cool, so I’ll explain how it works.

Objects with mass have gravitational pull between them. A basketball has a gravitational interaction with the Earth (because they both have mass). It is this gravitational interaction that makes the balloon accelerate as it falls towards the ground if you let go of it. But of course, everyone has always known that if you drop an object, it will fall. However, it was during Newton’s time that people realized that this interaction also worked with astronomical objects like the Earth, the moon, and the sun. This gives us this model of force – it’s often called Newton’s law of universal gravity, but like most great ideas, it probably had a lot of contributors.

Illustration: Rhett Allain

Let’s review this model of gravitational force. First, the magnitude of this force depends on the product of the two interacting masses (m1 and M2). Second, the magnitude decreases with the square of the distance between the two objects (r). Finally, there is this G. It is the universal gravitational constant. This is the key to finding the mass of the Earth.

So take a step back for a moment. When we measure things, we always have to make some type of choice. If we want to have a mass in kilograms, we have to decide how to specify the value of 1 kg. One way would be to say that a kilogram is the mass of a liter of water. Of course, this is not the best definition (we have better methods now). OK, what about the measurement of force? We use a unit called Newton where 1 Newton is the force required to accelerate 1 kilogram to 1 meter per second per second. Yes, things are getting out of hand, but the key is that you can make those definitions and build one unit on top of another unit.


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