Can You Explain Kepler's Third Law?


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Steve Theunissen Profile
By means of his first two planetary motion laws, Kepler had derived formulas for the shape and the speed of a planet's orbit. The answer to another perplexing question remained: What relation is there between a planet's distance from the sun and the time it takes to complete a circuit. He knew that planets that are closer to the sun travel at greater speeds than those farther away. After nearly 10 years of labour he discovered a formula that expressed this relationship. This came to be known as his Third Law. This law states that the squares of the periods of revolutions of any two planets are in the same ratio as the cubes of their average distances from the sun.

An example of this relationship can be seen in the case of the planet Jupiter. Jupiter is approximately 5.2 times as far from the sun as is the Earth. Correspondingly, it takes Jupiter about 11.8 earth years to make one orbit around the sun (called its "period" in the chart below), which is one Jupiter year. Let us prove the accuracy of the Third Law by applying it in the case of the planet Jupiter.

To square a number is to multiply it by itself; to cube a number is to multiply this result again by the original number. So going back to the example of Jupiter, what do we find? If we square the period (Jupiter's period of orbit around the sun is 11.8 earth years), we get 11.8 times 11.8, which equals nearly 140. Now, if we cube the distance, we get 5.2 times 5.2 times 5.2, which also equals approximately 140. This equality holds true for each one of the planets. You can easily prove this for yourself by carrying out the same calculation for the rest of the planets.
Tariq Habib Profile
Tariq Habib answered
Using production functions, we can understand one of the most famous laws in all economics, the law of diminishing returns:
The law of diminishing returns holds that we will get less and less extra output when we add additional doses of an input while holding other inputs fixed. In other words, the marginal product of each unit of input will decline as the amount of that input increases, holding all other inputs constant.
The law of diminishing returns expresses a very basic relationship. As more of an input such as labour is added to a fixed amount of land, machinery, and other inputs, the labour has less and less of the other factors to work with. The land gets more crowded, the machinery is overworked, and the marginal product of labour declines.
Putting ourselves in the boots of a farmer performing an agricultural experiment can flesh the law of diminishing returns out. Given a fixed amount of land and other inputs, assume that we use no labour inputs at all. With zero labour input there is no corn output. Hence, there will be zero products when labour is zero. Diminishing returns are a key factor in explaining why many countries in Asia are so poor.

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