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Calculating Earth's Gravity Using Newton's Law of Gravitation

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  • 0:04 The Same But Different
  • 0:27 Newton's Law of Gravitation
  • 2:27 Equation 1
  • 3:06 Equation 2
  • 3:46 Finding Acceleration…
  • 5:04 Lesson Summary
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Lesson Transcript
Instructor: Matthew Bergstresser
Gravity is more or less constant on Earth. There are two ways we can derive the gravitational acceleration constant on Earth. In this lesson, we will look at the two ways and show how they are the same thing.

The Same but Different

Have you ever noticed there are some words that mean the same thing, but are spelled differently? For example, ''check'' and ''cheque'', or ''theatre'' and ''theater''. Interestingly, this applies to physics as well in terms of gravitational force on Earth. Let's investigate this and see how we can derive acceleration due to gravity on Earth.

Newton's Law of Gravitation

Newton quantified the attractive force between two masses to be:


Newton_law_grav


where:

  • Fg is the gravitational force between two masses in newtons (N)
  • G is the universal gravitational constant, which is 6.67 x 10-11 (N⋅m2 / kg2)
  • m1 and m2 are masses 1 and 2, respectively
  • r is the distance between the centers of mass of m1 and m2

This equation is known as Newton's law of gravitation. It's the formal way to calculate the force of attraction between two objects. If we're on or near Earth's surface, this equation represents the weight of an object. This equation can be cumbersome to use because of the multiple terms in it along with the extreme values of G and the radius of Earth. There is a shortcut, which is:


mg


where:

  • m is the mass of the object in kilograms (kg)
  • g is the acceleration due to gravity, which is 9.8 m/s2

It is important to note that since the gravitational attraction between Earth and an object decreases by the square of the distance between their centers of mass, the mg shortcut only applies at or near Earth's surface. For our purposes in this lesson, we'll be on Earth's surface when doing any calculations relating to gravity.

Let's say we have a 5 kilogram mass on the surface of Earth, which has an average radius of 6.38 x 106 m and a mass of 5.98 x 1024 kg. Let's prove that these two equations will give us the same weight of the 5 kilogram mass.

Equation 1

Using Newton's law of gravitation to determine the weight of the 5 kg mass, we get:


e1


This gives us a gravitational force of 48.9954403 N.

Equation 2

Now we'll use mg to calculate the weight of the 5 kg mass, which is:


e2


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