Lesson 9: The Force of Gravity

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Somewhere, something incredible is waiting to be known.

Carl Sagan

Stop before you proceed

Download the Anticipation Guide and complete part one. Complete part two as you read the content pages.

Stories about Sir Isaac Newton discovering gravity after getting hit on the head by an apple while sitting under an apple tree are common, though likely not altogether true. What is more probable is that while seeing an apple fall out of a tree, Newton was inspired to view gravity in a different way than anyone had before him. He started to think that maybe gravity, a force that pulls objects together, wasn’t just limited to the Earth.

Mass versus Weight

Although we often use these terms interchangeably in everyday conversation, in science the terms mass and weight have very different, very specific meanings.

Mass is the amount of material in an object. It is measured in kilograms (kg). It is a measure of the number of protons, neutrons, and electrons that make up an object. Consequently, the mass of an object remains the same and does not change.

The more mass an object has, the more difficult it is to get that object to either start, or stop, moving. Consider the diagram to the right – it is much more difficult to change the motion of the bowling ball than it is to change the motion of the soccer ball. This resistance an object has to changing its motion is called inertia. The more mass an object has, the more inertia it has. Mass, therefore, is a measure of an object’s inertia.

Weight is the effect of gravity acting on a mass. It is measured in Newtons (N). The stronger the force of gravity is, the greater the weight of the object. While mass is a constant, weight is directly proportional to the force of gravity. The force of gravity on the Earth is about six times stronger than the force of gravity on the moon. This means that you weigh six times as much on Earth as you would on the moon. Of course, the total amount of “stuff” inside of you would not change, so your mass would be the same on both the Earth and the moon.

The Force of Gravity

The force of gravity is a force that acts to pull all matter together. It acts between the Sun and the Earth to keep the Earth orbiting at a constant distance from the Sun. It acts between us and the Earth to keep us from flying off into space. It even acts between the pencil on your desk and your calculator. Gravity, however, is actually a very weak force, which is why we only notice its effect when one or both of the objects has a very, very large mass. When we see objects that we have thrown fall back to the ground, or watch buildings come crashing down, we tend to think that gravity must be very strong. However, consider how little effort you need to exert to overcome the force of gravity acting on a textbook when you lift it over your head.
Thanks to the work of Newton, we are able to calculate the force of gravity acting between any two objects anywhere in the universe.

Newton’s Law of Universal Gravitation

F_g = the force of gravity (N)

G = universal gravitational constant (6.67 x 10^-11 Nm²/kg²)

m₁ and m₂ = masses (kg)

Δd = distance between the centre of the objects (m)

Some equations contain constants that are necessary to make the left and right sides of the equation equal. G is called the universal gravitational constant, which is equal to 6.67 x 10^-11 Nm²/kg².
From the equation, we can see that both mass values are found in the numerator. This means that as the mass of one of the objects increases, so does the force of gravity. For this reason, planets that have a greater mass than the Earth have a greater force of gravitational attraction.
The Δd value is found in the denominator. This means that the force of gravity is smaller at the top of a very high mountain than it would be at sea level.

Enrichment

Enter the phrase “shape of the Earth” into an internet search engine such as Google or Yahoo and read one or two of the resulting articles.

How could you use this information to make predictions about the force of gravity at various points around the globe?

Watch the video below to see what happens when you jump on an elevator!

Example 1

The Earth has a mass of 5.98 x 10²⁴ kg and a radius of 6.38 x 10⁶ m. What is the force of gravity acting on a 70 kg person standing on the ground?

Given:

G = 6.67 x 10^-11 Nm²/kg²

m₁ = 5.98 x 10²⁴ kg

m₂ = 70 kg

Δd = 6.38 x 10⁶ m. Even though the person is standing on the ground, we need to measure their distance from the centre of the Earth.

Required:

F_g

Solution:

F_g = 686 N

The force of gravity acting on a 70 kg person standing on the ground is 686 N.

Practice Questions

1. Using the above example as a model, calculate the force of gravity acting between a 70 kg student and their 18 kg desk if they are placed 50 cm apart.

Answer

Given:

G = 6.67 x 10^-11 Nm²/kg²

m₁ = 70 kg

m₂ = 18 kg

Δd = 0.50 m (remember to convert the distance to metres)

Required: F_g = ?

Solution:

F_g = 3.36 x 10^-7 N

The force of gravity acting between a 70 kg student and their 18 kg desk if they are placed 50 cm apart is 3.36 x 10^-7 N.

This force is incredibly small. The force of gravity is only noticeable when one or both of the objects have a very large mass.

A Simplified Gravity Equation

The beauty of Newton’s Law of Universal Gravitation is that it will always work. Its drawback is that it is a difficult equation to have to use whenever we need to calculate the force of gravity. Most of the questions or problems that you will encounter deal with the force of gravity acting on objects that are very close to the surface of the Earth.

Work though the following problem to derive a simplified equation for finding the force of gravity here on Earth.

 Check Your Understanding

1. The radius of the Earth is 6.38 x 10⁶ m (6380 km). This value does not change significantly if we walk up a hill, travel up an escalator, or even take a flight on an airplane. The mass of the Earth is 5.98 x 10²⁴ kg.
   1. Substitute these values into the Law of Universal Gravitation.
   2. Simplify the expression so that Force (F) and mass (m₂) are the only variables.
   3. What is the new constant found in the expression that you have derived?
   4. Where have you seen a value very close to this before?
      

Answer

1.

.

2. F_g = 9.799m₂.

F_g = 9.8m₂.

3. 9.8 N/kg.

4. This is the same value that we used for acceleration due to gravity.

You should have calculated a value very close to g = 9.8 N/kg. This is the same value that we used for acceleration due to gravity. This is not a coincidence. As you can see below, N/kg and m/s² are equivalent units.

This gives us a simplified gravity equation shown to the right.

Example 2

Ivan is a very strong weightlifter. The weights he is lifting have a force of gravity of 1 470 N. What is the combined mass of the weights?Given:g = 9.8 N/kgF_g = 1470 NRequired:m = ?

Solution:

F_g = mg

m = 150 kg

The weightlifter is lifting a mass of 150 kg.

Practice Questions

1. What would the force of gravity acting on the 150 kg weights be if the weightlifter was on Mars where the acceleration due to gravity is only 3.7 N/kg?

Answer

Given:

m = 150 kg

g_mars = 3.7 N/kg

Required:

F_g = ?

Solution:

F_g = mg

F_g = (150 kg)(3.7 N/kg)

F_g = 555 N

The force of gravity acting on the 150 kg weights is 555 N.