Episode 401: Newton's law of universal gravitation

This episode introduces Newton’s law of universal gravitation for point masses, and for spherical masses, and gets students practising calculations of the force between objects. The meaning of inverse square law is discussed.

Lesson Summary

• Discussion: Introduction to Newton’s law of universal gravitation (5 minutes)
• Discussion: Newton’s law of universal gravitation: F_gravity = GMmr²
(10 minutes)
• Worked examples: Using F_gravity = G  ×  m₁  ×  m₂r²(25 minutes)
• Student questions: More practice with F_gravity = Gm₁m₂r² (20 minutes)

Discussion: Introduction to Newton’s law of universal gravitation

Here are some questions and answers which lead towards Newton’s law of universal gravitation.

What causes the weight that each student feels? (gravitational attraction by the Earth.)

What affects the size of the Earth’s pull on you? Why would you weigh a different amount on the Moon? (Your mass, and its mass.)

If the Earth is pulling down on you, then what else must be occurring, by Newton’s 3rd Law? (You must be pulling up on the Earth with a force equal to your weight.)

What happens to the strength of the pull of the Earth as you go further away from it? (It gets weaker – most students guess this correctly from the incorrect assumption that in space, astronauts are weightless!)

So, in summary the force depends upon the masses of the Earth and you, and weakens with distance. This is all embodied in Newton’s law of universal gravitation

Discussion: Newton’s law of universal gravitation

Present the equation which represents Newton’s law of universal gravitation.

F_gravity = Gm₁m₂r²

F_gravity is the gravitational force of attraction in newton

m₁ , m₂ are the interacting masses, in kilogram

r is the separation of the two masses in metre

G is known as the universal gravitational constant ( not to be confused with little g). It sets the strength of the gravitational interaction in the sense that if it were doubled, so would all the gravitational forces.

G = 6.67 × 10^-11 N m² kg^-2

Show how the units can be worked out by rearranging the original equation.

This law applies between point masses, but spherical masses can be treated as though they were point masses with all their mass concentrated at their centre.

This force is always attractive. In some texts you will see a minus sign in the equation, so that F_gravity =  −  Gm₁m₂r². This minus sign is there purely to indicate that the force is attractive (it’s a relic from the more correct, but well beyond the syllabus, vector equation expressing Newton’s Law of universal gravitation). It’s simplest to calculate the magnitude of the force using

F = Gm₁m₂r² , and the direction is given by the fact that the force is always attractive.

Every object with a mass in the universe attracts every other according to this law. But the actual size of the force becomes very small for objects very far away. For example, the Sun is about one million times more massive than the Earth, but because it’s so far away, the pull on us from the Sun is dwarfed by the pull on us from the Earth (which is around 1650 times greater). As the separation of two objects increases, the separation² increases even more, dramatically. The gravitational force will decrease by the same factor (since separation² appears in the denominator of the equation). This is an example of an inverse square law, so called because the force of attraction varies in inverse proportion to the square of the separation.

Worked examples: Using F = Gm₁m₂r²

You can work through these examples, or you can set them as a task for your students if you feel they will be able to tackle them.

Episode 401-1: Worked examples; Gravitation forces (Word, 29 KB)

Student questions: More practice with F = Gm₁m₂r²

Episode 401-2: Newton’s gravitational law (Word, 53 KB)