Universal Gravitation and the Equation

Learn how to use Newton's Law of Universal Gravitation. The force of gravity a weak force resulting from two objects having mass and the distance between them.  Universal gravitation is not just planetary but attracts you to everything around you as well.

Isaac Newton's Big Idea

Isaac Newton had a lot of brilliant ideas in his lifetime and a massive one was universal gravitation.  Universal gravitation is not just the fact that objects on earth fall to the ground.  His big idea was that all objects in the universe attract all other objects.  An apple may or may not have hit him in the head to cause this revelation.    While the apple is probably not true it's not far from the point.  Isaac Newton questioned two things.  Why would object on earth fall to the ground while the moon continued in its circular path?

Newtons Apple and The Law of Universal Gravitation

The Moon and Apple Both Fall

Newton realized that both the apple and moon are really falling.  The earth follows the rules of the rest of the universe.  The apple is starting from rest and being attracted to the ground.  The moon has inertia which normally would make it continue forward but the attraction to the Earth makes it fall.  The combination of forward motion and falling gives the moon its orbital path around the Earth.

Moon Falling

Newton's Law of Universal Gravitation Equation

Any object with mass (m1) is attracted to any other object with mass (m2).  This is universal as the name says meaning "all" objects.  The amount of force depends on distance (d) and related by the universal gravitation constant (G).  G is a very small number 6.67 x 10-11 Nm/kg2 and not the 9.8 m/s2 (g) for the acceleration due to gravity when objects are on earth.  Universal gravity is the weakest of the fundamental forces in the universe.  Because it is so weak, the force of gravity takes massive objects like the earth (5.972 × 1024 kg or 5,972,000,000,000,000,000,000,000 kg) to feel it.  You feel the earth’s attraction, your weight (FW), but no other object “also” attracting you in a room.  The weakness can be seen by the gravitation constant (G) being so small.  6.67 x 10-11 is a really small number 0.0000000000667.

Universal Law of Gravitation

Analyzing the Equation

One way to analyze the force of gravity equation, place a 1 in for everything representing no changes.  Then change the one thing you are trying to analyze to a bigger number 2 and see what happens.  The rule of ones (click here) is a more detailed way to see how multiple changes alter force.

Force of Gravity Relative to Mass

Force of Gravity to Mass

When you change out mass, the force of gravity of 1 becomes 2 or gets bigger.  Force of gravity is directly related to mass.

Universal Gravitation with mass increasing

Force of Gravity Relative to Distance

Force of Gravity to Distance

When you change out distance, the force of gravity of 1 becomes ¼.  This is not just smaller (inversely where 1 would become 2) but inverse squared 1 becoming ¼ with a much greater effect.  The force of gravity is inversely squared related to distance.

Universal Gravitation With Distance Increasing

Universal Gravitation Distance to Force Graph

Notice what the distance the masses apart does to the force.  The distance apart of two masses is inversely proportional to the square of the distances (inverse squared).  Observe how the graph line is not straight but cured because the biggest drops in force occur early.

Universal Gravitation Distance to Force Graph
Rearranging the Universal Gravitation Equation for Mass

Rearrange the Universal Gravitation Equation

You need to be able to rearrange the universal gravity equation to solve for a mass or distance between objects.  Our animations on the right show the process.

Rearranging the Universal Gravitation Equation for Distance

Example Problems:

Example 1: What is the force of gravity between the earth and the moon?

Earths mass: 6.0 x 1024 kg

Moons mass: 7.35 x 1022 kg

Distance between: 3.844 x 108 m

Givens

m1 = 6.0 x 1024 kg

m2 = 7.35 x 1022 kg

d = 3.844 x 108 m

G= 6.67 x 10-11

Equation and Work

Equation and Work 1

Example 2: How much will the force of gravity change if you double one mass and quadruple the distance?

Follow the link back to the Rule of Ones lesson to see why we did what we did here.

Fg = ?   m1 = 2   d = 4

subtituted numbers rule of one

An answer of 1/8 or 0.125 means the force of gravity will be 1/8 or 0.125 times the original.

Additional Problems Including Solving for a Mass and Distance

1. What is the force of gravity between earth (5.972 × 1024 kg) and mars (6.39 × 1023 kg) when they are at their minimum distance of 5.46 x 1010 meters?

problem 1 solution

2. What is the force of gravity between earth (5.972 × 1024 kg) and mars (6.39 × 1023 kg) when they are at their maximum distance of 4.01 x 1011 m?

problem 2 solution

3. What is the distance between earth (5.972 × 1024 kg) and mars (6.39 × 1023 kg) when the force of gravity is 5.1 x 1015?

problem 3 solution

4. What is the mass of a planet that has a force of 1.94 x 1017 N when away 1.0 x 1011 m from 5.972 x 1024 kg earth? Look on the chart and determine what astronomical body in the earths solar system it is.

problem 4 solution

(Click Here to See All Problems and Solutions Above)

Planets Sun and Moons Mass Table

Universal Gravitation Practice Quiz

Universal Gravitation Quiz

1 / 11

How is the force of gravity related to mass?

2 / 11

How is the force of gravity related to distance between object?

3 / 11

What does a G having a value of 6.67 x 10^-11 tell you about the force of gravity between any two object?

4 / 11

What would happen to the force of gravity if you tripled the distance between the objects?

5 / 11

What would happen to the force of gravity if you doubled the mass of one object?

6 / 11

What would happen to the force of gravity if you doubled the mass of one object and doubled the distance?

7 / 11

What is the force of gravity between 96kg Joe and 87kg Jim when the distance between them is 0.20 meters?

8 / 11

Mass Distance Graph

Which graph correctly shows how mass relates to force using the Law of Universal Gravitation?

9 / 11

Force Distance Graph

Which graph correctly shows how distance relates to force using the Law of Universal Gravitation?

10 / 11

What is your mass if the attractive force from the 55 kg person 2.0 meters across the table is 4.586 x 10^-8 N? (when rounded to two significant figures)

11 / 11

What is the force of attraction between Isaac Newton and his 0.15 kg apple the moment it is 0.3 meters away if his weight was 83.6 kg?

Your score is

0%

Links