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Universal Gravitation

Gravity and the Universal Gravitation 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?

Isaac Newton Apple Falling Gravity Animation

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 at a constant velocity but the attraction to the Earth makes it fall.

The moon acts like a projectile with its inertia taking it forward as it also falls to the earth.  This combined action keeps the moon in orbit around the earth.

Isaac Newton Moon Following a Projectile Path

Q1: What was Isaac Newton’s big idea related to gravity?

All objects attract all other objects

Q2: Why does the Moon’s not collide with the Earth if there is an attraction?

The moon’s inertia keeps it going forward as it is falling as a projectile.  This creates an orbital path around. 

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 which have mass.

The amount of force depends on distance (d) and related by the universal gravitation constant (G).

Universal Law of Gravitation
Variable Name Unit Unit abbreviation
Fg Force of gravity Newton N
m1 Mass of one object kilograms kg
m2 Mass of a second object kilograms kg
d Distance between the objects meters m
G Universal Gravitation constant Newton∙meters/kilograms squared Nm2/kg2
g Acceleration due to gravity Meters per second squared m/s2
Universal G

Universal Gravitation Constant (G)

The universal gravitation constant (G) is a very small number 6.67 x 10-11 Nm2/kg2 and not the 10 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 gravity is such a weak force, 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.

Weight and Force of Gravity

You attract the earth with a force equal to that which the earth attracts you.

Weight (Fw) is equal to the force of gravity (Fg)

  • Using your mass (m1)
  • Earth’s mass (m2)
  • The distance between you and the center of the Earth (d) (Earth’s radius)
Weight and Force of Gravity

Analyzing the Universal Gravitation 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.

How do you answer a question like this:

How does the force of gravity change when you double the distance between two objects?

You set up a ratio following the steps below:

1. First find the equation most related.  This question asks about force when you are changing distance between.

force of gravity equation

2. Then find what you are trying to analyzes equivalent.

Fg is equivalent or equal to equivalent to the force of gravity

3. Then create ration of after (top) vs. before (bottom)

force of gravity ratio

4. Plug in ones for everything that did not change.  This is the same as saying it is one times itself or in other words, itself.

Rule of Ones Ratio

Once you solved for this you know the ratio of how many times different the force now is compared to the original.

5. If you knew the original value, you would multiply it by this ratio.

Q3: What happens to the force of gravity when you double the mass of both objects?

solution 1

Q4: What happens to the force of gravity if you double the mass of one object and double the distance between objects?

solution 2

Force of Gravity Relative to Mass

  • Gravitational force is directly proportional to mass of an individual object

Q5: Draw the shape of the graph of increasing mass of one object to the force of gravity

force of gravity versus mass

Q6: If the mass of one object increases what happens to the force of gravity and at what rate?

It increases at a constant rate (seen by the straight line)

Force of gravity directly related to mass graph animation

Force of Gravity Relative to Distance

  • Gravitational force between two objects is inversely proportional to the square of the distances (inverse squared).

 

Q7: Draw the shape of the graph of increasing distance to the force of gravity

FOrce of Gravity Distance Graph

Q8: If the distance between two objects increases, what happens to the force of gravity and at what rate?

It decreases at a diminishing rate (seen by the negative sloped line that is becoming flatter)

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:

Q8:  What is the force of gravity between the Earth (mearth = 6.0x1024 kg) and the Moon (mmoon = 7.35x1022 kg) when they are 3.844x108 m apart?

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

Q9: 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?

9

Q10: 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?

10

Q11: What is the distance between Earth (5.972 × 1024 kg) and Mars (6.39 × 1023kg) when the force of gravity is 5.1 x 1015?

11

Q12: 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?

12

Universal Gravitation Practice Quiz

Universal Gravitation Quiz

1 / 11

How is the force of gravity related to mass?

Mass to the force of gravity

2 / 11

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

gravity 7

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?

G is a very small number 0.0000000000667 and therefore it takes huge masses like planets to feel the attraction to something.  You cant feel the attraction to your neighbor, but there is one.  You can feel the attraction to earth.

4 / 11

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

gravity 6

5 / 11

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

Gravity Problem

6 / 11

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

gravity 4

7 / 11

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

Gravity3

8 / 11

Mass Distance Graph

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

Mass is in the numerator of the force of gravity equation below and directly related to the force of gravity.  If the mass doubles, the force of gravity would double.

gravity equation

9 / 11

Force Distance Graph

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

Distance is in the denominator and squared in the universal gravitation equation below.  There is a inverse squared relation.  Force decreases not proportionately more early since the d is squared. If you double distance the force goes down to 1/4.

gravity equation

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)

Universal Gravitation Problem

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?

Universal Gravitation

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Unit 1: One Dimensional Motion
Unit 2: 2D Motion
Unit 3: Newton’s Laws and Force
Unit 4: Universal Gravitation and Circular Motion
Unit 5: Work, Power, Mechanical Advantage, and Simple Machines
Unit 6: Momentum, Impulse, and Conservation of Momentum
Unit 7: Electrostatics
Unit 8: Current and Circuits
Unit 9: Magnetism and Electromagnetism
Unit 10: Intro to Waves
Unit 11: Electromagnetic Waves
Unit 12: Nuclear Physics

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