Mini Math LabExponential Decay Of Gravitational Force

Between Earth and the Sun.

I know gravity is always a force of attraction between objects with mass

Science: I can calculate the force of gravity between objects dependent on the distance between the objects using Newton’s Law of Universal Gravitation and graph the results.

Math: I can plug values into a formula and solve a single equation with a single variable.

Objectives

Gravity

Newton’s Law of Universal Gravitation

The mass of the Sun is 1.99 x 10^30 kg

The mass of the Earth is 5.97 x 10^24 kg

Because these numbers are so large and our graph paper is so small, I have reduced the masses of the Earth and Sun into relative units.

Actual Masses of Earth and the Sun

Relative mass of the Earth is 6 units.

Relative mass of the Sun is 2,000,000 units.

How much more massive is the Sun than the Earth?

More than 330,000 times.

Relative Masses of the Earth and Sun

An astronomical unit is the average distance from the Earth to the Sun or 93,000,000 miles.

We are going to use the AU as our unit of distance so we can see what happens to the Gravitational Force each time we double the distance.

Astronomical Unit (AU)

Astronomical Unit (AU)

The Constant

We will not calculate with the constant because we are looking for relative data, not actual data. Because this number is constant regardless of all other values, we can leave it out for now.

The manipulating variable (the one we are changing) is?

Distance between objects on the x axis. So the responding variable is? The gravitational force between the

objects on the y axis. What are the constants? The masses of the two objects and

Newton’s number.

The Variables

The Formula

F=m1 x m2d^2

F=6 x 2,000,0001 ^ 2

This point will be well off the top of the graph because there is no room on the X axis. Just record it in your data.

The First Equation

The First Point

Now calculate the force for: 2 AU 4 AU 8 AU 16 AU Don’t forget to square the

denominators!!!!

The Rest of the Equations

6 x 2,000,0002^ 2

The Formula

The Second Point

6 x 2,000,000

d^ 2

The Graph

Plot the 5 points that will fit on the graph.

Y axis interval is 250,000 N.

The X axis is labeled.

Once the points are plotted, carefully connect the dots.

6 x 2,000,000

d^ 2

Does Your Graph Look Like This?

Describe the actual line. It is curved not a

straight line. Exponential

relationships appear as curves on a graph.

Therefore the relationship between Gravity and Distance must be:

Indirect and exponential.

Slope

Positive-Direct Relationship

Negative-IndirectRelationship

Is it a direct or indirect relationship? Indirect. What does that mean? It means that as one value increases

the other Decreases. So as the distance between objects

increases, what happens to the force of gravity between them?

It decreases!!!!

What Does The Relationship Look Like?

How can we tell from the graph this is an indirect relationship?

It has to do with the slope of the line or curve.

The slope of the line is negative. Remember, an uphill from left to

right is a positive slope and a down hill is a negative slope.

What Does The Relationship Look Like?

As distance between objects increases, gravity

decreases As distance between objects

decreases, gravity increases

Conclusion

Mass and Distance Affect Gravity

Let’s Compare Changing Mass

M1=1 M2=1 M1=1 M2=2 M1=2 M2=2 M1=2 M2=4 M1=4 M2=4 M1=4 M2=6 d^2=1.25

F=(6.67 x 10^-11) Nm x (5.97 x 10^24) kg x 50 kg

kg x kg (6,378,000 ^2) m

Why a distance between you and Earth? Isn’t it zero?

NOOOOO!!!!! Never zero! Distance is measured to centers.

Gravity Acts Like It All Comes From An Imaginary Point In The Center

F=(6.67 x 10^-11) Nm x (5.97 x 10^24) kg x 50 kg

kg x kg (6,378,000 ^2) m

490 N

Calculate Force Between You and Earth

F=(6.67 x 10^-11) Nm x (7.35 x 10^22) kg x 50 kg

kg x kg (384,400,000 ^2) m

.oo169N

Calculate Force Between You and The Moon

FEG=140 N FMG=.00169 N Which is stronger? Earth’s: How much? (Divide) 289,941 times stronger!

Why Doesn’t the Moon’s Gravity Pull Us Off The Earth?

F=(6.67 x 10^-11) Nm x (1.99 x 10^30) kg x 50 kg

kg x kg (149,600,000,000 ^2) m

FS=0.297N FE=490N Earth’s gravity is still (Divide) 1650 times stronger

How About The Sun?

Compare The Variables

Pos./Direct/LinearNeg./Indirect/Exponential

Gravity