Lab: Confirm that Newton's Law of Universal Gravitation is an inverse square law
Given: the following data
Average Earth -Moon Distance (m) |
Radius of Moon (m) |
Radius of Earth (m) |
Mass of Earth (kg) |
Mass of Moon (kg) |
Mass of Sun (kg) |
Average Earth -Sun Distance (m) |
Radius of the Sun (m) |
3.84 x 10 6 | 1.74 x 106 | 6.38 x 106 | 5.98 x 1024 | 7.35 x 1022 | 1.99 X 10 30 | 1.496 x 10 11 | 6.96 x 108 |
Task: use graphical techniques to confirm that Newton's Law of Universal Gravitation is an inverse square law.
Solution:
Note: The center-to-center distance between the Moon and the Earth is the distance between the two celestial bodies plus their respective radii. i.e.
Gathering and Compiling additional Information:
For The Moon -Earth System
R1 = 1.74 x 106 m + 6.38 x 106 m + 3.84 x 10 6 m = 1.196 x 10 7 m
For the Sun -Earth System
R2 = 1.496 x 1011 m + 6.38 x 106 m + 6.96 x 10 8 m = 1.50 x 10 11 m
Procedure:
Use the Equation for the Law of Universal Gravitation
- Find the forces for Moon -Earth System
F1 = 2.05 x 1023 N - Find the forces for the Sun -Earth System
F2 = 3.53 x 1022 N - Compile the data in a chart and Plot the data on a graph
System | Forces (N) | Distances (R) (m) |
Distances-squared (R)2 , (m2) | Product of Masses (kg2) |
Earth-Moon | 2.05 x 1037 | 1.196 x 10 7 | 1.430 x 10 14 | 4.57 x 1047 |
Earth-Sun | 3.53 x 1022 | 1.50 x 10 11 | 2.25 x 10 22 | 1.19 x 1055 |
Here is the same data and its analysis using spreadsheets
System | Forces (N) | Distances (R) | Distances-squared (R)2 , (m2) | Product of Masses | Inverse Distances-squared (1/R)2 , (1/m2) |
(m) | (kg2) | ||||
Earth-Moon | 2.05E+37 | 1.20E+07 | 1.43E+14 | 4.57E+47 | 6.99E-15 |
Earth-Sun | 3.53E+22 | 1.50E+11 | 2.25E+22 | 1.19E+55 | 4.44E-23 |
This shows that in the relationship F α 1/R2,Force is inversely proportional to the distance squared
To confirm the inverse square relationship we can plot F vs. 1/R2
Note: F vs. 1/R2 gives us a linear relationship.
Conclusion: the Law of Universal Gravitation is a an inverse square law