Things fall. Pencils, buildings, people, everything falls. The Earth is falling towards the Sun. Luckily, it is falling in a way that consistently misses the Sun.
Early Theories of Gravity
Physics often contradicts our feelings for how it should work. The history of gravity tells the story of humanity realizing we aren't the center of the universe.
The Greek philosopher Aristotle (384–322 BC) believed that the natural place for the element of earth and water was down. The natural place for the elements of air and fire was up. He also believed that heavier objects fell faster than light objects.
Like many early philosophers, Aristotle organized the sky into a geocentric model. Moving planetary spheres surrounded a stationary and spherical Earth.
The geocentric models showed flaws. Planets would inexplicably change in speed and direction. To correct these flaws several small circles called epicycles were added to the path of the planets. As telescopes and data collection methods improved, the number of epicycles grew.
Around 1543, Polish astronomer, Nicolaus Copernicus published a book on the heliocentric theory. The idea that the planets revolved around the Sun, not the Earth. His heliocentric theory marks the beginning of the the scientific revolution.
In 1609, German astronomer, Johannes Kepler published 3 laws of planetary motion based on the heliocentric theory. In his model orbits moved in an ellipse, not circles.
Italian natural philosopher, Galileo Galilei, discovered that falling objects all accelerate at the same rate as long as air resistance isn't a significant factor.
Galileo also championed Copernican heliocentrism. This blasphemous idea upset the Catholic church so much they held a trail and found him guilty of heresy. They forced him to recant his findings, they banned his work, and they sentenced him to house arrest from 1633 until his death in 1642.
Universal Gravitation
In 1687 Isaac Newton published his book, Mathematical Principles of Natural Philosophy. The book contained his laws of motion and his law of universal gravitation. His reasoning was based on geometric proofs and new mathematical techniques which are now called calculus.
Universal gravitation states that every particle in the universe is attracted to every other particle. Amazingly, universal gravitation connects the earth and sky with one equation. The same force that makes apples fall also makes stars, planets, and moons orbit each other!
$$ F = \frac{GM_{1}M_{2}}{r^{2}} $$
\(F\) = force of gravity [N, Newtons, kg m/s²]
vector
\(G\) = 6.67408 × 10
^{-11} = universal gravitation constant [N m²/kg²]
\(M\) = mass [kg, kilograms]
\(r\) = distance between the center of each mass [m, meters]
In 1916 universal gravitation was improved on by Einstein's general relativity, but universal gravitation is still used for low mass and low speed situations, like the Earth and its satellites.
Each mass feels an equal but opposite force as predicted by Newton's 3rd law. This means that the same force of gravity you feel towards the Earth the Earth feels towards you. So why don't we notice the Earth accelerate towards you?
Notice how small G is (6.674 x 10 ^{-11}). Out of the four fundamental forces gravity is by far the weakest. If gravity is so weak why do we notice its effects so easily?
Local Massive Objects Data Table
name | mass (kg) | radius (km) | density (g/cm ^{3}) |
---|---|---|---|
Sun | 2.00 × 10 ^{30} | 695,700 | 1.408 |
Mercury | 3.301 × 10 ^{23} | 2,440 | 5.427 |
Venus | 4.867 × 10 ^{24} | 6,052 | 5.243 |
Earth | 5.972 × 10 ^{24} | 6,371 | 5.515 |
Moon | 7.346 × 10 ^{22} | 1,737 | 3.344 |
Mars | 6.417 × 10 ^{23} | 3,390 | 3.933 |
Jupiter | 1.899 × 10 ^{27} | 70,000 | 1.326 |
Saturn | 5.685 × 10 ^{26} | 58,232 | 0.687 |
Uranus | 8.682 × 10 ^{25} | 25,362 | 1.270 |
Neptune | 1.024 × 10 ^{26} | 24,622 | 1.638 |
solution
$$ F = \frac{GM_{1}M_{2}}{r^{2}} $$ $$ F = \frac{(6.674 \times 10^{-11})(5.972 \times 10^{24}) (7.346 \times 10^{22})}{(3.844 \times 10^{8})^{2}}$$ $$ F = \frac{(6.674 \times 10^{-11})(5.972 \times 10^{24}) (7.346 \times 10^{22})}{14.78 \times 10^{16}}$$ $$ F = \frac{6.674 \times 5.972 \times 7.346}{14.78} \times \frac{10^{-11}10^{24}10^{22}}{10^{16}}$$ $$ F = 19.798 \times 10^{19} \, \mathrm{N}$$solution
$$ F = \frac{GM_{1}M_{2}}{r^{2}} $$ $$ F = \frac{(6.674 \times 10^{-11})(5.972 \times 10^{24}) (2.00 \times 10^{30})}{(149.6 \times 10^{9})^{2}}$$ $$ F = \frac{(6.674 \times 10^{-11})(5.972 \times 10^{24}) (2.00 \times 10^{30})}{22380 \times 10^{18}}$$ $$ F = 3.561 \times 10^{22} \, \mathrm{N}$$solution
$$ F = \frac{GM_{1}M_{2}}{r^{2}} $$ $$ F = \frac{(6.674 \times 10^{-11})(5.972 \times 10^{24})(100)}{ 10,000,000^{2}}$$ $$ F = \frac{(6.674 \times 10^{-11})(5.972 \times 10^{24})(100)}{ 10^{14}}$$ $$ F = 398.57 \, \mathrm{N}$$solution
$$ F = \frac{GM_{1}M_{2}}{r^{2}} $$ $$ M_{1} = \frac{Fr^{2}}{GM_{2}}$$ $$ M_{1} = \frac{(100)(10,000,000)^{2}}{(6.674 \times 10^{-11})(5.972 \times 10^{24})}$$ $$ M_{1} = \frac{(100)(10^{14})}{(6.674 \times 10^{-11})(5.972 \times 10^{24})}$$ $$ M_{1} = 25.01 \, \mathrm{kg}$$solution
$$ F = \frac{GM_{1}M_{2}}{r^{2}} $$ $$ r^{2} = \frac{GM_{1}M_{2}}{F} $$ $$ r^{2} = \frac{(6.674 \times 10^{-11})(2.00 \times 10^{30})(2.2 \times 10^{14})}{3.65 \times 10^{12}} $$ $$ \sqrt{r^{2}} = \sqrt{8.04 \times 10^{21}} $$ $$ r = 8.97 \times 10^{10}\,\mathrm{m} $$Gravitational Acceleration
It is useful to adapt the universal gravitation equation to predict acceleration. To find acceleration we just need to divide an object's gravitational force by its mass.
derivation of universal gravitational acceleration
$$ F = mg $$ $$ \frac{F}{m} = g $$Newton's second law tells us we can replace F/M with acceleration.
$$ F = \frac{GM_{1}M_{2}}{r^{2}} $$ $$ \frac{F}{M_{2}} = \frac{GM_{1}}{r^{2}} $$ $$ g = \frac{GM_{1}}{r^{2}} $$$$ g = \frac{GM}{r^{2}} $$
\(g\) = acceleration of gravity [m/s²]
vector
\(G = \small 6.674 \times 10^{-11}\) = universal gravitation constant [m³/kg/s²]
\(M\) = mass of the body pulling [kg, kilograms]
(not the body experiencing the acceleration)
\(r\) = distance between the center of each mass [m, meters]
The acceleration vector is pointed towards the center of the mass producing the acceleration.
The mass of the body being accelerated isn't used in this equation. Use the mass of the body producing the acceleration. To find the acceleration of objects on Earth use Earth's mass.
The simulation below shows a vector field. Each vector shows the gravitational acceleration potentially felt at that location. These diagrams are helpful for predicting how a particle will accelerate.