masses =

Things fall. Pencils, buildings, people, everything falls. The Earth is falling towards the Sun, but luckily it is falling in a way that doesn't hit the Sun.
An orbit.

Early Theories of Gravity

The history of gravity tells the story of humanity realizing we aren't the center of the universe.

The Greek philosopher Aristotle (384–322 BCE) 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 to correct for flaws in the geocentric model.

In 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 scientific revolution.

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 the heliocentric model even though it contradicted Catholic scripture. The leaders of the church asked him not to publish information that supported the heliocentric model, but Galileo continued. In response, the church forced him to recant his findings, they banned his work, and they sentenced him to house arrest from 1633 until his death in 1642.

Question: What advantages did the heliocentric model have over the geocentric model?

solution

The heliocentric model was able to predict the locations of planets more accurately than the geocentric model. The heliocentric model also didn't have any epicycles to calculate.

Kepler's Laws of Planetary Motion

In 1609, German astronomer, Johannes Kepler published 3 laws of planetary motion based on the heliocentric theory.

The orbit of a planet is an ellipse with the Sun at one of the two foci. (F₁, F₂)
A line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time. (A₁ = A₂)
The square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit.

$$ T^2 = r^3$$

$T$ = orbital period, total time for one orbit [yr, years]
$r$ = semi-major axis [AU, astronomical units, 1.5 × 10¹¹ metres]

Kepler's laws mostly agreed with the Copernican model, but with a few improvements. In Kepler's model orbits moved in an ellipse, not a circle. The Sun wasn't the center of a planetary orbit, it was a foci of an elliptical orbit. Also, the planets would speed up and slow down as they orbited.

The semi-major axis of an ellipse is the distance from the center to the longest edge. For the Earth that distance is 149 597 870 700 m, which is called 1 astronomical unit (AU).

The orbital period only equals the semi-major axis for units of years (yr) and astronomical units (AU). The relationship is more complex with other units.

modified equation for units of meters and seconds

$$T^2 = \frac{4 \pi^2}{GM} r^3$$

$T$ = orbital period, total time for one orbit [yr, years]
$r$ = semi-major axis [AU, astronomical units, 1.5 × 10¹¹ metres]
$M$ = Mass of central body [kg, kilograms]
$G$ = 6.67408 × 10^-11 = universal gravitation constant [N m²/kg²]

planet	semi-major axis	orbital period
Mercury	0.387 AU	0.240 yr
Venus	0.723 AU	0.615 yr
Earth	1.000 AU	1.000 yr
Mars	1.524 AU
Jupiter	5.204 AU	11.86 yr
Saturn	9.582 AU	29.46 yr
Uranus	19.22 AU	84.02 yr
Neptune	30.11 AU	164.8 yr

Example: Calculate the time it takes Mars to orbit the Sun in years.

solution

$$T^2 = r^3$$ $$T^2 = (1.524)^3$$ $$T^2 = 3.540$$ $$T = 1.88 \, \mathrm{yr}$$

Each planet in the solar system lands somewhere on this graph.

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 his new mathematical techniques which are now called calculus.

Universal gravitation states that every particle in the universe is attracted to every other particle. It connects the heavens and the Earth with one equation. The same force that makes apples fall also controls the motion of stars, planets, and moons.

$$ F = \frac{GM_{1}M_{2}}{r^{2}} $$

$F$ = 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]

Universal gravitation improved on Kepler's laws of planetary motion because it had a more universal application. Kepler's laws only applied to orbits where one body was much more massive, like the Sun and it's planets. Newton's gravitation applied to all matter.

No current theory perfectly describes reality. They all have the potential for improvement. In 1915 universal gravitation was improved on by Einstein's general relativity. It is hoped that one day we will improve on general relativity by combining it with quantum mechanics.

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.

Question: Why don't we observe the Earth accelerating towards you?

answer

The Earth and you both have the same force, but not the same acceleration.

$$F=ma$$ $$a = \frac{F}{m}$$

Acceleration equals force divided by mass. The Earth has a large mass, so the acceleration from you pulling on the Earth is small.

name	mass (kg)	radius (km)
Sun	2.00 × 10³⁰	695 700
Mercury	3.301 × 10²³	2440
Venus	4.867 × 10²⁴	6052
Earth	5.972 × 10²⁴	6371
Moon	7.346 × 10²²	1737
Mars	6.417 × 10²³	3390
Ceres	9.384 × 10²⁰	470
Jupiter	1.899 × 10²⁷	70 000
Saturn	5.685 × 10²⁶	58 232
Uranus	8.68 × 10²⁵	25 362
Neptune	1.024 × 10²⁶	24 622

Example: Find the force of gravity between the Earth and the Moon. The distance between them is 384 403 km.

solution

$$ F = \frac{GM_{1}M_{2}}{r^{2}} $$ $$ F = \frac{(6.674 \times 10^{-11})(5.972 \times 10^{24}) (7.346 \times 10^{22})}{(3.844 \times 10^{8})^{2}}$$ $$ F = \frac{(6.674 \times 10^{-11})(5.972 \times 10^{24}) (7.346 \times 10^{22})}{14.78 \times 10^{16}}$$ $$ F = \frac{6.674 \times 5.972 \times 7.346}{14.78} \times \frac{10^{-11}10^{24}10^{22}}{10^{16}}$$ $$ F = 19.798 \times 10^{19} \, \mathrm{N}$$

Example: Find the force of gravity between the Earth and the Sun. The distance between them is 149.6 billion m.

solution

$$ F = \frac{GM_{1}M_{2}}{r^{2}} $$ $$ F = \frac{(6.674 \times 10^{-11})(5.972 \times 10^{24}) (2.00 \times 10^{30})}{(149.6 \times 10^{9})^{2}}$$ $$ F = \frac{(6.674 \times 10^{-11})(5.972 \times 10^{24}) (2.00 \times 10^{30})}{22380 \times 10^{18}}$$ $$ F = 3.561 \times 10^{22} \, \mathrm{N}$$

Example: Find the force of gravity between the Earth and a 100 kg person that is 10 000 000 m from the center of the Earth.

solution

$$ F = \frac{GM_{1}M_{2}}{r^{2}} $$ $$ F = \frac{(6.674 \times 10^{-11})(5.972 \times 10^{24})(100)}{ 10\,000\,000^{2}}$$ $$ F = \frac{(6.674 \times 10^{-11})(5.972 \times 10^{24})(100)}{ 10^{14}}$$ $$ F = 398.57 \, \mathrm{N}$$

Example: How massive must an object be in order to feel a force of 100 N at a distance of 10 000 000 m from the center of the Earth?

solution

$$ F = \frac{GM_{1}M_{2}}{r^{2}} $$ $$ M_{1} = \frac{Fr^{2}}{GM_{2}}$$ $$ M_{1} = \frac{(100)(10\,000\,000)^{2}}{(6.674 \times 10^{-11})(5.972 \times 10^{24})}$$ $$ M_{1} = \frac{(100)(10^{14})}{(6.674 \times 10^{-11})(5.972 \times 10^{24})}$$ $$ M_{1} = 25.01 \, \mathrm{kg}$$

Example: When it is closest to the Sun on its 75 year orbit, Halley's Comet feels a force of gravity from the Sun of 3.65 × 10¹² N. Calculate its distance from the Sun using the comet's mass of 2.2 × 10¹⁴ kg.

solution

$$ F = \frac{GM_{1}M_{2}}{r^{2}} $$ $$ r^{2} = \frac{GM_{1}M_{2}}{F} $$ $$ r^{2} = \frac{(6.674 \times 10^{-11})(2.00 \times 10^{30})(2.2 \times 10^{14})}{3.65 \times 10^{12}} $$ $$ \sqrt{r^{2}} = \sqrt{8.04 \times 10^{21}} $$ $$ r = 8.97 \times 10^{10}\,\mathrm{m} $$

$$G = 6.674 × 10^{-11} $$ Question: Notice that universal gravitation constant is very small . 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?

solution

The force of gravity scales with distance and mass. The Earth is very close and very massive, so it's gravity is large enough to feel.

The force of gravity is very weak though. If gravity were stronger we might notice the pull from human sized objects.

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.

masses =

Example: Find the acceleration of gravity for an object on the surface of Earth.
Is it really 9.8 m/s²?

Local Massive Objects Data Table

name	mass (kg)	radius (km)
Sun	2.00 × 10³⁰	695 700
Mercury	3.301 × 10²³	2440
Venus	4.867 × 10²⁴	6052
Earth	5.972 × 10²⁴	6371
Moon	7.346 × 10²²	1737
Mars	6.417 × 10²³	3390
Jupiter	1.899 × 10²⁷	70 000
Saturn	5.685 × 10²⁶	58 232
Uranus	8.68 × 10²⁵	25 362
Neptune	1.024 × 10²⁶	24 622

solution

$$ r = 6\,371\,000 \, \mathrm{m} $$ $$ g = \frac{GM}{r^{2}}$$ $$ g = \frac{(6.674\times 10^{-11})(5.972\times 10^{24})}{(6.371\times10^{6})^{2}}$$ $$ g = \frac{(6.674\times 10^{-11})(5.972\times 10^{24})}{40.590\times10^{12}} $$ $$ g =\frac{(6.674)(5.972)}{40.590}\times 10^{-11+24-12}$$ $$ g=0.981\,95 \times 10^{1}$$ $$ g=9.8195 \, \mathrm{\frac{m}{s^{2}}}$$

Example: Find how far away from the Earth you need to be to only accelerate at half of 9.8 m/s².

solution

$$ g = \frac{GM}{r^{2}} $$ $$ r^{2} = \frac{GM}{g} $$ $$ r^{2} = \frac{(6.674\times 10^{-11})(5.972\times 10^{24})}{9.8 \times 0.5} $$ $$ r = 9\,160\,226 \,\mathrm{m} = 9160 \,\mathrm{km} $$ $$\text{distance above Earth's surface}$$ $$ 9160\, \mathrm{km} - 6371\,\mathrm{km} = 2789 \,\mathrm{km}$$

Example: Mars has two very small moons, Phobos and Deimos. Phobos has a surface gravity of 0.0057 m/s² and a surface radius of 11 266 m. Calculate the mass of Phobos. (Does your answer agree with wikipedia?)

solution

$$ g = \frac{GM}{r^{2}} $$ $$ M = \frac{gr^2}{G}$$ $$ M = \frac{(0.0057)(11\,266)^2}{6.674\times 10^{-11}}$$ $$M = 1.08 \times 10^{16}\,\mathrm{kg}$$

Inverse-Square Law (1/r²)

All particles are attracted to each other, but the attraction is divided by the distance squared. Physical laws that diminish at 1/r² are common in nature because of how signals spread out in 3 dimensions. If you threw darts in random directions, the chance of hitting your target would obey this rule.

As you can see in the graph below, a 1/r² function has some interesting results at radius = 0 and force = 0.

Question: What distance between masses produces a force of zero?

answer

As the distance approaches infinity, the force approaches zero.

The effects of gravity can be felt at any real distance. Astronomers see gravitation between our galaxy, the Milky Way, and the Andromeda galaxy at a distance of 2.5 million light years (10²² m)

Question: What happens to the force of gravity as the distance between masses approaches zero?

answer

The force rises towards infinity as the distance approaches zero.

When matter gets very dense, a black hole forms. Understanding gravity at distances near zero require using general relativity and quantum physics at the same time. Currently these two theories are compatible, but we know that some strange things would happen. Time would slow down, and it would become very difficult for anything to escape.

Question: What else, besides gravity, might follow an inverse square law?

answer

Anything that spreads out in 3 dimensions will follow this law.

Light

Sound

Randomly throwing things, like darts.

Example: If you triple the distance between two massive objects, what happens to their force of gravity? What about ten times the distance?

solution

This is easier to understand with some made up numbers. Set the masses and G to equal 1 since they aren't changing.

$$F = \frac{G M_1 M_2}{r^2} \quad \to \quad F = \frac{1}{r^2}$$

$$\text{\underline{triple}}$$ $$r = 3$$ $$ F = \frac{1}{r^2}$$ $$ F = \frac{1}{3^2}$$ $$ F = \frac{1}{9}$$

$$\text{\underline{ten times}}$$ $$r = 10$$ $$ F = \frac{1}{r^2}$$ $$ F = \frac{1}{10^2}$$ $$ F = \frac{1}{100}$$

As the distance between two masses increases, the force of gravity decreases.

name	mass (kg)	radius (km)	g (m/s²)
Mercury	3.301 × 10²³	2440	3.7
Venus	4.867 × 10²⁴	6052	8.9
Earth	5.972 × 10²⁴	6371	9.8
Mars	6.417 × 10²³	3390	3.7

Question: Which terrestrial planet is being graphed?

strategy

Hover your mouse over the graph. Move your mouse horizontally to check the radius for each planet in the chart above. If the acceleration and the radius match, it is the correct planet.

answer

name	mass (kg)	radius (km)	g (m/s²)
Jupiter	1.899 × 10²⁷	70 000	?
Saturn	5.685 × 10²⁶	58 232	?
Uranus	8.68 × 10²⁵	25 362	?
Neptune	1.024 × 10²⁶	24 622	?

Question: Which gas giant is being graphed? You need to do math for this one.

answer

Questions: Imagine you could to dig into the center of the Earth. How would the acceleration of gravity change as you went deeper? Would it increase, decrease, or stay the same?

answer

As you go deeper into the Earth you fell less gravity. When there is mass above and below, they pull in opposite directions. You can't feel opposing forces of gravity, because gravity pulls on your whole body evenly.

Using a model of the Earth with constant density the gravity decreases linearly. At 50% to the center of the Earth the gravity is 50% of 9.8 m/s². At the center gravity is zero.

Earth's actual gravity according to the Preliminary Reference Earth Model (PREM) is more complex. This is because the density changes with each layer, and increases towards the center.

Gravitational Potential Energy

Our old gravitational potential energy equation, U=mgh, can be made more accurate if we replace g = 9.8 with a calculated gravitational acceleration. This version of gravitational potential energy now works beyond Earth's surface.

derivation of universal gravitational potential energy

$$U_{g} = mgh \quad g = \color{blue}{\frac{GM}{r^{2}}} $$ $$U_{g} = m{\color{blue}{\frac{GM}{r^{2}}}}h $$

If we define h to be zero when the two masses are zero distance apart, we can rename h to r.

$$U_{g} = \frac{GM_{1}M_{2}}{r^{2}}r$$ $$U_{g} = \frac{GM_{1}M_{2}}{r} $$
When distance is very big the energy goes to zero, so it makes sense to choose the zero of this gravitational potential energy at an infinite distance away. This means that as we bring a mass closer to another mass we have negative potential energy. $$U_{g} = -\frac{GM_{1}M_{2}}{r} $$

$$U_{g} = -\frac{GM_{1}M_{2}}{r} $$

$U_g$ = gravitational potential energy [J, joules]
$G$ = 6.67408 × 10^-11 = universal gravitation constant [N(m/kg)²]
$M$ = mass [kg, kilograms]
$r$ = distance between the center of each mass [m, meters]

What is the gravitational potential energy when two masses are at zero distance away?

What is the gravitational potential energy when two masses are at an infinite distance away?

masses =

Energy is a scalar, not a vector. This means that when we calculate gravitational potential energy it has no direction. You can see the potential energy in the simulation below as a scalar field. Think of the reddish regions as being deeper into the gravity well.

Example: Calculate the gravitational potential energy between two 100 000 kg masses when there is only 2 m between the center of each mass.

solution

$$U_{g} = -\frac{GM_{1}M_{2}}{r}$$ $$U_{g} = -\frac{(6.674 \times 10^{-11}) (100\,000)(100\,000)}{2}$$ $$U_{g} = -0.33\,\mathrm{J}$$

The energy is negative because it would take positive work to separate the masses.

Example: Two 100 000 kg masses are 2 meters apart. How much work would it take to bring them to 10 m apart?

strategy

Calculate the gravitational potential energy at each distance. The change in energy is the work.

solution

$$U_{i} = -\frac{GM_{1}M_{2}}{r}$$ $$U_{i} = -\frac{(6.674 \times 10^{-11})(100\,000)(100\,000)}{2}$$ $$U_{i} = -0.33\,\mathrm{J}$$
$$U_{f} = -\frac{(6.674 \times 10^{-11})(100\,000)(100\,000)}{10}$$ $$U_{f} = -0.068\,\mathrm{J}$$

Work is a change in energy, so we can subtract the final energy from the initial.

$$W = \Delta U = U_{f} - U_{i} $$ $$W = (-0.068) - (-0.33)$$ $$W = 0.262 \, \mathrm{J}$$

Example: A 2000 kg spaceship starts at rest 20 000 km from the surface of Earth. How much kinetic energy will the ship have after it falls down to 10 000 km from the surface of Earth?

strategy

Use conservation of energy. Set the initial kinetic and gravitational potential equal to the final kinetic and gravitational potential.

solution

$$E_{i} = E_{f}$$ $$U_{gi} = U_{gf} + K$$ $$ K = U_{gi} - U_{gf}$$ $$K = -\frac{GM_{1}M_{2}}{r}+\frac{GM_{1}M_{2}}{r}$$ $$K = -\frac{(6.674\times 10^{-11})(5.972\times 10^{24})(2000)}{6\,371\,000 + 20\,000\,000}$$ $$+\frac{(6.674\times 10^{-11})(5.972\times 10^{24})(2000)}{6\,371\,000 + 10\,000\,000}$$ $$K = -3.02 \times 10^{10}+4.86 \times 10^{10} $$ $$K = 1.85 \times 10^{10} \, \mathrm{J}$$

Example: Find the minimum escape velocity for a 100 kg person on Earth.

strategy

Use conservation of energy to find the energy needed to bring both the kinetic and potential energy to zero.

solution

$$E_{i} = E_{f}$$ $$U_{gi} + K = U_{gf} + K$$ $$U_{gi} + K = 0$$ $$K = -U_{gi}$$ $$(1/2)M_{2}v^2 = \frac{GM_{1}M_{2}}{r}$$ $$(1/2)v^2 = \frac{GM_{1}}{r}$$ $$\boxed{v^2 = \frac{2GM_{1}}{r}}$$ $$v = \sqrt{\frac{2(6.674\times 10^{-11}) (5.972 \times 10 ^{24})}{6\,371\,000}}$$ $$v = 11\,183 \, \mathrm{\tfrac{m}{s}}$$

Example: Find the minimum escape velocity on Phobos.

solution

$$v^2 = \frac{2GM_{1}}{r}$$ $$v^2 = \frac{2(6.674\times 10^{-11})(1.1 \times 10^{16})}{11200}$$ $$v = \sqrt{\frac{2(6.674\times 10^{-11})(1.1 \times 10^{16})}{11200}}$$ $$v = 11.45\, \mathrm{\tfrac{m}{s}}$$

Use the mouse to move the camera. Double click for full screen.

In the simulation above, gravitational potential energy is represented as warping a 2-D fabric into a 3rd dimension. This is similar to how gravity is thought of in general relativity.