Gravity, it's everywhere.

Newton proposes that $F_g = \frac{Gm_1 m_2}{r^2}$ applies not only to earthly objects, but also objects in the heavens.

• One explanation for both 'natural' and 'celestial' motion.
• Implies elliptical orbits of planets around the sun.
• Explains slight deviations of planets from elliptical: planets feel the sun's gravity, but also each other's gravity.

Escape velocity

An object launched around the earth with a speed of ~8 km/s will orbit around the earth.

To completely escape Earth's gravity, an object has to either get an assist (from fuel, or other means) or be launched from the surface of the earth with an initial speed greater than the escape velocity of...

$$v_e = \sqrt{\frac{2GM}{R}} \approx 11 {\rm km/s\ (Earth's\ surface)}.$$

This is about 10 times the speed of a bullet.

Earth's rotational speed at the equator is ~0.5 km/s, so countries often locate their launch facilities as close to the equator as possible.

Black holes

If our sun, which currently has a radius of some 700,000 km were to collapse down to a radius of just 3 km, then the escape velocity on the surface would be...

$$v_e = \sqrt{\frac{2GM}{R}} =$$
$$=\sqrt{\frac{2*6.67 \times 10^{-11}*1.99\times 10^{30}{\rm kg}}{3000 {\rm m}}} = 3 \times 10^8 {\rm m/sec}.$$

This is the speed of light. An object with an escape velocity greater than the speed of light is called a black hole.

How can you see a black hole?

There two kinds of evidence that seem to point to black holes:

• Visible stars circling around nothing that you can see: From the radius of a circular orbit and a good guess of the mass of the visible star, you can guess the mass of the thing it's circling.
• Strong x-ray emissions: As matter is sucked into a black hole it speeds up massively, and hits other matter. Models of black holes predict that this will cause strong x-ray emissions.

The picture is a false color x-ray image of the center of the Andromeda galaxy.

Sun->black hole?

If our Sun suddenly collapsed into a black hole (it probably won't) would Earth suddenly get sucked in??

$$F_g = \frac{Gm_1 m_2}{r^2}.$$

All the pieces

But really, the force of "Earth's gravity" is some sort of great sum of all the pieces of the Earth acting on us... Including each other.

If you stand over an empty basement, (compared to standing over solid ground, might you feel...

• ever-so-slightly heavier?
• the same?
• ever-so-slightly lighter?

Writing

Interpersonal (gravity) attraction

1. How much do you "weigh" in Newtons? Use the formula for the force of gravity at the surface of earth: $f_g=mg$ where your mass $m$ is in kilograms, and $g=9.8 m/s^2$. There are approximately 2 lbs in each 1 kg. This is the force between you and the earth.

Let's say that you weigh 150 lbs. This is approximately $$150\text{ lbs }*\frac{1\text{ kg }}{2\text{ lbs }}\approx 75 \text{ kg}$$ Your weight (a force) will come out in Newtons if $m$ is in kilograms, and $g$ has units of m/s^2: $$f_g=mg=75*9.8 \approx 750 \text{ N}.$$
2. What is the gravitational force between you and the person next to you in Newtons?
Estimate your masses (kg) and the distance between your centers (in meters), then use the "full-blown" gravitational law:
$$f_g =6.7 \times 10^{-11}\,\text {[Nm^2/kg^2] } * \frac{m_1 m_2}{d^2}$$

Let's say that you have a mass of 75 kg, and you're sitting next to someone with a mass of 60 kg (~120 lbs), and that the centers of your two bodys are about 0.5 m (one-and-a-half feet) apart: $$f_g = 6.7 \times 10^{-11} * \frac{60*75}{(0.5)^2}=1.2\times10^{-6}\text{ N}.$$ This force is less than a hundred millionth of your weight, so no wonder you don't notice it.

The force of gravity on an 80 kg person on Earth's surface is ...
$$f_g = 6.7 \times 10^{-11}{\text N m^2/kg^2}\frac{80 {\rm \ kg\ } * 5.97\times10^{24}{\rm \ kg\ }}{(6.4\times10^6{m})^2}=780 N.$$

The mass of the moon is 1/100 of Earth's mass, and the moon's radius is 1/4 Earth's radius. What would the force of gravity acting on an 80 kg person be on the moon?

Gravity mapping

Earth's gravity varies by about 0.5% over the surface of the earth.

You can fly around with a gravimeter, essentially a very sensitive scale, that measures slight changes in how much a test weight weighs, and create a gravity map.

Below is a gravity map of New Jersey (left) and a geological map (right).

Below is a the gravity map that Antonio Camargo and Glen Penfield made while looking for oil for PEMEX near the Yucatan Peninsula.

This is now called the Chicxulub crater, and is probably the impact crater from a massive asteroid that wiped out the dinosaurs at the end of the Cretaceous period 65 million years ago.

Supporting evidence for this includes:

• shocked quartz and other minerals associated with impacts rather than underground pressures,
• An iridium-enriched layer world-wide at the K-T boundary. (Meteorites are much higher in iridium than Earth's crust).

You might be interested to read more about...

Mass vs. weight

Mass: resistance to acceleration: $a=f/m$.

Weight: (net) force of gravity on an object, $mg$.

Example

On the surface of the moon, the force of gravity is less than on earth. Would it require more / less / the same force to lift a baseball up off the surface of the moon compared to Earth? Would it require more / less / the same force to throw (accelerate) a baseball horizontally to a speed of 90 mph?

Units

Metric: weight [Newtons] = 9.8 m/s^2 $\times$ mass [kg];

English: weight [pounds] = 32 ft / s^2 $\times$ mass [stones]\$

Wondering about the solar system...

Would Newton's law prohibit planets that orbited the sun like this?

And yet...

The planets all go around the sun in the same direction, and very roughly in the same plane...

Our origins

This strongly suggests that the solar system coalesced from a large cloud of 'gunk' all rotating together, with gravity driven lumpiness.

Cultural echoes - equality

Both humans and mud are subject to the same gravity, a principal uncovered by human effort, not decreed by princes.

Cultural echoes - determinism

The clockwork universe could be wound up and released without daily intervention from the creator.

Once you know the starting positions, speeds of planets, their motion is determined precisely from the gravitational force.

Isaac Newton, who wrote more about theology than science, celebrated this as evidence of a Creator who loved geometry.

A hymn in that spirit:

The spacious firmament on high, with all the blue ethereal sky, and spangled heavens, a shining frame, their great Original proclaim. The unwearied sun from day to day does his Creator's power display; and publishes to every land the work of an almighty hand.

Soon as the evening shades prevail, the moon takes up the wondrous tale, and nightly to the listening earth repeats the story of her birth: whilst all the stars that round her burn, and all the planets in their turn, confirm the tidings, as they roll and spread the truth from pole to pole.

What though in solemn silence all move round the dark terrestrial ball? What though no real voice nor sound amid their radiant orbs be found? In reason's ear they all rejoice, and utter forth a glorious voice; for ever singing as they shine, "The hand that made us is divine."

-The Spacious Firmament on High, Joseph Addison, 1712

What about Free will?

[It is unbelievable] that all nature, all the planets, should obey eternal laws, and that there should be a little animal, five feet hight, who in contempt of these laws, could act as he pleased, solely according to his caprice. -Voltaire

Here is some additional content on estimation and scaling

Suggested Exercises

Conceptual Exercises, Chapter 5: 2, 3, 4, 11, 15, 17, 20, 38

Problems, Chapter 5: 3, 4