Newton's universal law of gravity

Galileo's Law of Falling

falling in vacuum

In the picture, the rock and the feather are falling in vacuum: all the air in the cylinder has been eliminated.

What might the arrows represent? Which of these are equal (not equal) for the feather and the rock?

  • Acceleration,
  • Force of gravity,
  • Mass.

g-factor

This dragster accelerates from 0 to 100 mph in ~0.9 sec. 100 mph in one second ~ 5 *(22 mph increase in one second) = 5g.

There is a formula that relates distance and time (starting from rest) to acceleration. It is...

$$d = \frac{1}{2}at^2.$$

Solve it for $a$:

$$\frac{2 d}{t^2}= a.$$

World class sprinter: 40 m dash in 4.4 seconds:

$a = (2*40)/(4.4^2) \approx 4 $m / sec / sec = 0.4 * 10 m / sec / sec = 0.4 g.

John Stapp survived horizontal acceleration of $\approx 22g$ for more than a second (1954 USAF sled trial).

The human body can withstand only about $5g$ vertically before blood flow away from the head leaves you unconscious. The space shuttle is designed to limit accelerations to $3g$.


What about force pairs?

A force results from an interaction between two bodies. If the earth pulls on a rock, isn't the rock also pulling on the earth too?

In fact the forces are equal and opposite. Why don't we notice that the earth is getting pulled on??



So, it seems reasonable that the gravitational force $F_g$ is proportional to the masses of both bodies involved in the interaction. There's nothing more special about one or the other.

Newton's equation for gravity

Newton's full equation for the gravitational force between any two bodies of mass $m_1$ and $m_2$ does indeed depend on the masses of both:

With masses in kilograms and distances in meters:

$F_g = 6.7 \times 10^{-11}\frac{m_1 m_2}{d^2}$

Distance, $d$ is measured from the center of mass of one body to the center of mass of the other. (But this is just an approximation for a body with a uniform density).

Weight

Newton's equation for gravity means that the force of gravity at the surface of the earth on some object of mass $m_o$ is:

$$F_g \approx 6.7 \times 10^{-11} \frac{m_E \cdot m_o}{r_E^2} = 6.7 \times 10^{-11}\frac{5.97 \times 10^{24} {\rm kg}\cdot m_o}{(6.38 \times 10^6 m)^2} = 9.8 (m/sec^2) * m (kg)$$

So this ought to change...

  • On earth... at different heights.
  • On earth... depending on whether there's a lot of mass close by (you're standing on a bolder) or very little (you're standing over a basement).
  • On other planets... depending on whether they are bigger or smaller, more or less massive.

Calculate the gravitational force between you and the person sitting closest to you!

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).

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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?

Exercises

Conceptual Exercises in Chapter 5: 3, 4, 11, 12, 18, 22.
Problems in Chapter 5: 2, 3, 4