Astronomy 161:
An Introduction to Solar System Astronomy Prof. Richard Pogge, MTWThF 9:30 |

Newtonian Gravity

- All falling bodies experience the same gravitational acceleration

- Gravity is an attractive force between all pairs of massive objects
- Gravitational force is proportional to the masses, and inversely proportional to the square of the distance between them.

NOTE:

This and the following lecture are probably the most mathematical of all the lectures that will be given in this class. I encourage you all to read these notes in advance and try to follow the arguments in them. In will make it easier to follow along during lecture. [rwp]

Explored the rate of falling bodies by dropping different weights, or sliding them down inclined planes.

**Law of Falling Bodies**:

- In the absence of air, heavy objects and light objects fall at the same, constant rate of acceleration.

Gravity is an **Attractive** force:

- Works to bring massive objects closer together.

Gravity is a **Universal** force:

- Works everywhere in the Universe.

Gravity is a **Mutual** force:

- Works between
**pairs**of massive objects.

The **masses** of the two objects:

- More massive objects exert a
**stronger**the gravitational force.

The **distance** between them:

- The force gets
**stronger**as the two objects move closer together. - The force gets
**weaker**as the two objects move farther apart.

It does not depend on the shapes, colors, or compositions of the objects.

The force of gravitational attraction between any two massive bodies is proportional to their masses and inversely proportional to the square of the distance between their centers.

The Force of Gravity is an example of an "Inverse Square Law Force"

Stated Mathematically:

- Where:
- F = force due to gravity.
- M
_{1}= mass of the first object - M
_{2}= mass of the second object - d = distance between their centers.
- G = "Gravitational Force Constant"

G is very small, in metric units:

G=6.7x10

^{-11}Newtons meter^{2}/ kilogram^{2}

The **Newton** is the metric unit of force:

4.41 Newtons = 1 pound

G has to be measured experimentally.

What is the force of the Earth on the apple?

F = GM

_{E}M_{apple}/R_{E}^{2}

What is the apple's acceleration (2nd Law):

a

_{apple}= F/M_{apple}= GM_{E}/R_{E}^{2}= 9.8 meters/sec^{2}

Note that the mass of the apple (M_{apple}) had divided out
of the equation. This means that the acceleration due to gravity is
**independent** of the mass of the apple, just like Galileo had shown
earlier.

What force does the the apple apply in return upon the Earth?

F = GM

_{E}M_{A}/R_{E}^{2}

How much does the Earth accelerate towards the apple?

a = F/M

_{E}= GM_{A}/R_{E}^{2}

= 9.8 m/sec

^{2}x (M_{A}/M_{E}) = very small amount (about 10^{-25}meters/sec^{2})

a = 9.8 meters/sec

^{2}

We can also measure the radius of the Earth using geometry (Eratosthenes):

R

_{E}=6378 kilometers = 6,378,000 meters

Combining these together using Newton's formula for the Gravitational Force allows us to estimate the mass of the Earth, as follows:

This is an example of one of the powerful implications of Newton's Law of Gravity: It gives us a way to use the motions of objects under the influence of their mutual gravitation to measure the masses of planets, stars, galaxies, etc.

**What keeps the Moon in orbit around the Earth?**

The Law of Inertia (Newton's 1st law) predicts:

If there were no gravitational force acting between the Moon and the Earth, the Moon would travel in a straight line at a constant speed.

But, of course the Moon really moves along a curved path:

- It is deflected from a straight-line path by the force of gravity.

- Newton computed this. In order to stay in its orbit, the Moon
must fall by
**0.00136 meters (about 1.4 mm)**each second.Call this quantity

**x**, the deflection of the orbiting moon in 1 second._{moon}

How far does an apple fall on the Earth during the first second?

- Newton also knew this (he could measure it directly), it falls
**4.9 meters**in the first 1 second.Call this quantity

**x**, the deflection of a falling apple in its 1 second of motion._{apple}

Newton also knew that:

- Moon is about 60 Earth Radii from the Earth.

Summarizing the numbers:

**The Moon:**

- Distance that the Moon falls towards Earth in 1 second:
**x**_{moon}= 0.00136 meters - The distance of the Moon from the center of the Earth:
**d**_{moon}= 60 R_{E} - Acceleration of the Moon:
**a**_{moon}= GM_{E}/d_{moon}^{2}= GM_{E}/ (60R_{E})^{2}

**The Apple**:

- Distance the an Apple falls on Earth in 1 second:
**x**_{apple}= 4.9 meters - The distance of the Apple from the center of the Earth:
**d**_{apple}= 1 R_{E} - Acceleration of the Apple:
**a**_{apple}= GM_{E}/d_{apple}^{2}= GM_{E}/R_{E}^{2}

The ratio of the deflections of the Apple and the Moon in 1 second will be proportional to the ratio of how much they have been accelerated during that 1 second.

Putting all the info we have together, we get the following:

This predicts that the deflection of the moon in 1 second
necessary to keep it in orbit around the Earth should be 1/3600^{th}
the deflection of an apple during the first second of its fall
to the Earth.

Is this right?

Recalling that from before, we found that the deflections of the moon and apple in 1 second are:

**x**= 0.00136 meters_{moon}**x**= 4.9 meters_{apple}

Gravity predicts that

**x**_{apple}/3600 = 4.9 meters/3600 = 0.00136 meters!!

The agreement is essentially perfect!

This demonstrates that the same law of gravity applies to both the apple and the Moon! Both feel the gravity of the Earth in the form of a force that gets weaker as the square of their distance from the center of the Earth.

The way to answer this question is to first consider what would happen if there was no gravity acting:

Question: How far would the Moon travel in a straight line in 1 second if there were no gravity acting?

Answer:About 1000 meters.

At the same time, the Moon's motion along this straight-line path would also cause it to move away from the Earth.

Question: How far away from the Earth would the Moon move in 1 second if no gravity were acting?

Answer:About 0.00136 meters!

In round numbers, the amount the Moon falls towards the Earth due to gravity is just enough to offset the straight-line path it would take if gravity were not acting to deflect it. This balance effectively closes the loop.

We have therefore reached a startling conclusion:

The Moon is really perpetually fallingaroundthe Earth!

This is a totally different way of looking at an "orbit" under the influence of gravity.

While at first sight the fall of an apple and the orbit of the Moon appear to be two completely different phenomena, viewed in light of Newton's laws of motion, they are in fact different manifestations of the same thing! The fall of the Moon around the Earth is the same kind of motion as the fall of an apple to the Earth. Both are described by the same three laws of motion, and both feel a gravitational force described by the same, universal force law.

- Governs the fall of an apple on the Earth.
- Governs the fall of the Moon around the Earth.
- Governs the fall of the Earth/Moon system around the Sun.
- Governs the fall of the Sun around the center of the Milky Way Galaxy.
- Governs the fall of the Milky Way and Andromeda Galaxies in their mutual orbit...

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Updated: 2006 October 14

Copyright © Richard W. Pogge, All Rights Reserved.