Notes to a video lecture on http://www.unizor.com
Gravity
We all know a lot about gravity, weight, weightlessness, rockets flying on orbits calculated based on the laws of gravity etc.
But what is gravity?
A short answer is: we don't know. It's like most of us can use a smart
phone and Global Positioning System (GPS), but don't know how and why it
works. It just works, we know what to do to effectively use it, but
have no idea about the real mechanism that allows us to use it.
Obviously, designers and engineers who created these technological
marvels know, but most of people don't.
With gravity it's similar. We know it exists, we can use it, we feel it,
but we don't know the underlying reason why it is what it is.
To be more precise, physicists have certain ideas about the source of
gravity, but they are rather vague, on the level of hypothesis.
Therefore, we skip this foundational discussion about why gravity
exists, what is an underlying mechanism of its work. We will just use it
as we use GPS without getting much deeper.
To use a computer game, we just have to know its rules and controls, we don't have to know what software is inside.
To use gravity, it's not necessary to know its underlying mechanism, we
just need to know its properties, and that's the subject of this
lecture.
The first fundamental property of gravity is that all objects we deal
with attract other objects. This effect of attraction is called gravity.
Attraction is a force.
Since we usually model physical objects as points, this force is
directed along the line connecting these point-objects and pushes them
towards each other.
It is also important to note that the Newton's Third Law says that the force point-object B attracts point-object A is paired with the same in magnitude and opposite in direction force point-object A attracts point-object B.
In more complicated cases of objects that cannot be considered as
points, we can assume that every tiny peace of each object, which can be
modeled as a point-object, is attracted to every other tiny peace. Then
some process of integration of all these forces might be used to
determine the resultant forces. But we will rarely deal with this type
of gravitation, most of cases we will consider will involve
point-objects.
Forces change the velocity. Therefore, gravity, which is the force
observed for any type of object, causes change of motion of objects. If
there is only one point-object in the Universe, it will maintain its
inertial motion along a straight line with constant velocity. As soon as
another object appears somewhere, the force of gravity will cause a
change in the inertial movement of the first object.
Our next question is: how exactly forces of gravity change the motion of objects?
Different objects attract differently.
Consider some probe object A in inertial motion along a straight
line with constant velocity. For example, it flies in our
three-dimensional space along the X-axis in positive direction, going
through point of origin of coordinates
Let's measure the degree of the change of its motion, when at the later moment of time t=1 another object B appears at the origin of coordinates
will decelerate. Measuring this deceleration and knowing the mass of
objects involved, we can measure the force of attraction between objects
A and B using the Newton's Second Law.
Our observations show that different objects B will cause different decelerations of probe object A and the same object B causes different decelerations of probe object A at different distances between them. We conclude then that gravitational force of attraction between objects A and B depends on gravitational properties of objects themselves and on distance between them.
Our purpose is to analyze what is the gravitational property of any
object, how to measure it and how the force of gravity depends on it and
the distance between objects.
The situation with distance is easy.
Experiments with the same objects showed that the gravitational force of
attraction between them weakens with distance in inverse proportion to a
square of this distance. In other words, if the distance between any
two objects A and B doubles, the gravitational force of attraction weakens by a factor of 4.
So, it is sufficient to establish the gravitational force between two
objects at a unit length (say, 1 meter), after which the gravitational
force between these objects at any distance D will be that force at a unit distance divided by a factor D2.
Let's discuss now the gravitational property of an object, its ability to attract other objects, which in Physics is called gravitational mass of an object.
An experimental fact is that two identical objects, B1 and B2 combined together, attract twice as strongly as only one of them, say B1, providing they attract the same probe object A, and the relative position of participating objects is the same.
That means that gravitational mass is additive and the gravitational force is proportional to gravitational mass.
Let's choose one particular probe object A and assign it a gravitational mass of a unit and another identical object B. Since they are identical, the gravitational mass of object B is also a unit.
Then, comparing the attraction between this unit probe object A and identical unit object B at the unit distance with the attraction of any other object C to the same unit probe object A on the same unit distance, we can assign a gravitational mass to that other object C. Since gravitational mass is additive, the stronger the gravitational force of object C - the proportionally greater is its gravitational mass relative to a unit object B.
Notice, that additive property of gravitational mass is similar to a property of inertial mass, which is also additive. This is precisely the reason why both properties are call mass.
The analogy goes further. Another experimental fact is that two different objects of the same inertial mass have exactly the same gravitational mass,
that is they attract equal probe objects on equal distance equally.
From this follows that the quantitative difference between inertial mass and gravitational mass is just in units of measurement.
Based on this, it was decided to measure the gravitational mass in exactly the same units as inertial mass and, by definition, say that an object of 1 kilogram of inertial mass has 1 kilogram of gravitational mass, which, quantitatively, fully characterizes the gravitational properties of an object.
When we talk about gravity, 1 kilogram is a measure of gravitational attraction of an object, its gravitational mass. When we discuss inertia, motion, force, 1 kilogram is a measure of an object's inertial mass.
Let's derive the formula that expresses the force of gravity between two
point-objects in terms of their gravitational masses and distance
between them.
We already know that the force of gravity is proportional to a
gravitational mass, but, since we always deal with two point-objects,
the force must be proportional to a gravitational mass of each of them,
that is it is proportional to their product.
We also know that the force of gravity is inversely proportional to a square of a distance between objects.
These two factors lead to the following formula for the force F of gravity between two point-objects with gravitational (and inertial, as we defined) masses M1 and M2 at distance r between them:
F = G·M1·M2 /r²
where G - a constant of proportionality, since the units
of force (N - newtons) have been defined already, and we want to measure
the gravitational force in the same units as any other force.
This formula was presented by Sir Isaac Newton in 17th century, though other scientists, like Robert Hooke claimed it as well.
Physicists call this formula the Newton's Law of Universal Gravitation.
To determine the constant G in this formula, all we need
to do is to place two objects of inertial (and gravitational, as we
defined) mass of 1 kilogram each at the distance of 1 meter and measure
the force of gravity between them by measuring an acceleration they
develop as a result of gravitational force. This force (in newtons) will
be quantitatively equal to a gravitational constant G.
This measurement shows a very weak force, and the gravitational constant equals to
G = 6.674·10−11 N·m²/kg²
Finally, let's attempt to explain the phenomenon of weakening of the
gravitational force inversely proportional to a square of a distance
from the gravitating object.
This is not really a theoretical proof, but a reasonable explanation of this fact.
Assume that the source of gravitational force around an object is
something similar to tentacles of an octopus with objects of larger
gravitational mass corresponding to greater number of tentacles. The
gravitational grip, presumably. depends on the density of tentacles per
unit of covered area.
To propel gravity on a longer distance the tentacles should be longer, while their quantity remains the same.
Now, the longer these tentacles - the more is the area they have to spread around. This area for tentacle of the length r is a surface of the sphere of this radius, that is 4r².
So, area these tentacles are supposed to cover is increasing as a
square of their length, which causes a gravitational grip to be weaker
in exactly the same proportion.
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