In order to test his ideas
about uniformly accelerated motion, Galileo found a clever way to slow the effect
of gravity. He used inclined planes to study the accelerations of objects when
those accelerations were small enough to allow him to make careful measurements
with time. For him, the important feature of the dynamics was "uniform
acceleration" -- equal changes in speed over equal intervals of time. This
is a feature of the motion of a ball rolling down the plane, as well as of the
motion of a freely falling body.
An "inclined plane"
is a tilted platform on which a heavy ball is free to roll (with as little friction
as possible) in a straight groove. It is useful for studying the effect of gravity
because, by varying the tilt, the acceleration along the plane may be controlled.
One way to picture this effect
is to imagine that, at each height above the Earth, there is an imaginary surface,
parallel to the surface of the Earth, called a "gravitational energy surface."
To these surfaces are assigned numbers that increase with the distance from
the center of the Earth. While, globally, these surfaces are spheres (centered
at the center of the Earth),locally, they look like planes. The actual numbers
assigned to such surfaces have no physical meaning. What counts is the "spacing"
of the surfaces (which amounts to something like an "intensity") and
their orientation, or direction.
The following discussion is
a bit abstract and you needn't take it too seriously to understand how an inclined
plane works. Galileo's book: Two New Sciences makes a few of the later
points clearer. But the following discussion represents the best mathematical
model we have at present of the process by which the ball acquires this acceleration.
When an object is falling
freely, or rolling or tumbling under the influence of gravity it moves in such
a way that the change of its velocity is, at any instant, cutting these
surfaces as quickly as it can, given the constraints it is subject to. This
determines the direction of acceleration: that is, velocity changes in the direction
that cuts these surfaces most quickly in decreasing order. The actual amount
of acceleration (or change) is determined by choosing a unit change of velocity,
in the determined direction, viewed as a directed arrow (vector), counting the
number of these surfaces penetrated by that arrow, and magnifying (multiplying)
by the length of the time interval. The number of surfaces cut depends on the
"intensity" or "spacing" of the surfaces. The picture is
as follows:
The velocity changes from
initial to final velocity in a short time in a direction determined by the direction
and order of the energy surfaces (here represented as lines). It cuts those
lines as quickly as it can. Once the direction is determined, a vector of unit
length (the red vector on the left) is drawn in the same direction, and the
number of lines it cuts multiplied by the length of the time interval, determines
the length of the change in velocity vector. Thus, the change in velocity in
a short time is determined both by the orientation of the surfaces, and by their
intensity. Now all of this is an approximation and assumes the surfaces are
flat and equally spaced. They are neither flat nor equally spaced, but, for
short intervals of time, it is a good approximation. In the actual process,
one "sums" these changes in velocity over a succession of very short
time intervals. This is called solving a differential equation. Remember, however,
that the vectors (arrows) we are drawing are velocity vectors and the object
moves in such a way that at every point it follows the velocity vectors - while
those themselves are changing as it moves because of the direct influence of
gravity, as depicted in our picture.
Thus, if an object, such
as a heavy ball, is simply falling, its velocity increases downward, perpendicular
to those planes. What happens if it is rolling down a hill, or better, an inclined
plane? It tends to accelerate downward, but since it may not be free to move
straight down, the acceleration (change in velocity) proceeds in the direction
that decreases its height most quickly. The plane itself pushes against it,
cancelling some of that acceleration, and it goes accelerating in the direction
free to it that decreases the height most quickly. The amount of acceleration
is determined by cutting the energy surfaces (lines) in the available direction
by a vector of unit length. Then in short intervals of time, the change of velocity
is determined by the number of surfaces cut multiplied by the length of the
time interval. This may be pictured in the following way:
The green arrow represents
the actual acceleration (rate of change of velocity - not of position) along
the plane. It is the projection of the blue arrow onto the plane. The red arrow
represents the acceleration that cancels the gravitational acceleration caused
by the supporting plane. It is clear that the more nearly level the plane is,
the smaller will be the residual (or actual) acceleration. In this way, Galileo
was able, with the aid of these planes, to "slow" the effect of gravity
enough to measure it carefully. In fact, it is merely a matter of using trigonometry
to calculate the residual acceleration, and that is the way it is usually done
for flat inclined planes.
Exploration: A curious constant
In this exploration, you
may construct your own inclined plane by clicking on the slider and, holding
the left mouse button down while the cursor looks like a "pointing hand"
to a new position.
When you are satisfied with
the new position, press the right mouse button, and the cursor should change
to a pointer. A plane of height 5 with your new base will be drawn:
Now, you may roll the ball down that plane by pressing the 
button. You will see the dials change as they measure time elapsed and speed:
Notice the final "speed".
This is the absolute speed of the ball (not just the downward speed) when it
reaches the bottom. Notice also the time elapsed. As you vary the length of
the base of the inclined plane, you may notice a remarkable thing. The final
speed is always the same! Why should this be true? Consider the picture below.
There, "a" is the actual acceleration along the plane, "g"
is the gravitational acceleration, the height is 5 meters, "w"
is the width of the base (that you determined), and "L" is
the length of the inclined plane.
It may be clear from the picture that 
and that
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Now the final speed will be 
,
and our question is: why should 
be independent of w ?
Using the ratio above, we may deduce that
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Now it happens that 
is constant (independent of w).
We can see this in two ways.
The system also records in the MathEdit on the right the statistics of each
trial. You see three columns. The first column is the time, the second
column, the width (w) of the base. The numbers that appear in the third
column called "ratio" are the values of 
for each of the trials. These numbers should be essentially equal. Now, why
are those ratios equal? What are they equal to?
Galileo argues in his Two
New Sciences that when an object accelerates uniformly from rest, the distance
it travels in time t is the same distance that the object would travel
if it moved at a constant speed equal to 
the speed it attained at the end of its accelerated trip. This is essentially
an "integration" as we will see later when we study the Calculus.
This means that in time t, if the object moves with acceleration a
from rest, then it covers a distance equal to 
We will see this in another way later. From this, we conclude that the time
it takes to roll down the plane is:
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Therefore, 
and from our ratio above, 
so,
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and this is constant.
Question 1: Distance is measured in meters, and time in seconds. From the value of time/L you approximated in your table, estimate the value of g (meters per second per second).
Question 2: We observed that the final speed was constant. In fact it is 
Suppose that "h" is the height of an inclined plane.Show that the
final velocity of a ball rolling down the plane depends only on h and g and
is in fact equal to 
If v is the final velocity
of a ball rolling down an inclined plane of height h, then Question 2 shows
that:
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If m is the mass of the ball, then 
is called the "Kinetic energy" or "energy of motion" of
the ball. 
is called the "potential energy" or "energy of position"
of the ball. What we have shown may be interpreted in the following way: Initially,
there is 0 kinetic energy (the ball is not moving initially) and the potential
energy is 
.
Later, some of the potential energy is "transformed" into kinetic
energy. As it loses height, the ball gains speed, and always in such a way that
the speed gained depends only on the height lost.
In the end, this is the justification
for the interpretation of things in terms of energy surfaces. It is called by
several names: As a "principal of reciprocity" it expresses the reciprocal
nature of the two types of energy: energy of motion, and energy of position.
It is also a conservation principle, in that it asserts that the sum Kinetic
energy + Potential energy is constant (conserved). In this form, it is an instance
of the general Principal of Conservation of Energy. The latter is a very
powerful tool for understanding the nature of gravity, and we shall see and
use it in many forms in this and later Microworlds.
