After Galileo's conviction in Rome, publicly at least, he lost
interest in astronomy and he concentrated his efforts on his
other life long work in the pioneering field of mathematical
physics. In 1638, he managed to have his final work,
Discourse on Two New Sciences smuggled out of Italy and
published in the Netherlands. This work detailed his findings on
the correct understanding of
dynamics, gravity and, putting the two together, of
projectiles.
Yet again Galileo was to challenge the long held theories of
Aristotle.
In his early work in this field he performed inclined plane experiments, in which he studied both gravity and inertia.
Galileo's Law of Fall| Galileo's findings | |||||
| Time | 1 | 2 | 3 | 4 | 5 |
| Distance | 1 | 3 | 5 | 7 | 9 |
| Total Distance | 1 | 4 | 9 | 16 | 25 |
| Time2 | 1 | 4 | 9 | 16 | 25 |
He was studying the Law of Fall from as early as 1604 and was using inclined planes because in his initial experiments he found that objects in free fall accelerate too quickly for accurate measurement so he began using these planes to effectively slow down the rate of fall, therefore allowing him to make accurate time and distance measurements.
Aristotle had previously said that bodies falling in the same
medium will fall at a speed proportional to their weight.
Galileo believed this to be incorrect he suggested that all
objects in a medium without resistance (e.g. on the Earth's
surface but ignoring the effects of air resistance) will all gain
equal amounts of velocity in equal intervals of time (uniform
acceleration), regardless of weight (in other words all objects
will fall at a constant rate and further to this, this rate will
be the same for all objects no matter their weight).
After long experimentation he found that as the seconds past the
distance the object would travel would increase in ascending odd
numbers i.e. 1, 3, 5 ,7 ,9 etc (in other words in the third
second the object will travel a distance of 5 units, but the
total distance travelled in all 3 seconds of the experiment is
5+3+1=9units) from this he realised that the distance covered is
directly proportional to the square of time taken (see
table).
This was one of his startling discoveries, that laid down the
groundwork for some famous work to be done in the not too distant
future.
On inertia Aristotle had formulated the principle of impetus from his observations that most objects do not remain in motion after a force that is acting upon them is moved is removed. So he suggested that any object in motion will not remain so unless the force that is acting upon it does so constantly, if a force was removed the impetus would run out. What Aristotle actually believed was that the natural state of any object is at rest and so any object at rest will remain so unless acted upon by a force. Using just observation it is clear why Aristotle would think this way, although this concept clearly has shortcomings.
Take, for example, an arrow travelling
through the air, how could this arrow continue on its path after
the force (the bow string) was removed?
Aristotle's principle should mean that the arrow would not remain
in motion after the force acting on it was removed. This is an
often quoted example because it this was particularly troublesome
to the Greeks (their actual reasoning for why it remained in
motion had something to do with the arrow creating a vacuum
behind it and air rushed into the vacuum to push the arrow
along!).
With use of inclined planes Galileo realised that this was
wrong because Aristotle had failed to take into account a hidden
force (of course frictional force). He worked out that this
force was acting in the opposite direction to motion and that if
this force was decreased (by using oil, grease etc) then the
object in motion would move further before stopping.
From this Galileo was to formulate his Law of Inertia:
An object in a state of motion possesses an inertia causing it to remain in a state of motion unless acted on by an external force.
Another way to state this is: if the frictional force is
reduced to zero and a force is applied to an object so that it is
pushed at a constant velocity after that force is removed the
object will continue at that velocity forever, unless of course a
new force acts on it at a later time.
Clearly then Galileo demonstrated that a object's natural state
was not at rest, as Aristotle had believed, but in fact in
motion, and rest was just a special case where velocity was zero,
though there was still forces acting on it.
These principles are simply taken for granted now, just as many of Galileo's achievements are, but it must be remembered at the time, as foolish as it seems to us now, everyone believed Aristotle's principles and Galileo's work was totally revolutionary. Clearly his work laid down the principles of the modern day understanding of dynamics.
Galileo and Projectiles
|
Then Galileo went on to study
projectiles, where he brought together his work on falling
bodies and inertia and added the principle of Superposition. This
stated that if a body is subject to two influences, each
producing a characteristic type of motion, the object will
respond to each, without modifying its response to the other. Or
in simpler terms, referring to the diagram (right), a diagonal motion (V) may be spilt into
its vertical (Vy)
and horizontal (Vx) vectors and these two vectors can be treated
separately.
Before Galileo it was believed that when a projectile was
launched it would continue until its impetus (horizontal motion)
was lost and then fall towards ground. (Aristotle believed
that the projectiles were pushed along (horizontally) by an
external force transmitted through the air, see the example of
the arrow above).
For Galileo though projectiles move both horizontally and
vertically at the same time, and that the motion can be separated
into these two components (principle of Superposition). So what
Galileo achieved here was to realise that the object was subject
not only to horizontal force that caused acceleration but also to
a vertical acceleration.
The horizontal component is described by Galileo's principle
of inertia (contrast to impetus) so, providing no force acts on
it after launch (in this case no air resistance) the horizontal
velocity will remain constant (i.e. horizontal acceleration is
zero), so the horizontal distance is clearly proportionate to the
time taken to cover it.
The vertical component causes constantly accelerated motion
(don't forgot it is a vector so the direction is important, in
this case downwards towards the Earth) so the vertical distance
is proportional to the square of time taken (from his law of
fall).
By treating the components separately when Galileo combined the calculations for the components he predicted that the path of a projectile would be a gently curving arc called a parabola and he proved this by experimentation. He showed that the projectile would always follow this path regardless of launch angle and launch velocity. (Obviously there is resistance in real world experiments, but the effect of air resistance is not that great and experimentation results are very similar to what would be predicted using the mathematics).
Galileo's Projectile work Galileo's original sketch |
Different parabolicpaths original source unknown |
Galileo
Galileo stands as one of the all time great scientists, not only for his discoveries and his pioneering techniques but also for the fact that he had the courage to stand up and fight against tradition, public opinion and the church for what he believed in.
Science truly owes Galileo Galilei a debt of incredible gratitude.
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