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In this lesson we will see how Galileo's combination of inductive and deductive logic with observations and experiments allowed him to forge a new paradigm. Along the way we will see how Galileo made assumptions, tested suppositions, made inferences and conclusions as he rewrote the book of motion, proved Aristotle to be nonsense, discovered the principle of inertia practically by accident, and set the final props on the stage for Newton's gravitational act.
In the previous program we tried to establish some insight to Galileo and his times. We also learned about the different tools that Galileo used in his work. Now it is time to see how he used those tools to analyze motion and ascertain its true nature.
The goal of disproving Aristotle's views on motion, Galileo met successfully. Through his use of logic, mathematics, experiments, and observations, he not only refuted Aristotle, but along the way he made some other surprising discoveries of the properties of matter and motion set the stage for Newton's work on gravity and the subsequent development of conservation laws. Even today Galileo's discoveries define our understanding of the nature of motion
In addition to his motion studies, and his telescope observations, Galileo used logic and mathematics in ways that had not been tried before to discover relationships which were quantitatively testable in a laboratory. He defined motion in an unambiguous way by introducing the concept of time and its relationship with space or location. From the data he collected in cleverly designed, carefully controlled and repeated experiments, he was able to show that pure freefall acceleration is uniform and constant for all objects regardless of their own size and weight.
The air, regarded by Aristotle as the controlling factor in natural motion, was shown by Galileo to be a interference to motion rather than a cause of it.
After observing the motion of balls of various textures as they rolled on inclined planes of various angles, he concluded that without gravity to speed the descent and slow the ascent, and without friction, objects would not start or stop moving at all.
Recognizing that falling objects accelerate downward at the same uniform rate even if they are moving horizontally allowed him to describe the motion of projectiles. algebra.
Describing projectile motion as a combination of horizontal and vertical motion and in the same terms and with the same relationships destroyed Aristotle's concept of different types of motion.
Through his efforts, and despite the political suppression of his ideas and writings, Galileo established a new paradigm of motion. In this new view, the idea that motion ceased unless actively maintained was replaced by the notion that motion continued unless interfered with.
Galileo used a number of methods to attack Aristotle's conjecture that objects freefall at a rate proportional to their gravity, or heaviness. Recall that Aristotle thought that freefall motion, which he called natural motion, was caused by the desire for a substance to attain its rightful place in the concentric hierarchy of matter.
So the heavier an object was, the more desire it had, and therefore the greater necessity to find that place, and therefore the faster it would fall. (Or rise in the case of lightness, like air bubbles in water.)
Galileo applied the logical style of Plato, while using both inductive and deductive logic to theorize about and observe the behavior of objects in freefall. He applied his own methods of mathematical generalization to derive new relationships from precise definitions, which described motion as a rate of change. He performed controlled experiments repetitively and understood the role of errors in measurement. Of all of the methods used by Galileo, it was his ability to creatively generalize and extrapolate which led to the real discoveries.
Imagine the feeling to not only disprove two thousand years of thinking wrong, but also to see how simple it all becomes when certain verifiable assumptions are made. The key word here is "verifiable". Why is that a significant word?
Galileo's logic did not prove conclusively that Aristotle's freefall was incorrect. It only showed it to be logically inconsistent. That's all it takes logically, but as Aristotle unknowingly demonstrated to us, logic alone is not enough. Logic is enough to cast serious doubt, but in his attempt to maintain logical consistency he had painted himself into an intellectual corner. Using the same methods of discourse, but employing both inductive and deductive logic in a complimentary way, Galileo took a different approach to reality.
He attacked the logic Aristotle's conjecture about natural motion being dependent upon weight and composition in a very clever and insightful way.
3.1.1. two objects of different weight tied together
Suppose two weights of different gravities were tied together and dropped.
Assuming the rope holding them together does not stretch, then one of the following might occur, but which one?
1. The heaviness of one interacts with the lightness of the other, but they fall at a reduced rate which is somewhere in between the rate that either would fall alone.
2. The two objects as a unit have more gravity than either of them alone, so they fall at a faster rate than either would fall alone.
3. They fall at the slower rate of the lighter weight.
4. They fall at the faster rate of the heavier weight.
You see, if objects fall at a different rate according to their gravity, then there is no clear way to predict which one of these might actually occur. You would think that it would be easy to go outside and try it. Why do you think more people didn't?
The only explanation which does not produce a series of paradoxical possibilities, is that all of the objects fall at the same rate regardless of their weight. Consider this for a minute.
If the two objects fall at the same rate then they will do so whether or not they are tied together. There is no ambiguity, no confusion.
This is a brilliant and simple argument, which proves nothing. What it does is to suggest that it makes more sense if everything falls at the same rate, regardless of why it might be so.
It also sets up the desire to actually watch things drop, which Galileo did, and he demonstrated it to others. Most of them thought his demonstrations to be parlor tricks, or thought that he was somehow making them see things that didn't happen.
That's what magic is, right?
3.1.2. very large vs. very small weight
And what about a very large and a very small weight. Galileo asserted that it was preposterous to assume that if one object weighed a thousand times more than another, then it might reach the ground before the lighter weight had appreciably moved.
This might be true for a marble and grain of dust, but it is certainly not true for a marble and a bowling ball. Try it if you have one.
3.1.3. inductive and deductive
We have already noted, in program thirteen, how Galileo used both inductive and deductive logic to derive and test specific relationships. We will see this more clearly in the next section when we study Galileo's experiments.
For instance, you use deduction, in the form of algebra. Start with definitions of velocity and uniform acceleration, then derive a relationship between distance and time when an object is uniformly accelerated. This is using a principle to predict outcome, deductive logic.
Next, you design an experiment which will allow you to observe the behavior that you wish to observe, in this case, distance and time. You observe and record the behavior and you have a bunch of numbers. Then what?
Then you compare the numbers to see if they match what your deduction predicted they should be. That's deduction. What about induction?
For induction you observe that a ball will roll uphill after it rolls downhill. Then you use induction to discover a general rule that applies in both cases. In this case, you decide that the principle is that the same cause applies in both the uphill and the downhill case, so the amount of influence is the same in both cases.
Now you can take this conclusion as a general rule and go deductive with it again and predict that if the induction is correct, then the ball should roll uphill to the same height from which it was rolled downhill. Now you have deduced a testable behavior which is true if the induced rule is true.
Do you see how it works. You might need to read the preceding paragraph over several times. If you do not see it, try drawing a diagram like the one at the end of lesson thirteen, you know the triangular one.
3.1.4. what about rolling?
And what about objects which roll downhill? Is that like or unlike freefall? In what ways is it different?
For one thing, Galileo noted, downhill motion is horizontal and vertical at the same time, kind of like a bug crawling sideways on a moving board. This was contrary to Aristotle's notion that violent and natural motion were different in nature.
We have already seen in lessons twelve and thirteen the way in which Galileo used mathematics to define velocity and average velocity. From those definitions he derive a testable description of uniformly accelerated motion in terms of the mathematical relationships between distance and time.
This analysis led to the conclusion that in uniformly accelerated motion a linear relationship exists between distance and the square of time. If you are so inclined you may want to go back and look at this analysis and note that acceleration is a rate of a rate. Thus the appearance of time twice, as in time squared.
The concept of a rate of change is abstract, enough to have escaped thousands of years of great minds who thought about change. The concept of a rate of change of a rate of change is one step removed, like moving from flatland to three dimensions. Like people who cannot see three dimensions in flat drawings, we have difficulty perceiving such things.
Although it is an abstract concept there are other examples which are easier to grasp, interest rates for example. Suppose you invest at a time when banks are paying three percent interest. This is a rate of growth which will allow you to know how much your investment will be worth after a certain time has passed.
Suppose that you knew that the rate of interest increased one half percent per year, so that next year it would pay three and one half percent, and the following year four percent, and so on. This is a rate of a rate.
From this you could know two things. First you could know the interest rate in any year from a simple addition of one half percent per year. Second you could know how much the investment is worth at any time because you know the interest rate at any time. Even here, time is involved twice, once to calculate the interest rate and again to calculate the value once the interest rate is known.
So, even if you are not mathematically inclined, note that the concept of rate and changing rate are not just useful in obscure calculations of distance and time.
Equally important was his use of mathematics as description of change, specifically change in relation to time. Time had not entered into previous notions of motion, except in the sense of clock time. We use time and rely upon it without thought of its properties. Maybe that is why no one thought to use it before Galileo. It is an abstract concept that events happen in space, but also in time.
You might even say that Galileo discovered time in the same way the the Greeks discovered the mind. We will explore these aspects of time bit by bit in upcoming lessons, just because we are curious, and to remind us that there may be other things like time, and air for another, that we take for granted.
As a way to prepare our minds for the future, think and write about this one:
What is time? Not what time is it, but what is time? Try to describe its properties. Is it matter, or energy, or something else? Does it really exist, or is it just a convenience used to describe change the way Ptolemy's devices described the motions of the planets?
The Macintosh defines time: "Time is a nonspatial continuum in which events occur in apparently irreversible succession from the past through the present to the future."
3.2.1. mathematical definitions and derivations
3.2.2. description of motion as rate of change
As for the experiments, they were not just well designed in terms of the defined variables. They were also designed to test the theoretical relationships Galileo had so laboriously defined and derived.
If the numerical relationship of the measured distances and times was of the same type as the theoretically derived relationships, then it made a strong case that reality was actually defined in some general way by the relationship. Specifically the assumption is that if the relationship between distance and time turns out to be as the mathematics predicted, then it is safe to conclude that this relationship does apply and that the falling object is indeed uniformly accelerated.
Just in case you might be getting a sense of "So . . .", remember that this is not really about balls rolling down hills, it is about truth and authority. Galileo's purpose in doing this was to prove that Aristotle was wrong about motion because if he was then he might be wrong about geocentrism.
Is the fact that it is the type of relationship between the numbers and not the numbers themselves consistent with Pythagorean beliefs or against it? Write a short essay that argues both sides convincingly.
Another important aspect of the experiments was the number of repetitions. Galileo recognized that the more measurements which were taken, the closer the results approximated reality. After hundreds of trials over several years, any irregularities tend to be smoothed out and overwhelmed by the averages.
The formalization of the concept of averages and statistics was still two hundred years away at this point, but we will see it again elsewhere in our travels, downstream in the river of scientific heritage.
3.3.1. collected numerical data and compared it with theory
3.3.2. performed experiments to test theories
3.3.3. introduced time into motion studies
3.3.4. did many repetitions for precision
Aristotle had done experiments where he correctly induced that stones and other smooth objects falling through water do indeed reach a terminal velocity that is proportional to their weight (or more appropriately, their density).
What Aristotle had failed to understand was that the extrapolation from water to air does apply, but water is millions of times thicker (more viscous) than air. When objects fall through a medium such as air or water they accelerate until the effort required to move through the medium at a certain speed is great enough to match the weight of the object in the medium. When that happens they no longer accelerate but move at a constant speed, called the terminal velocity.
The time it takes for objects falling through water to reach their terminal velocity is small because water retards their motion extremely. In air, very dense objects such as lead weights are not affected much by air resistance and could fall great distances before reaching terminal velocity.
Over the short distances of the laboratory, the motion of very dense objects might not be affected much at all. If we could measure the distance they fell in various time intervals we could analyze their motion to see if it is uniformly accelerated, then we could quantitatively determine the rate and see if it is the same.
OK, here we go drop the ball and . . .Oops! The problem is that when you drop things everything happens too quickly to measure accurately, much below the reflex time of starting and stopping a stopwatch, even assuming you had a stopwatch, which Galileo didn't.
On the other hand, dropping them from heights great enough to time their fall with any precision requires that they be in the air long enough to be more significantly affected by it as their velocity increases.
Aristotle solved this problem by dropping things through water because he thought that the medium was the prime factor in the motion. His paradigm drove the way in which he tested it.
Galileo chose a different approach. He decided to slow down the motion through the air in some way which did did not affect the nature of the distance-time relationship. He correctly reasoned that a ball rolling downhill on a flat surface moved downhill due to the same cause as a ball in freefall, regardless of the nature of the cause. Galileo adopted Aristotle's terminology and simply said it was due to the gravity of the ball. Here the word gravity is used in the context of a quality of the ball, as in "the ball falls because it has heaviness." Note that the cause of the motion is independent of the quality of heaviness, which exists as a driving force for motion regardless of the cause.
3.4.1. air resistance masks true nature of freefall motion
Here's one of those "wrong" questions that Galileo asked.
Aristotle taught that the ball fell to attain it's appropriate place and that in the absence of a medium such as air or water, it would fall at infinite speed. This is an interesting assumption, but is it a warranted assumption. What if air is not the controlling factor of motion at all, as Aristotle taught, but rather air is just a minor hindrance to the motion which masks the true motion rather than controlling it? Then what?
If we could get rid of the air, just pump it out we could see the pure uninterrupted motion before air had a chance to alter it.
The problem was that the vacuum pump would not be invented for another one hundred and fifty years so they could not do the thing with the tube that we do on the tube.
18.104.22.168. Video Demonstration
In the video you will see a demonstration of the effects of the air on the rate of fall of a feather and a coin. First we will see it when the tube is full of air, then again when the air is removed with a vacuum pump.
You will also see a video taken on the surface of the moon. One of the early manned missions to the moon included, as part of the designed experiments, a hammer and a feather. The airless surface of the moon truly allowed this experiment to be tested for the first time in a nearly perfect vacuum. It also demonstrated that the effects of gravity on the moon were the same as here on earth.
How insightful of Galileo to be able to conclude that which could be truly tested only on arrival at the surface of another planet, earthlike in its gravity as well as in its mountains and craters.
22.214.171.124. remove air resistance to see true nature
In Galileo's time he could not remove the air physically, but he could do it mentally, through the use of logical inference. With this idea he unknowingly introduced the notion of limits which Newton would use to great advantage in analyzing planetary motion sixty years later.
The concept of a limit is easy in principle, although difficult in practice. Suppose you start with a lot of interference and gradually reduce it, noting the effect on motion. Then you extrapolate the results to the case of no resistance. Simple, no?
This is a difficult concept at first. Certainly it is a difficult one to have thought of in the first place. What a concept! Minimize and generalize to the ideal case of no air at all. Then the true nature of motion will be understood. Later the effects of the air and other resistance to motion can be studied separately. Do you see how this is an example of model-building? It is like wiping the dust off an old photograph, or cleaning the patina off the silverware. The model-building process tries to find the essence of the problem by removing those things which are not essential to its understanding. Galileo's insight and intuition into the nature of motion allowed him to recognize the true role of air resistance, and it's similarity to rolling resistance.
This process of breaking the problem of motion into parts, or smaller problems, and then solving them separately is a form of reductionism which Galileo originated and which led to the writing of algorithms in the construction of computer codes which drive the programs which we used to produce this study guide.
126.96.36.199.1. can't be done in practice, but can be done in principle
188.8.131.52.2. minimize effects and see how motion is affected
184.108.40.206.3. generalize to ideal case of no air
Galileo sought a practical method to minimize air resistance while maximizing the time over which the measurement could be made and at the same time controlling the amount of interference encountered by the moving object.
He discovered in church one Sunday that the swinging chandelier took the same amount of time for one swing regardless of how widely it was swinging. He used his heartbeat to verify this, although it is likely that his heart sped up a little when he realized it. Some scholars have claimed that this is what gave him the idea to use time as a variable. We will probably never know, but, we do know that he used a pendulum as a steady timing device instead of a heartbeat which was variable depending on mood and state of exertion.
Although he later used a more precise clock made of water, his recognition and use of the pendulum as a timekeeping device stimulated the development of mechanical clocks and initiated the practice of time keeping. This alone, in the absence of Galileo's other contributions, would have have a profound impact on history.
Later, in Holland, Christopher Huygens would design an accurate mechanical clock, the pendulum clock, based on Galileo's observation which would revolutionize the science of physics as well as navigation and astronomy. A century and a half later the accuracy of the clock and Newton's law of gravity would motivate the voyage of Captain James Cook. This voyage, which brought European culture into the Hawaiian islands for the first time, was prompted by the desire to verify Newton's laws with the help of an extremely accurate pendulum clock.
3.5.2. inclined plane
With the idea that motion down the inclined plane is both horizontal and vertical, Galileo saw that the flat, sloping plane slows down the vertical part of the motion without changing the nature of the relationship between distance and time. He reasoned that if freefall acceleration was constant, then so would the acceleration of the ball down the ramp, but at a slower rate which was somehow related to the slope or steepness of the incline. This assuming that the effects of rolling resistance could be minimized.
In other words it changes the rate without changing the relationship. The experiments that he conducted also were designed to test this hypotheses. The relationship and the hypotheses will be explored as is part of a laboratory exercise and will be covered separately.
The details of the experiment will become involved as we analyze them. We do this partly in respect for Galileo's genius, but mostly we do it because it has become second nature to us in our science to conduct ourselves in this manner. It is so much a part of our paradigm that we do it almost subconsciously, like riding a bicycle. We do these things automatically today in science. It is our paradigm. Like any cultural behavior it needs no justification, other than it works. If we are not careful we are likely to forget that the paradigm arose out of formalized efforts like those of Galileo's. Unlike us, he had to be aware of these details in order to formulate them.
You will have the opportunity to do similar experiments as part of the laboratory exercises.
The design of the experiment is the most important part. In the same way that asking the right question is important in getting the right answer, the results of an experiment are no better than its design.
Although there is no recipe for designing an experiment, Galileo's method left little room for improvement. Other than a statistical error analysis, the methods of which would not be available for two hundred years, there is little that we have added to the method since Galileo did it first.
A hypothesis is a conjecture which is to be tested, logically, mathematically, or quantitatively. In an experiment it is a deductive statement about the results of the experiment to be expected if certain relationships are present. The best hypotheses are those which are simply stated, with few qualifying statements or special conditions.
220.127.116.11. Objects in freefall are uniformly accelerated in the absence of air resistance.
Galileo's hypothesis is actually composed of several parts. One assumption is that the same rate of freefall applies to objects regardless of their weight, or gravity as Galileo called it.
Another assumption is that the rate o does not depend on the size, the shape, or the composition.
18.104.22.168.1. rate of acceleration does not depend upon size, shape, or substance
22.214.171.124.2. acceleration of gravity follows a general natural law, not specific and different laws for each object.
4.1.2. Define Terms
In order to measure the rate of something, we must know what that thing is. We have seen how Galileo defined the terms in such a way that they depended upon measurable quantities.. Go back and review lesson 2.3 if you do not understand it or need to review it. It might also help to review the laboratory exercise entitled "Describing Motion", if you have received it.
126.96.36.199. velocity = rate of change of position or location
188.8.131.52. acceleration = rate of change of velocity
4.1.3. Derive Relationships
We also saw in lesson 12.2.3, Describing Motion, how Galileo derived relationships that could be used to determine whether or not acceleration is uniform. We know of two ways because there are two relationships.
1. Rate of change of velocity with respect to time is constant <==> Graph of velocity versus time is linear.
2. Distance changes with the square of time <==> Graph of distance verses time to the second power is a straight line.
Galileo did not have our ability to use graphical analysis, but he could make measurements and compare them with calculated values, or compare ratios.
184.108.40.206. can't measure speed or acceleration directly
In Galileo's time there were no speedometers or accelerometers. He could measure average speed over a long time interval, but that would not show the details of the accelerated motion.
220.127.116.11. distance and time are measurable quantities
What he could measure were distance and time, so he found it necessary to define uniform acceleration in terms of distance and time alone. This, as we have seen, involves algebraic logic, which was known to Galileo, although he did not have the benefit of our modern notation.
18.104.22.168. derive relationships in terms of distance and time
Here is the relationship again, the way Galileo derived it.
This is really a simple substitution logic, so don't get lost in it. In words it would sound something like this.
1. We define average velocity as distance divided by time so it must be true that distance is equal to velocity multiplied by time.
2. We define acceleration as the rate of change of velocity, or the change in velocity divided by time. If we start from rest then the instantaneous velocity at any time is the acceleration multiplied by the time.
3. In uniformly accelerated motion the average velocity is one half of the instantaneous velocity at the end of a time period, assuming an initial state of rest.
combining yields the final relationship
Distance equals average velocity multiplied by time, but average velocity is one half of final instantaneous velocity. Furthermore velocity is acceleration multiplied by time so distance must be equal to one-half times the acceleration multiplied by time multiplied by time.
Since the factor of time appears twice, multiplied by itself, then the distance traveled must be proportional to the second power of time, with a constant of proportion equal to one half of the acceleration, assuming the initial velocity was zero.
Which one is easier to follow. What makes this difficult to follow? Is it the math, or is it the fact that we are discussing the rate of a rate, or is it something else entirely?
The fact that the concept of time, which we don't really know what it is anyway, could appear twice in the relationship in this very algebraic way and accurately describe the way something falls to the earth is simply amazing, if you think about it. Why should it be like this? Is this Pythagorean, does it mean something?
22.214.171.124. uniform acceleration ==> distance proportional to square of time
We may never know the reasons why these relationships exist. It may take forever if it can be done at all. But to understand how it works is very simple.
The direct proportion between distance and time to the second power can be expressed in terms of a ratio.
For a given rate of acceleration, doubling the time results in quadrupling the distance, for any distance and time.
For example if the ball rolls ten centimeters in two seconds, in four seconds it will roll forty centimeters. Twice the time but four times the distance. Four is two squared. Distance is proportional to the square of time.
In six seconds the ball will roll ninety centimeters. So how far will it roll in six seconds if the acceleration is constant?
Six is three twos, or three times two. The time is tripled so the distance will be nine times greater because nine is three squared. Again the distance is proportional to the square of time.
126.96.36.199.1. doubling time results in quadrupling distance
188.8.131.52.2. for any distance and time
184.108.40.206.3. example: in 2 seconds ball rolls 10 cm
in 4 seconds ball rolls 40 cm
in 6 seconds ball rolls 90 cm
4.1.4. Measure distance and time accurately
Measuring the distance and time as accurately as possible will make it easier to interpret the results and to see the relationships.
220.127.116.11. peg board plane
Distance is relatively easy to measure and easy to control. Galileo used a grooved board with small holes at even distances into which he inserted pegs to stop the rolling ball at a certain distance and to give an audible clue as to when to stop the timer.
He tried using the pendulum clock as a timer but found it difficult to measure the distance accurately while counting the swings of the pendulum.
18.104.22.168. water clock
To solve the problem of the clock he invented a water clock.
Water flows at a uniform slow rate from a large tank if the tank is large compared to the flow of water. Why is this so? Does it have something to do with the level of water in the tank and how fast it changes?
He found that he could easily start and stop the watch by holding it in the stream and removing it. This he could do without looking at it. It is the equivalent of the button on the modern stopwatch.
4.1.5. Keep careful and accurate records
In addition to everything else you must keep accurate and organized records of the data and conditions for each run of the experiment. This requires data tables of distance and times with notations of the angle of the ramp, the type of surface on the ball and on the board and other information which might be relevant.
Keeping good records is a skill that does not come easily to most of us.
22.214.171.124. of distances, times, angles, surfaces, types of balls
4.1.6. Change variables
The variables involved in the inclined plane include the angle of slope of the plane, the height from which the ball is rolled, and the surfaces over which the ball roll and the surface of the ball itself, like a tennis ball on asphalt compared to a baseball on astroturf.
To do this requires that the distance and time are measured for various angles, heights, and surfaces and the results compared with the predictions made by the derived relationships.
126.96.36.199. angles, heights, surfaces
4.1.7. Many repetitions
You must also do many repetitions in order to be sure of the relationships. Just because you can find numerical relationships does not mean you have a physical relationship. There are many numerical coincidences which do not express physical relationships. For example 2 plus two equals two times two and also equals two squared. That is a numerical relationship but not a physical relationship.
188.8.131.52. precision is improved with many trials
184.108.40.206. averages out errors in measurement
4.1.8. Analyze results
Last but not least is t0 analyze the results and compare them with predicted relationships. In Galileo's case that would be to compare the measured distances and time with the calculated ones. This is not as easy as it might seem, even with our modern tools of graphical and numerical analysis.
220.127.116.11. compare measured data with expected results
Now we are ready to consider what Galileo discovered as a result of his studies. Basically what he did was to prove the hypothesis but also to discover some corollaries which led him to an unexpected explanation of projectile motion.
4.2.1. acceleration down the inclined plane is uniform
The distance traveled is directly proportional to the second power of the time. This is in accord with the derived relationship between distance and time. A ball rolling down an inclined plane undergoes uniform acceleration.
4.2.2. rate of acceleration on incline depends on ratio of height to length
The rate of acceleration increases when the angle is steeper, but the direct proportion is not a relationship between acceleration and angle. Rather acceleration is proportional to the ratio of the height of the plane to its length.
From the figure you can see that the length of the plane forms the hypotenuse of a right triangle.
This ratio is known as the sine of the angle of incline, so we can say that uniform acceleration on an inclined plane is directly proportional to the sine of the angle of incline.
From this relationship Galileo could extrapolate to discover the rate of freefall. He reasoned that if the height of the ramp was increased while the length remained constant, the limit would be reached when the length equaled the height. At that angle, 90deg., the rate of acceleration would equal the rate of freefall.
In modern terms we could write a = g sin ø where a is acceleration, g is the rate of freefall acceleration and ø is the angle of incline. We could also write this as a = g (h/l) where h is the height and l is the length of the plane.
18.104.22.168. ratio is a measure of the angle called "sine"
22.214.171.124. acceleration is proportional to the sine of the angle of incline
126.96.36.199. limits of the angle of incline are a horizontal surface and a vertical surface
4.2.3. speed does not depend on weight or size of ball
Just as hypothesized, balls of all sizes and compositions show the same uniform acceleration on a ramp of a given slope. Since freefall acceleration is the vertical limit of acceleration on a ramp, it must be true for freefall as well.
188.8.131.52. balls of various weights and sizes show the same relationships
4.2.4. speed depends on resistance of surface
As expected, the effect of the roughness of the ball or the surface is to slow the motion and cause deviations from the distance-time squared relationship. The rougher the surface, the greater the deviation. For smooth balls rolling on smooth surfaces, the deviation is very small.
184.108.40.206. rougher surface or balls slows speed more
4.2.5. speed depends on height, not slope
This observation and the next was were entirely unexpected, that is to say they were not part of the hypothesis.
Galileo was surprised to discover that the smooth balls reached the same speed when released from a given height, regardless of the angle of incline. At first this result seems surprising, but upon closer examination it makes good sense. The results can be be predicted deductively from the definition, but the definitions can also be inductively derived from the behavior.
There are several different ways to look at this result. It's significance will become apparent soon, but to give a hint, it has to do with the principle of inertia, which Galileo discovered as a result of these experiments.
Consider two flat sloping surfaces A and B, each beginning at the same height, as shown in the figure.
Galileo's observation was that balls rolled from rest would acquire the same speed upon reaching the bottom of either slope. This does not imply that they took the same amount of time to reach the bottom. In fact the ball on slope B will reach the bottom in a shorter time than the ball on slope A.
Now let's look at this in terms of the parameters of motion and their relationships. The ball on slope B will accelerate at a greater rate, but the length of the hill is short compared to slope A. The ball rolling down slope B gains velocity faster but it has a shorter distance and a shorter time to accumulate the velocity than the ball which rolls down slope A. There the rate of acceleration is slower, but there is a longer distance and a longer time to accumulate speed. The distances and times for the two inclines are different, but the ratio of distance and time is the same in each case. The ratio of distance to time is the average velocity, which is exactly one half of the final velocity, in this case the speed at the bottom of the plane. So in one case a small rate applies for a long time and in the other case a larger rate applies for a shorter time, but the end result is the same.
We could sum this up by saying that the extra length of the gentler slope exactly compensates for the slower rate of acceleration and vice versa.
As you can see, we can explain this and we could have predicted it, had we only known to expect it. We did, Galileo didn't, but he was perceptive enough to recognize it from his data.
220.127.116.11. balls reach same speed if rolled from equal heights
18.104.22.168. gentler slope => longer time, greater distance, smaller acceleration
22.214.171.124. ratio of total distance to time remains constant
4.2.6. uphill height = downhill height
Having watched the pendulum swing back and forth, each time rising to nearly the same height on either side of its swing, like Galileo, we can now ask what, if any similarity there is between the motion of the ball on the slope and the motion of the pendulum bob.
Galileo reasoned it this way. The pendulum bob is moving downhill similar to the ball on the incline, except that it moves in a curved path. The experiments suggest that it is not the shape of the path that matters, but rather it is the height from which the motion started.
Whatever it is that causes the downhill motion to speed up also causes the uphill motion to slow down. Whatever role the gravity of the object plays, it should affect it equally whether it is moving uphill or downhill. It is as if the gravity of the ball somehow is exchanged for a certain amount of speed. If that is the case then that amount of speed should be just sufficient to allow the pendulum bob to rise to the same height as it loses exactly the amount of speed it acquired on the way down.
If this hypothesis is correct, then the ball should roll uphill to the same height from which it was rolled downhill, and it should reach that height regardless of the slope of either the uphill or downhill slopes.
What Galileo had done at this point was to generate a hypothesis for the observed behavior of the ball inductively: The ball or the pendulum bob acquires a certain amount of speed when falling from a certain height and will lose that same amount of speed when rising to an equal height. This is true regardless of the slopes of the inclines.
Note how he used deduction in the first place to successfully predict the distance-time relationship and uniform acceleration. Then he used the observed behavior of the ball, specifically the speed-height relationship, and induced a general relationship from it.
You might guess the next step is to test the induced hypothesis. Set up two opposing ramps and let the ball roll down one and up the other to see if the heights match on the uphill and downhill.
They did. Not exactly, but Galileo found they matched more closely the smoother the surfaces. From there he made the inference that in the absence of any interference, the heights would indeed match. Although he could never manufacture a completely frictionless surface (neither can we), he could still idealize the concept and extrapolate to it.
As we will see in the next section, this will lead Galileo inductively to the principle of inertia, and the coup de grace, his analysis of projectile motion.
126.96.36.199. regardless of uphill or downhill slope angle
188.8.131.52. not exactly, but closer when less resistance
184.108.40.206. leads to principle of inertia
From these experiments Galileo drew three major conclusions which would lead him to his explanation for projectile motion, replacing Aristotle's weakly justified notion of antiperistasis, and combining the concepts of violent and natural motion into a common set of rules and terms. It would also put into place a new paradigm and stimulate a new era of mathematical analysis which Isaac Newton would nail down a generation later.
4.3.1. interaction with air and surfaces interferes with purity of motion
Motion in the absence of air resistance is the pure state and the effect of air and other substance is to interfere with the motion. This specifically contradicts Aristotle's view that the medium plays a causal role in the motion along with the nature of the object in motion.
4.3.2. freefall is uniform acceleration
Galileo determined that freefall acceleration is uniform. The data he collected supported the distance vs. time hypothesis on inclined planes at various angles. The limit of a vertical plane represents freefall. The rate of acceleration is the same for all objects in the absence of air friction. The rate of freefall (in modern units) is 10 meters per second squared, or [ten meters per second] per second.
This photograph shows a falling weight captured on film by a strobe light which is flashing thirty times per second. You can clearly see that the distance traveled by the falling weight in each successive interval is increasing. Your eye will probably even tell you that it is increasing at a regular rate, although you cannot quantitatively determine what the rate it.
220.127.116.11. data supports distance vs. time squared hypothesis
18.104.22.168. limit of 90 deg slope is freefall
22.214.171.124. constant rate for all objects
126.96.36.199. independent of mass, size or composition
188.8.131.52. rate is (10 meters per second) per second
4.3.3. inertia is a property of matter which tends to remain in motion or at rest
Galileo's use of induction and inference is no better illustrated than by the principle of inertia. You will recall that Aristotle had claimed that motion was sustained only by cause or effort. Galileo was forced to disagree.
Suppose you had a completely frictionless surface and there was no resistance whatsoever to the movement of the ball. This is an idealized model, one that can not actually be constructed in the laboratory. Armed with the rules of behavior for the ball on the ramp, Galileo was now in a position to reach logical conclusions based upon verifiable postulates, not upon speculations as were those of his predecessors. We have said before that logic can produce a result no truer than its starting assumptions or statements. All the logic in the world can not produce truth from untrue postulates.
The figure below illustrates the logic. The less steep the angle of incline the greater distance the ball must roll to reach a certain height. In the absence of interference from the surface, the only thing that acts to slow the ball is its gravity. So how far will the ball roll before it reaches its starting height on the level surface?
Forever? It will never reach the starting height, so what will stop it from rolling forever? The evidence suggests that there are only two things which act to stop the ball from rolling. One of them is the roughness of the surface. The other is the weight of the ball which slows it down as it goes up the ramp. If both of those things are missing, then what will stop the ball?
Galileo answered those questions with a revelationary as well as a revolutionary idea.
It is the tendency of the ball to remain in motion unless something acts to stop it. Normally something acts to stop motion. We now call it friction. It is the sum of all the interferences which tend to "steal" speed from moving objects. We will deal with the topic of friction sporadically throughout the course. But in the absence of friction, or hills, or running into something solid, there is no cause for the ball to slow down. It will keep moving in a straight line at a constant speed until something acts on it to cause it to slow down.
This is contrary to our senses, at first. But when you think about it, there is no longer any reason for the prime mover to turn the planets. This will open the door for a new question to replace the old question of what keeps the planets moving. The new question will become, what keeps them from moving in straight lines at a constant speed.
184.108.40.206. height is the same
220.127.116.11.1. amount of speed gained or lost depends on height
18.104.22.168.2. foreshadowing conservation of energy
22.214.171.124. lower angle = longer uphill distance
126.96.36.199. how far will the ball roll before it reaches the starting height on a level surface?
188.8.131.52. what will slow the ball down if there is no hill and no rolling resistance?
We can now summarize the new paradigm which emerged from Galileo's studies. Don't forget that Galileo published these results in two books, Dialogues Concerning the Two Chief World Systems (1632) and Discourses on Two New Sciences (1638). Both of these books were widely read by scholars in Italy, all over Europe, and in England.
Change occurs according to natural laws. Change does not occur according to moral principles in an attempt to make the universe a better place.
The same laws apply to all substances, and their motion does not depend on the character, intent, or desires of the material.
5.1.1. natural laws as opposed to moral laws
5.1.2. same laws describe motion of different substances
5.1.3. motion not dependent on character or desires of material
The natural laws regarding motion can be expressed as mathematical relationships which apply both generally and specifically to motion of all types, horizontal or vertical, natural or violent.
We are interested in how nature behaves, not why it behaves that way. We do not need to know why motion occurs in order to describe how it occurs. We will be able to describe it in terms of relationships between measurable quantities.
5.3.1. How vs. why nature behaves
5.3.2. Description instead of cause
5.3.3. relationships between measurable quantities
With the principle of inertia, the Scholastic Physics is rendered invalid. Aristotle's idea that motion is an unnatural state which will continue only by cause is replaced by Galileo's inference the motion is a natural state which will continue until something acts to stop it.
This is an important principle and it is important that you understand the difference. Read these statements over and over again until they make sense.
5.4.1. "Motion is a natural state which continues until altered"
"Motion is an unnatural state which ceases unless maintained"
5.4.2. Aristotle's local and violent motions are the same
The next step in Galileo's analysis of motion was the realization that the principle of inertia removes the difficulty in understanding projectile motion. You may recall that Aristotle's explanation, which he termed antiperistasis, was unsatisfying to practically everyone who bothered to think about it, including Galileo.
Actually it is the principle of inertia in combination with the law of freefall that solves the problem.
Here is Galileo's revelation: In the absence of air or other resistance, an object will accelerate downward at a certain rate regardless of its composition, shape, weight, surface, size, etc. Shouldn't this also be true even if the object happened to be moving forward while it was falling?
According to Aristotle, a ball dropped from the mast of a moving ship should fall straight down and land behind the ship, like this.
From the point of view of the observer on the ship the ball would appear to land behind the ship, because it's "true" motion must be in a straight line and straight down. Aristotle used a ship for this analogy because it was about the fastest and smoothest motion he could imagine. A chariot behind a galloping horse was faster, but it's really hard to do physics there.
It is almost certain that Aristotle never actually tried this because it obviously is not the way things happen. Suppose they did happen this way. Suppose the flight attendant on an airliner dropped a can of soda from the cart. What would happen to it from the perspective of the passengers?
Air travel would be very dangerous in Aristotle's world.
What actually does happen when you drop something inside a moving vehicle, assuming it's moving at a constant speed and not turning a corner. If you don't know, try it, but not while driving, please. Sit in the car, bus, or airplane seat and toss something into the air or drop it from one hand to another. People might stare. So? You're doing science, it's like art.
I'll bet you it does exactly what it would do if the car, bus or plane was standing still. That is, it would appear to you to fall straight downward.
OK, but how would it appear to someone outside the airplane? Let's use Aristotle's ship to look at the situation from the point of view of inertia. The thought process goes something like this. When the ball is held in the hand at the top of the mast, it is moving at the same speed as the ship. When it is dropped, it continues to move at the same speed as the ship, ignoring the small amount of wind and air resistance it might encounter. Since it is still moving forward while it is accelerating downward, it will land in the same place on the deck that it would had the ship been standing still, directly below the observer. But it will not fall straight downward. To an observer outside the ship it will follow a curved path, as shown in the figure.
So, what does this have to do with projectiles? We will show you that the motion of the ball dropped from the mast of the ship is exactly analogous to a ball rolling off a tabletop. In the video you will see an animation of a ball rolling off a flat surface with gravity turned off and then on. Here's how it works.
Here's a strobe photograph which clearly shows the paths of two balls which were released at the same time. One was stationary, and one was launched horizontally. Notice that the two balls are always at the same height, although one is moving forward while it falls.
6.2.1. inertial motion horizontally
6.2.2. freefall motion vertically
6.2.3. objects freefall at the same rate even if they are moving forward
184.108.40.206. horizontal motion is separate
220.127.116.11. horizontal motion does not affect vertical motion
18.104.22.168. air resistance alters motion in a predictable way
Galileo was quick to recognize that if a ball is thrown vertically upward it will lose speed against its gravity. It will eventually come to rest instantaneously, then begin to gain speed as it falls. The upward and downward part of the motion are symmetrical. The time to reach the top is exactly the same as the time to fall back to the starting position.
The same is true even if the ball is thrown upward while moving forward, say in the car, bus, or plane. This situation is exactly analogous to launching the projectile at an angle, as if hit by a golf club or a baseball bat. The only variables are the speed and angle of launch. The rate of freefall acceleration does not change just because something is moving horizontally.
The strobe photograph below shows the path of a projectile. You can not help but notice the symmetry.
6.3.1. upward part of projectiles motion is symmetrical with downward part
6.3.2. time up = time down
Here is a good example of parsimony. The apparently complex motion of projectiles which had stumped the minds of Aristotle and one hundred generations who followed him, is easily resolved into two separate pieces, each of which is relatively simple. Two simple principles produce a complex motion, and it is much easier to understand the individual components.
6.4.1. two simple principles combine to produce complex motion
6.4.2. easier to understand individual components
Galileo had been convinced of the truth of heliocentrism by his telescopic observations after he had convinced himself of Aristotle's errors through his own studies of motion in the laboratory.
We noted before that the Scholastic cosmology really contained two different and equally strong paradigms. One of these was geocentrism, the other was circular perfection. Copernicus had proposed a heliocentric system, but it was circular. Brahe proposed a combination heliocentric and geocentric system. It too was circular. Kepler bent the circular paradigm when he showed inductively that the planetary orbits were elliptical.
Galileo, for all of his revolutionary thinking and his genius for persuasion, never embraced the idea of elliptical orbits, even though he had heard of them. Interesting isn't it that the one man upon who we focus as piercing the paradigm with his stubborn intelligence and clever discourse, still could not break free of the circle even while knowing the truth of heliocentris..
He was able to forge a new paradigm through a combination of genius, perseverance, curiosity, and stubbornness. That he did it at all is amazing. That he did it virtually single-handedly in the midst of the Scholastic thinking of his peers is incredible.
But he did, and we owe him much, as scientists and as human beings.
Following is an outline of Galileo's accomplishments. Review it to make sure you covered everything in the lesson
7.1.1. experience is not only the best teacher, it is also the best authority
7.1.2. authority does not equate with knowledge
7.2.1. observed the heavens and knew what he was seeing
7.2.2. discovered many new facts which supported heliocentrism
22.214.171.124. moon craters
126.96.36.199. crescent of Venus
188.8.131.52. moons of Jupiter
184.108.40.206. new stars
7.3.1. motion depends on both distance and time
7.3.2. invention of water clock
7.3.3. discovered regular motion of pendulum
7.4.1. vertical and horizontal motion can be described in the same terms
7.4.2. precise definitions of velocity and acceleration
7.4.3. relationships between distance and second power of time for uniformly accelerated motion
7.5.1. Galileo's Law: In the absence of air resistance, all objects near the surface of Earth accelerate uniformly at a rate of 10 meters/s2 regardless of shape, composition, size, or weight.
7.6.1. an object in motion remains in motion unless something acts on it to change its motion
7.6.2. an object at rest requires action to cause motion
7.7.1. complex motion is combination of simple motions
7.7.2. motion can be in more than one direction at a time
7.7.3. motion in one direction can be independent of motion in another direction
7.8.1. theory - analysis - observation
7.8.2. mathematical modeling
7.8.3. experimental design
7.8.4. inductive and deductive logic work together
7.9.1. formulate theories
7.9.2. test hypotheses
7.9.3. design controllable experiments
7.9.4. collect and analyze data
7.10.1. Aristotle was wrong about a lot of things
7.10.2. Ptolemaic system is not consistent with observations
7.10.3. natural laws apply to a range of objects
7.10.4. objective and repeated measurements
7.10.5. use mathematical relationships to test hypotheses
7.10.6. goal is description of rather than cause of phenomena