Factors Affecting the Swing of a Pendulum

SAM BOOTH COURSE WORK January 2001

FACTORS AFFECTING THE SWING OF A PENDULUM

RESEARCH

A pendulum is described as a weight suspended on a length of string oscillating constantly. The pendulums mechanics were first discovered by Galileo he discovered that pendulum could keep time even when the oscillations dyed away. A simple pendulum is defined as a small heavy body suspended by a light Inextensible string. We can understand that if you change certain variables e.g. Length of string, weight on string, or the pull back angle of the string. These can vary the speed of a single swing of the pendulum. To get the time taken for the pendulum to swing (From one point back to the same point) It is more accurate to time 10 swings and then divide the time taken by 10 to get your answer.

The formula for periodic time of a pendulum is T = 2 L/g

L = Length in M

G =m acceleration due to gravity in m/s

The square of the periodic time I think should be proportional to the length of the pendulum.

Galileo was taught Aristotelian physics at the university of Pisa. But he quickly began questioning this approach. Where Aristotle had taken a qualitative and verbal approach, Galileo developed a quantitative and mathematical approach. Where the Aristotelians argued that heavier bodies fell faster than lighter ones in the same medium, Galileo, early in his career, came to believe that the difference in speed depended on the densities of the bodies. Where Aristotelians maintained that in the absence of the resisting force of a medium a body would travel infinitely fast and that a vacuum was therefore impossible, Galileo eventually came to believe that in a vacuum all bodies would fall with the same speed, and that this speed was proportional to the time of fall.

Because of his mathematical approach to motion, Galileo was intrigued by the back and forth motion of a suspended weight. His earliest considerations of this phenomenon must be dated to his days before he accepted a teaching position at the university of Pisa. His first biographer states that he began his study of pendulums after he watched a suspended lamp swing back and forth in the cathedral of Pisa when he was still a student there. Galileo's first notes on the subject date from 1588, but he did not begin serious investigations until 1602.

Galileo's discovery was that the period of swing of a pendulum is independent of its amplitude--the arc of the swing--the isochronism of the pendulum. Now this discovery had important implications for the measurement of time intervals. In 1602 he explained the isochronism of long pendulums in a letter to a friend, and a year later another friend, Santorio Santorio, a physician in Venice, began using a short pendulum, which he called "pulsilogium," to measure the pulse of his patients. The study of the pendulum, the first harmonic oscillator*, date from this period.

The motion of the pendulum ...

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The motion of the pendulum bob posed interesting problems. What was the fastest motion from a higher to a lower point, along a circular arc like a pendulum bob or along a straight line like on an inclined plane? Does the weight of the bob have an effect on the period? What is the relationship between the length and the period? Throughout his experimental work, the pendulum was never very far from Galileo's thought. But there was also the question of its practical use.

A pendulum could be used for timing pulses or acting as a metronome for students of music: its swings measured out equal time intervals. Could the device also be used to improve clocks? The mechanical clock, using a heavy weight to provide the motive power, began displacing the much older water clock in the High Middle Ages. By incremental improvement, the device had become smaller and more reliable. But the accuracy of the best clocks was still so low that they were, for instance, useless for astronomical purposes. Not only did they gain or lose time, but they did so in an irregular and unpredictable manner. Could a pendulum be hooked up to the escape mechanism of a clock so as to regulate it?

In 1641, at the age of 77, totally blind, Galileo turned his attention to this problem. describes the events as follows, as translated by Stillman Drake:

One day in 1641, while I was living with him at his villa in Arcetri, I remember that the idea occurred to him that the pendulum could be adapted to clocks with weights or springs, serving in place of the usual tempo, he hoping that the very even and natural motions of the pendulum would correct all the defects in the art of clocks. But because his being deprived of sight prevented his making drawings and models to the desired effect, and his son Vincenzio coming one day from Florence to Arcetri, Galileo told him his idea and several discussions followed. Finally they decided on the scheme shown in the accompanying drawing, to be put in practice to learn the fact of those difficulties in machines which are usually not foreseen in simple theorising.

Viviani wrote this in 1659, seventeen years after Galileo's death and two years after the publication of Christiaan Huygens's Horologium , in which Huygens described his pendulum clock. It is from Huygens's construction that we date the practical development of the device.

[1]Strictly speaking, a simple pendulum is not isochronous, the period does vary somewhat with the amplitude of the swing. This was shown by Christiaan Huygens, in the 1650s. Huygens installed cycloidal "cheeks" near the suspension point of his pendulums and showed that as a result the bob now described a cycloidal arc. And he proved that when this is the case the pendulum is truly isochronous. In practice, the swing of the bob was kept very small and the amplitude as constant as possible, as in the long-case clock or our familiar grandfather clock. Under these conditions the simple pendulum is isochronous for all practical purposes

PREDICTION

I predict that the shorter I make the string the faster each single oscillation will be because there is less distance for it to move also if the mass is increased there will be more gravitational pull so creating a down force moving the pendulum faster. If the pull back angle was increased the oscillations would stay the same to a certain extent because you may increase the pull back angle which would mean it moving faster. But there will also be a longer distance for the bob to travel so they would equal each other out.

PLAN

Start by getting a tall clamp stand a clamp a you then clamp a cork into it. The cork must have a small split half way into it.
Put the string into the cork, we use a cork because there is minimal friction produced therefore the string should oscillate freely.
You then attach a protractor to the cork so that you can measure the pullback angle equally every time (as shown in the diagram).
You then extend the string to a certain length and attach a 100g mass to the string.
I am using 10cm, 20cm, 25cm, 30cm, 35cm, and 40cm lengths (measured by a metre ruler). and keeping the mass at 100g for each different length.
I pulled the string back to angle of 45degrees and released it and counted 10 oscillations (from one point and back to the same point). After the 10 you stop the stop clock and take down the results. This is repeated 3 times on each length.
Next you change the mass instead of the length so I kept the length constant at 20 cm and the pullback angle at 40degrees for each experiment.
I used 10g, 30g, 50g, 70g, 90g, and 100g and repeated everyone 3 times and recorded my results.

FAIR TEST

RESULTS FOR LENGTH VARIBLE (Time for 10 oscillations).

RESULTS FOR MASS VARIBLE (Time for 10 oscillations).

The next two pages contain my graphs.

GRAPH 1:

This is a graph plotting length of sting against time taken for one oscillation to occur and it is directly proportional.

GRAPH 2:

This is plotting weight against time taken for one oscillation to occur and there is no immediate pattern.

Looking at the results we can see some patterns first of all we can see that in the first experiment were I varied the length of the string which suspended the bob, that as the string got longer it took a greater amount of time to Oscillate. This has also produced a directly proportional graph, which means that if you, double the length of a pendulum you can half its time taken for one oscillation. We can also see from the results in the first experiment that every time I experimented with a particular length the results of the 3 tests were very close in results within 10ths of a second. Looking at the experiment I did on weight and its affect on the pendulum the results somewhat differed from my expectations. We can see that they are extremely close in result within a few 10ths of a second so the graph does not portray the relationship as well as hoped. Even though the range was from 10grams to 100grams the results were still close in fact the result I got from measuring 100grams took more time to oscillate than that of 10grams which was the opposite to what I was expecting. The graph for this is just a straight line so therefor we could say, if my results are accurate, that the weight suspended from the pendulum does not affect its oscillation rate. As expected in my prediction as you lengthen the material suspending the bob (string in this case) it took a longer time for the pendulum to oscillate once. On the tables as you can see I have taken the averages of the 3 attempts I had at each experiment and then divided them by 10 to get the time taken for one oscillation of the pendulum. I have drawn lines of best fit on both of my graphs to show the median of the three attempts.

In conclusion we can say that the longer the pendulum the longer it takes to swing and the weight has no affect on the swinging time. This is probably because the longer the string the longer the distance is for it to travel to get back to the same position therefore one oscillation. I shouldn’t think the minor alterations I was making to the weight were large enough to make a difference.