There was this question involving a pendulum expriment: the five values of the square of the period (T^2) was plotted against the 5 values of the distance y in centimetre from the floor to the centre of the bob. I was then asked to determine the intercept on the vertical axis and slope of the graph as well; I have determined the intercept and slope of the graph. I was then asked to use the value of the intercept to determine the height H of the point of attachment of the pendulum to the floor. How do I go about that? I have gotten the intercept and the slope of the graph as well.
The period depends only on the length (l) of the string and the acceleration (g). A physics text should have the equation: \( P\;=\; 2 \pi\sqrt {\frac {l} {g}} \), where P is the period. Note this only holds for small angles (i.e. displacements). So, the acceleration under gravity doesn't depend on the nature of the mass or how much it weighs, since all bodies are accelerated equally.
You might try to research Galileo's efforts in this regard. He is reported to be the first to do this experiment.
Physics is about the confrontation of reliable, precise, and communicable frameworks of predicting domains of phenomena (physical models) with reality. Here the two models you have are the period of a pendulum, \(T \approx 2 \pi \sqrt{ \frac{\ell}{g} }\), and the relation between the height of the attachment point, the length of the pendulum and the height of the pendulum above ground, \(H = \ell + y\). In an advanced physics class you might have additional information, like the angle of the measured arc, and in an advanced physics lab, you would also want to work with your error budget and model how good your estimate of H is. As it is, you have some formulas (formulae), but need to recognize that these physics formulas are models and that it is entirely legal for you to use algebra to rewrite them to suit your needs. Here \(T\) is the mesaured period of a full back-and-forth pendulum cycle, presumably measured in seconds \(\ell\) is the distance from attachment point to pendulum bob center-of-mass, measured in cm \(g\) is the local acceleration due to gravity, measured in cm / s^2, about 980.5 cm/s^2 \(y\) is the measured height of the pendulum bob center-of-mass above the ground, measured in cm \(H\) is the desired height of the attachment point, measured in cm, \(b\) is the intercept of the plot of T^2 versus y on the T^2 axis where y=0 and has units of seconds^2 \(m\) is the slope of the plot of T^2 versus y and has units of s^2/cm \(\tilde{b}\) is the intercept of the plot of y versus T^2 on the y axis where T^2=0 and has units of cm \(\tilde{m}\) is the slope of the plot of y versus T^2 and has units of cm/s^2 Using algebra, we establish: \(\ell = H - y\) and \(T^2 = 4 \pi ^2 \frac{\ell}{g} = \frac{ 4 \pi^2}{g} H - \frac{ 4 \pi^2}{g} y\). So if you have established \(T^2 \approx m y + b\) then \(m = - \frac{ 4 \pi^2}{g}\) and \(b = \frac{ 4 \pi^2}{g} H\) so \(H = - \frac{b}{m}\) without the need to determine what the local value of the acceleration due to gravity is. Or you can rearrange \(y = H - \frac{g}{4 \pi^2} T^2\) so that if you have \(y \approx \tilde{m} T^2 + \tilde{b}\) then \(\tilde{m} = - \frac{g}{4 \pi^2}\) and \(\tilde{b} = H\) so \(H = \tilde{b}\) The error for a swing of 1° is about 0.002%, for a 10° swing is about 0.2%, for a 30° swing is about 2%, for a 45° swing is about 4%. Galileo measured the height of attachments in the ceiling by timing pendulum swings with the bob set at different heights above the ground? I doubt that. For one, Galileo had no precision timing instruments. What Galileo did was to establish the rough proportionality between \(T^2\) and \(\ell\) which was used in this exercise. A group replicated Galileo's experiments and found that his experimental procedure was good and concluded he probably reported actual experiments actually performed by him. http://galileo.rice.edu/lib/student_work/experiment95/galileo_pendulum.html And of course the Wikipedia page on Pendulums addresses Galileo. http://en.wikipedia.org/wiki/Pendulum
I think I got the gist now: H = l+y -> l = H-y the value of the intercept that I got from the actual graph that I plotted is 4.55 (sec)[sup]2[/sup] the period of oscillation of a simple pendulum is given as T = 2(pi)/w = 2(pi)(l/g)[sup]1/2[/sup] since the intercept is at T[sup]2[/sup] squaring both sides T[sup]2[/sup] = (2(pi)(l/g)[sup]1/2[/sup])[sup]2[/sup] we have (91/20)[sup]2[/sup] = 4(pi)[sup]2[/sup](l/g) = 4(pi)[sup]2[/sup](H-y/g) assuming the acceleration due to gravity g is 10m/s and that the value of pi = 22/7 (91/20)[sup]2[/sup] = 4(22/7)[sup]2[/sup](H-y/10) what do I subtitute in now as the value of the y?
A must see video. http://ocw.mit.edu/courses/physics/...-1999/video-lectures/for-the-love-of-physics/
Why do you think you were asked to determine the intercept, \(b\), and slope, \(m\)? Also, what values did you get and what were the units of these values? So since \(T^2 = \frac{ 4 \pi^2}{g} H - \frac{ 4 \pi^2}{g} y\) from physics formulas and algebra. And since \(T^2 = m y + b \) from your measurements and calculations. It follows that \( m y + b = \frac{ 4 \pi^2}{g} H - \frac{ 4 \pi^2}{g} y\) Or \(H = \frac{\left( m + \frac{ 4 \pi^2}{g} \right) y + b}{\frac{ 4 \pi^2}{g}} \) Since \(H, \, m , \, b \, \textrm{and} \, g\) are constants for this problem it follows that \(m + \frac{ 4 \pi^2}{g} = 0\) thus \( -m = \frac{ 4 \pi^2}{g} \) so we substitute in this value when we see \( \frac{ 4 \pi^2}{g} \) . So \(H = \frac{\left( m + (-m) \right) y + b}{(-m)} =\frac{\left( 0 \right) y + b}{-m} = - \frac{b}{m} \)
You should read up on Galileo. His timing instrument initially was his own heart-beat. This is easy to replicate, and I routinely have my middle level and high school students replicate the experiment (as do many others).http://suite101.com/article/galileos-pendulum-a15031 After he had established how to use a pendulum as a precise time-measuring instrument, he used pendulums as time-measuring instruments in his subsequent research. And, of course, his work was readily adopted into the clock-making industry of his age, ultimately leading to chronometers that allowed ships to measure longitude, thus allowing for accurate lengthy sea voyages.http://en.wikipedia.org/wiki/John_Harrison
My guess is rpenner is aware of this. You know 'the veritable wealth of information'. That's rpenner with a high level of precision.
Possibly. Though he claimed that Galileo had no precision timing instruments, when in fact he invented the pendulum as a timing device, and it can give a very high degree of precision. While it apparently was not incorporated into clock-making during Galileo's lifetime (he died 1642, first pendulum clock 1656), it was not necessary to have a clock to measure seconds, when one could simply count the number of swings and use that as a timing mechanism, including fractions of a swing. Galileo apparently conceptualized making clocks with a pendulum, but this was not satisfactorily accomplished during his lifetime. http://en.wikipedia.org/wiki/Pendulum_clock But my posts are not only for the person to whom I quote; rather, they are for the general reader primarily, and it is a fascinating story to learn of the efforts that were made to create a chronometer that could go to sea and maintain precision over the course of months, with the ship rocking, etc,. so as to measure longitude. Books have been written on it, and our younger readership could learn a lot from those stories, as they are what gave rise to the mapping of the globe and allowed the British Empire to become the dominant maritime power in the 1700s to 1800s and obtain global leadership. Incidentally, the mathematics of the pendulum was well developed ages ago, and rpenner repeats it for emphasis, which is good.
I wasn't criticizing your post I was paying a small tribute to rpenner for all his contribution to intellectual discourse. Sorry if it seemed like I was.
(Emphasis added.) This experiment is about using the known linear relationship between the square of the period (in the limit of small pendulum movements) as measured with precision timing devices with the length of the gravity-driven pendulum to estimate the height of the point of attachment from the linear relationship between the square of the measured period and the height of pendulum bob above the floor. It's deeply unclear (due to the passive voice in the description of the graph) that any actual timing was done by the student in this case, so this appears to be a pedagogical exercise (rather than true experimental lab assignment) in physical models, reading graphs and application of algebra. (Standard coursework regularly requires students to time pendulums with their own pulse? Citation required.) Rather than estimating height or length of pendulums from the period, Galileo described the period in terms of the length of the pendulum and the relative insenstivity to the angle of the swing. "Uno è, che le vibrazioni di un tal pendolo si fanno con tal necessità sotto tali determinati tempi, che è del tutto impossibile il fargliele far sotto altri tempi, salvo che con allungargli o abbreviargli la corda; ... L'altro particolare, veramente maraviglioso, è che il medesimo pendolo fa le sue vibrazioni con l'istessa frequenza, o pochissimo e quasi insensibilmente differente, sien elleno fatte per archi grandissimi o per piccolissimi dell'istessa circonferenza." (page 254 of Dialogo sopra i due massimi sistemi del mondo tolemaico e copernicano , fourth day)Later, in Discorsi e dimostrazioni matematiche intorno à due nuove scienze, Galileo would describe the use of a water clock to achieve some level of precision relative timing of the motion of balls on inclined planes. Water clocks, pendulums and heart beats have various levels of replicability, but until they are calibrated to a time standard the experimentalist can only speak of relative time, not a specific duration. Also, heart beat varies from person to person, with age, level of activity and emotional state, so while the heartbeat timing method might be good enough to establish a rough proportionality between length and number of heartbeats squared, that's a very personal relationship. It has little bearing on this experiment. John Harrison's most successful chronometer was an oversized pocket watch which did not rely on pendulums and so didn't have the problems Harrison's pendulum clocks had in rough seas. Notably, the need to stay upright and the equivalence principle leading to times where the local acceleration was quite different from the London standard. As Harrison was likely far more interested in keeping the period and therefore the length of the pendulum of his pendulum clocks constant, he's not particularly relevant to this experiment which is about variation of length and period. Precision compared to what? Pendulum periods depend on details of their construction, and ambient conditions. Water clocks also depend on construction and ambient conditions. Heartbeats vary. Sundials don't always mark off the same time noon to noon. So if a pendulum has a certain amount of discrepancy with another clock available to Galileo, was the discrepancy due to the clock or the pendulum? How would Galileo know? Well, one simple thing Galileo could do is start pendulums of 1,2,3,and 4 times a standard length. A careful observer would not take long to conclude that the longest pendulum made 1/2 swing in the time the shortest took to make one full swing. It would be considerably harder to establish the ratios for the middle two. For the four pendulums the ratio of number of swings would be something close to 1, 12/17, 11/19, 1/2. Experimental difficulties with the length including possibly part of the weight makes it harder to identify the pattern. -- My larger assertion was that I found your claim that Galileo did this experiment of estimating height of attachment point from pendulum swings of heights above the floor to be both nakedly asserted and dubious due to technology and extremely peculiar that one would find a true set of pendulums (and not chains or ropes) mounted and one wished to use timing (rather than geometry) to estimate height. It seemed like you were inserting yourself into a thread without taking any sense of the topic, this experiment. Fractions of a swing? Citation required. Half of a full swing is relatively easy. Quarter swings might be estimated with some certainty with marks on a board. But arbitrary fractions seem to be intrinsically subjective in the observer's imagination. Much better would be in estimating ratios of full pendulum swings, which is useful for timing periodic phenomena but less so for arbitrary intervals. Which makes your earlier post about the "clock making industry of his age" so hard to understand. In my opinion, your posts in this thread should be primarily for those interested in the topic of this thread. Also the equation of time and the analemma and Kepler's Laws have been featured in numerous books. But for this experiment, the student is already living in an age where the metrology of time is well-developed. If you wanted to start a thread where you wanted to tell the story of pendulum clocks for the general reader, perhaps you should start a new thread. But it looks like you misunderstood the nature of problem the student was having, recommended reading about an antique scientist who worked with raw numbers and geometry, didn't write textbooks, and whose original works are only to be found in larger libraries. You didn't even cite a specific text to read about. You performed contextomy on my post where I gave a balanced discussion of what I think Galileo did and did not do. From there you went on any number of digressions, never returning to this experiment.
What's so hard to understand about that? Mechanical clocks had been made for about 2 centuries when Galileo came along. While Galileo saw that his pendulum could/should be incorporated into clocks, and an effort was made by his son in that regard, the clock-making industry didn'c actually incorporate pendulums until about 14 years after Galileo's death. But it was Galileo who provided the impetus. Further, one's pulse can be relatively uniform. It is sufficiently uniform to obtain the result that Galileo obtained. Typically, when instructors present a pendulum experiment, they do two-in-one, which is what I was referring to as 'the pendulum experiment' even if the student raising this thread only referred to the varying the length vs. period portion, rather than varying the swing vs. period. I also use the experiment to teach how to manipulate data to obtain relationships, using best-fit lines on graphs, etc. If you ever have the chance to work with youngsters, you'll probably find as I do, that it is not intuitive to them that varying the swing distance doesn't appreciably affect the period. And they are amazed when you plot T-squared vs L and get a straight line.
