The Pendulum. Experiment #1 NOTE:

Transcription

1 The Pendulum Experiment #1 NOTE: For submitting the report on this laboratory session you will need a report booklet of the type that can be purchased at the McGill Bookstore. The material of the course that is relevant to this laboratory exercise is that of lectures 1, 2 and 3. References to the Material in Hecht can be obtained from the Index of that text under "Pendulum" and "Simple Harmonic Motion". Introduction This exercise is a study of the relationship of the period of oscillation of a pendulum (i.e. the time for one swing) to its length, mass and extent of the swing. You will be doing these experiments long before you will have studied the theory of the pendulum. (Unless, of course, you have studied it in a previous physics course, in which case some of the value of the exercise will be lost.) In this way you will be following the normal sequence of a scientific study; i.e the observation of an interesting, and perhaps useful, relationship followed by a study of possible reasons for this observed relationship. In this respect you will be in the position of Galileo when he first contemplated the motion of a pendulum, according to legend when he was bored by the church sermon and observed the swinging of the church chandeliers caused by the drafts in the church. Whatever, he concluded from his observations that a swinging pendulum could be an accurate clock, a necessary instrument for studying motion. To do this he had to identify the factors that influenced the period of a pendulum so that he could build an accurate one. If you believe that you should understand the theory of something before you study it then you should be aware of the fact that even Galileo never got to understand why a pendulum worked the way it did. For this experiment you are provided with four masses; 10, 20, 50 and 100 grams, a support stand and a string with which to suspend them from the support arm. You are also provided with an angle indicator and a stop-watch to time the swings of the pendulum. The period is to be determined by averaging over many swings. This is because the period will be at most about two seconds while the error in timing with the stop-watch will be about 0.1 second, or about 5%. Averaging over many swings will reduce this percentage error in inverse proportion to the number of swings.

2 P101 Laboratory Exercise Version 2 Procedure Exercise 1 - The Relationship of the Period to the Mass Insert the string in the clamp on the support arm lamp so that there is about 55 cm of string from the loop to the support clamp. Insert the hook of the 100-gram mass into the loop at the end of the string and, by momentarily loosening the clamp, adjust the string length so that the center of the mass is 50 cm from the edge of the support clamp. This distance is the "pendulum length". CLAMP 50 CM LOOP CENTER OF MASS Grasp the mass and pull it to one side so that the string is inclined about 10 degrees from the vertical. (Use the angle indicator for reference.) Release the mass and watch a few swings. Make sure that the mass is not spinning and that it swings without turning significantly from its original direction. Practice a few swings to get this right. For timing the oscillation we suggest that, with the mass at rest, you place yourself so that for your viewing eye, the string and the support stand are aligned. SUPPORT STAND STRING

3 Experiment #1 - The Pendulum 3 You then pull the mass side-ways and release it so that it swings perpendicular to your line of view. CLAMP This will give you a full view of the oscillation. The best way to then determine the period of the oscillation is to note the time at which the mass passes the post, headed in the same direction. Reset your stop-watch to zero and get ready to start it. Pull the mass to one side so that the string is again about 10 degrees to the vertical and release it so that it swings in a plane perpendicular to your line of view. It is recommended that you allow the mass to complete one swing before you start your stop-watch at the next time it passes the support post. Count this pass as "zero". Then count the subsequent passes of the post, each with the mass going in the same direction, until you get to twenty. It is recommended that you count out loud. At the 20th swing past the post abruptly stop the watch and note the time. The pendulum period is this time divided by the number of swings (i.e. 20). Repeat this exercise with the 50, the 20 and the 10-gram masses. Remember for each mass to adjust the length of the string so that the distance from the support point to the center of the mass is 50 cm. Enter the results in your lab report booklet.

4 P101 Laboratory Exercise Version 4 Exercise 2 - The Relationship of the Period to the Amplitude of the Swing Replace the 100-gram on the string, again adjusting the string length to 50 cm. Now repeat the exercise but this time with a 20 degree swing but for only 10 swings instead of 20. For this it is recommended that you hold the angle indicator against the support rod so that with a horizontal line of view its tip is in line with the support point. SUPPORT POINT CLAMP Repeat the exercise for swings of 30, 40, 50, 60 and 70-degree swings. For these tests it will be important to estimate the angle of the last swing. You will find, particularly for the larger swings, that this will be noticeably less than the angle at which the mass was released. Take the amplitude of the swing as the mean of the initial and final angles. Record your results in your lab report booklet. Exercise 3 - The Relationship of the Period to the Length of the Pendulum Keeping the 100-gram mass on the string, adjust the string length so that the distance from the support point to the center of mass is about 100 cm. Now determine the period with a 10-degree initial swing and for 20 swings. Repeat for string pendulum lengths of 75 cm, 25 cm and 15 cm. Together with the 50-cm result, which you have already determined, enter your results in tabular form in your report booklet. Analysis of Results From the data that you have tabulated draw four graphs as follows. (You may use the prepared graph sheets appended to this write-up or you can enter your data in a spread-sheet program and have it plot the graphs for you. If you like self-flagellation you can adapt the graph sheets of your workbook and plot them there.) 1. Period versus mass. 2. Period versus amplitude of angle. 3. The period versus the length of the pendulum. 4. The square of the period versus the length of the pendulum.

5 Experiment #1 - The Pendulum 5 The Relationship of the Period to the Length of the Pendulum First look at the period versus the length of the pendulum. It is clearly not a linear function. Some other functional relationship must therefore be tested. The relationship suggested here is that tested by the graph of the square of the period versus the length of the pendulum. What is this relationship called? (Enter your response in your report.) This graph of the square of the period versus the length of the pendulum should be close to a straight line. Determine the slope of this line and enter it in your booklet. Express this functional relationship by the equation T 2 = al where T is the period, l is the length of the pendulum and a is obtained from your graph. You will, of course, fill in the value you have obtained for a. Answer the following questions in your report booklet: 1. What are the units for a?. 2. How are the units of a related to the units for g? 3. What does this suggest concerning the nature of the relationship between a and g? (In mathematical terms, what is the form of the mathematical relationship between a and g? 4. (Tricky) Given that π is , can you find an equation between a and g that gives close to the actual value of a that you got in your experiment? (If you find this too difficult to come up with yourself, you might want to consult your textbook for some clues.) The Relationship of the Period to the Mass Study carefully your graph of period versus mass. It should be almost a horizontal straight line. What does this mean as a functional relationship? (Write your answer in your booklet.) This result was very important to Galileo in the development of his theories of dynamics. Since the motion of a released pendulum is essentially a falling of the pendulum toward the motion center the independence of the period of a pendulum on the mass means that all masses fall at the same rate. This was in direct violation of the theory of Aristotle, which had been believed by all philosophers up to the time of Galileo, that heavier objects fell faster than light ones. However, depending on the care with which your experiment was carried out there should be a noticeable dip in the period for the 10-gram mass, perhaps even for the 20-gram mass. This would at first seem to indicate that very small masses fall faster than heavy ones. Considering the results of the first analysis, i.e. the relationship of the period to the pendulum length, can you come up with a more reasonable explanation? (Write your answer in your booklet.) This can be a difficult puzzle for anyone beginning in physics and so a clue might be appropriate: The string supporting the mass also has, itself, a mass, albeit small.

6 P101 Laboratory Exercise Version 6 The Relationship of the Period to the Amplitude of the Swing The aspect of the swinging of the church chandelier that intrigued Galileo was that the period of the swing did not seem to depend on the amplitude of the swing. (It was this aspect of the pendulum swing that suggested it might make a good clock.) However, a look at your graph of the period versus the amplitude of the swing shows that this is only true for very small swings, there being an increase in the period for larger swings. Galileo himself had trouble understanding this but seemed to accept it as something to take into account when designing a pendulum clock. (If you stick with the course you should understand it later.) Draw a smooth curve through your data for period versus amplitude, making sure that your curve starts off horizontal at zero amplitude. From this graph, what will be the amplitude of the swing at which the period deviates from the very small amplitude value by 1%. (Write your answer in your booklet.) Conclusion From the above results what should be taken into account when designing a pendulum to be an accurate clock? (Write your answer in your booklet.) Submit your graphs with your report.