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Exp Simple Pendulum experiment-simple pendulum-writeup

simple pendulum experiment

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Physics I (8. 01)

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Massachusetts Institute of Technology

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MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics Physics 8.

Experiment Simple Pendulum

Section ______________________ Group Members _______________________ _______________________ _______________________

Learning Objectives: After completing this experiment, you should be able to 1. Understand the difference between precision and accuracy when making a measurement. 2. Quantify the precision of measurements (uncertainty) using error analysis. 3. Design a measurement scheme to minimize the uncertainty in measuring the period of a pendulum. 4. Understand small angle approximation. 5. Compare two sets of measurements of the period at different starting angles, to see if you can measure deviations from the small angle approximation for the period

Experiment Equipment Kit: String; Pencil; Washer; Nut A simple pendulum consists of string tied around a pencil, and a nut or washer attached to the other end.

Report: The format for your excel spreadsheet containing your data and statistical analysis, and along with answers t the report is contained at the end of the experiment document. Include a copy ofo the question contained in the report and submit with Problem Set 12 due Friday Dec 3 at 9 pm.

Introduction: Precision and Accuracy When measuring a physical quantity like the period of a simple pendulum, we need to distinguish between making a precise measurement and an accurate measurement. As an example suppose you are trying to hit the center of a target, a metaphor for our measured quantity. Precision is usually defined by the statistical uncertainty of the measurement, which means the spread in random directions with respect to the average. The accuracy on the other hand is related to systematic effects that have to be covered by the systematic uncertainties. It is important to remember that when you try to make a measurement. The target in the figures below we show four examples of a collection of data points (trials).

figure 1a figure 1b figure 1c figure 1d

Our goal is to make a measurement that has high accuracy and high precision (figure 1a). We may make a very precise (reproducible) measurement but it might not be accurate due to systematic effects in the measurement process (figure 1b). It is quite difficult to determine whether a precise measurement is really accurate because the experimenter does not know the outcome, so there is no target, because otherwise you would not need to measure. There might thus be systematic discrepancies between the real value and the mean of the measurements.

This is why it is essential for an experimental physicist to carefully document the measurement such that another experiment team can reproduce the result using the same procedure and possibly discover unrecognized systematic effects. Finally, we could make an accurate measurement but with low precision represented by data points that are widely spread out over the target but more or less centered (figure 1c). Finally we could make a measurement with a wide scattering of data points (low precision) that is also has low accuracy (figure 1d).

Section 2: Defining the statistical uncertainty of the measurement Now it’s time to test your predictions against reality! You will perform the single period measurements using the different schemes listed in section 1 and compare their precision. To do so, we need first to define our measure for the statistical uncertainty.

If a measurement (M) is repeated N times, giving values M 1 , M 2 , ... , MN, we define the mean

value M as the average over all measurements

M = ( M 1 + M 2 N+ ××× + MN )= N 1 i åi==N 1 Mi

The mean value is what we usually report as the final outcome of the measurement. However, to get a sense of how precise the measurements are, we need to have a measure of the spread of

the outcomes around that mean. The standard deviation S( M )serves this purpose and it is

defined as

S( M ) = N 1 - 1 i åi== N 1 ( Mi- M ) 2

Both the mean and excel spreadsheet built standard deviation of-in functions as will be detailed later. the repeated measurements can be calculated using

While S( M )is a measure of how the different measurements are spread around their mean, it

doesn’t give information about how the mean value itself is distributed as a random variable. If two different persons A & B perform ten measurements of the period each and calculate their

two means M Aand M B, these won’t necessarily match. So, in order to quantify the precision

of the measurements, we want to determine the spread (standard deviation S( M )) between

mean values obtained by different people. Fortunately, statistics tells us that this standard

deviation of the mean, S( M ), (called the standard error of the mean) can be estimated from

the standard deviation of the measurement, S( M ), using the formula

S( M ) = 1 N S( M )

The standard error is important because when statistical uncertainty of the mean value decreases. the number of measurements This will be our measure for the statistical increases, the the uncertainty of the measurement. We can finally report the final uncertainty of the outcome of

the measurement as S( M ).

Measurements: Now we’re ready to start! First measure the length of the pendulum.

Length of Pendulun: ____________________ Draw a sketch showing between which points you made your measurement. made your measurement.. Describe how you

Sketch:

Description:

Question 2 : How accurately can you measure the length of the pendulum? Answer:

Use excel functions AVERAGE() and STDEV() to calculate the mean and standard deviation

S(T )for each column. For example, if one column contains data in cells

A1:A10, you can obtain the mean and standard deviation in two new cells as =AVERAGE(A1:A10) and =STDEV(A1:A10). Make sure you label your columns properly.

3. Calculate the standard deviation of the mean, S(T ), of each scheme by dividing the

standard deviation S(T )by the square root of 10.

Question 4: accurate results Briefly describe. any improvements you made to your apparatus to produce more

Answer:

Question 5 standard deviation of the mean.: For your period measurement, report your mean value, standard deviation, and

RSB: T= _; S(T )= _; S(T )= ______.

SSSS: T= _; S(T )= _; S(T )= ______.

BB: T= _; S(T )= _; S(T )= ______.

Question 6 : Ranking Precision of the Methods Rank the three schemes according to their compare with your earlier predictions. Make sure to report your findings briefly and explain uncertainties (which method is more precise) and possible reasons favoring one method over the others in a few sentences. Reasons:

Section 3: Improving the precision of the period measurement

From this point on, you will only use the most section 2. However, we want you to think of ways to improve it even more. While this scheme precise measuring scheme that you obtained in is better than the other ones, we still get a statistical factor is the human accuracy/reaction time when starting/stopping the watch. If the period of uncertainty due to many factors. One main the pendulum is 1 second and you have a human error of 0 seconds to locate and respond to the pendulum, then this uncertainty represents 10% of the measurement time. One way to decrease the effect of human error is to make the measurement last longer such that the error is “diluted”.

Question 8 : Improving your Measuring Technique (IMT) Discuss with your group possible ways to measure the period of the pendulum while increasing the time between starting and stopping the stopwatch. Briefly describe your new procedure for measuring the period. Answer:

Once you settle on a modification of the scheme from section 2, perform a number of repeated measurements using it. Record your data in your excel spreadsheet and calculate the standard

deviation, S( Mm ), (don’t forget to divide by N). Compare the uncertainty with the one you

got from section 2. Question 9 value, standard deviation, and standard: For your period measurement deviation of the mean. for a 10 degree initial angle, report your mean

IMT: T= _; S(T )= _; S(T )= ______.

Optional you have time Section 4). Testing a hypothesis: Comparing data from two different initial angles (if

In this section, we will investigate the hypothesis that the period of the pendulum does not depend on the initial angle/amplitude. Remember that this is just a hypothesis that could be right or wrong. In order to be able to tell whether two measurements are different, we need to minimize the statistical uncertainty for both of them as large uncertainties can make two distinct outcomes overlap. For example, if the period at one angle is 1±0 s and 1±0 at another, then there is no significant difference between the two measurements since they heavily overlap. On the other hand, if the measurement were made using a more accurate scheme that results in periods of 1±0 s and 1±0, then we can be much more confident that the periods obtained are statistically different from one another. That’s why it is crucial that we tried our best in the previous sections to minimize the uncertainty in measuring the period. One way to quantify the statistical difference between two measurements is by the ratio of the difference between their mean values over their combined standard deviations. We can define a

t - test as follows. Let

t = (S(TT 1 - T 2 1 )) 2 + (S(T 2 )) 2

Larger values of the statistical uncertainty in the two measurements. Hence larger values of tcorrespond to a larger difference between the two mean values relative to

confidence that the two measurements are different. If the number of measurements ( tsignify more N) for

each is large enough (t < 1 , the two measurements are statistically indistinguishable. N ³ 5 ), we require t > 2 for 90-95% statistical confidence. While if

Now, calculate the horizontal distance you need to displace the pendulum to start at an angle of 30 degrees with the vertical. Release the pendulum from that angle and measure its period using the scheme you developed in section 3.

Optional report your mean value, standard deviation, and standard deviation of the mean. Question 10: Comparing Results: For your period measurements for 30 degrees,

Period for 30 degrees initial angle:

T = _; S(T )= _; S(T )= ______.

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Exp Simple Pendulum experiment-simple pendulum-writeup

Course: Physics I (8. 01)

181 Documents

Students shared 181 documents in this course

University: Massachusetts Institute of Technology

Was this document helpful?

MASSACHUSETTS INSTITUTE OF TECHNOLOGY

Department of Physics

Physics 8.01

Experiment Simple Pendulum

Section ______________________

Group Members _______________________

_______________________

Learning Objectives:

After completing this experiment, you should be able to

1. Understand the difference between precision and accuracy when making a

measurement.

2. Quantify the precision of measurements (uncertainty) using error analysis.

3. Design a measurement scheme to minimize the uncertainty in measuring the period of a

pendulum.

4. Understand small angle approximation.

5. Compare two sets of measurements of the period at different starting angles, to see if

you can measure deviations from the small angle approximation for the period

Experiment Equipment Kit:

String; Pencil; Washer; Nut

A simple pendulum consists of string tied around a pencil, and a nut or washer attached to the

other end.

Report:

The format for the report is contained at the end of the experiment document. Include a copy of

your excel spreadsheet containing your data and statistical analysis, and along with answers to

the question contained in the report and submit with Problem Set 12 due Friday Dec 3 at 9 pm.

Exp Simple Pendulum experiment-simple pendulum-writeup

Physics I (8. 01)