1.Pendulum

first name (print) last name (print) Brock ID (ab13cd) TA initials grade 

Experiment 1 

The pendulum 

A simple pendulum consists of a compact mass m suspended from a fixed point by a string of length L, as 

shown in Fig. 1.1. 

Figure 1.1: Pendulum experimental arrangement 

The equilibrium position of m is O. Here, the tension ⃗P exerted by the string on m is exactly equal and 

opposite to the weight m⃗g of the mass. This is not true however if m is anywhere else along the arc, say 

at an angle α (in radians) with respect to O. Then the weight will have a component tangent to the arc, 

whose magnitude is m⃗gsinα. If α is small, we may approximate sinα ≈ α, so that the force on m is equal 

to mgα. This force is a restoring force, as it will drive m back to its equilibrium position O. When a mass 

is acted on by a restoring force, whose magnitude mgα is proportional to the deviation α from O, then 

the resulting motion is an oscillation around the equilibrium position. In our case, m will swing back and 

forth around O. The time for one complete swing is defined as the period T. An equation for acceleration 

due to gravity g is given by: 

g = 4π2 L 

T 2 (1.1) 

This equation predicts that the period T is independent of the mass m of the ball, and that T is also 

independent of the angle α through which the ball swings, as long as the approximation sinα ≈ α is valid. 

For α = 15 ◦ ≈ 0.2618 radians, there is a difference of approximately 1.2% between α and sinα. 

11

12 EXPERIMENT 1. THE PENDULUM 

Introduction to error analysis 

The result of a measurement of a physical quantity must contain not only a numerical value expressed in 

the appropriate units; it must also indicate the reliability of the result. Every measurement is somewhat 

uncertain. Error analysis is a procedure which estimates quantitatively the uncertainty in a result. This 

quantitative estimate is called the error of the result. Please note that error in this sense is not the same as 

mistake. Also, it is not the difference between a value measured by you and the value given in a textbook. 

Error is a measure of the quality of the data that your experiment was able to produce. In this lab, error 

will be considered a number, in the same units as the result, which tells us the precision, or reliability, 

of that experimental result. Note that error value, represented by the Greek letter σ (sigma), is always 

rounded to one significant digit; the result is always rounded to the same decimal place as σ (see below). 

Error of a single measurement 

Consider the measurement of the length L of a bar 

using a metre stick, as shown in Figure 1. One can 

see that L is slightly greater than 2.1 cm, but because 

the smallest unit on the metre stick is 1 mm, 

it is not possible to state the exact value. We can, 

however, safely say that L lies between 2.1 cm and 

2.2 cm. The proper way to express this information 

is: 

L±σ(L) = 2.15±0.05 cm 

This expression states that L must be between, 

(2.15−0.05) = 2.10cmand, (2.15+0.05) = 2.20cm, 

Figure 1.2: Measurement with a metre stick which is our observation. The quantity σ(L) = 

±0.05 cm is referred to as the maximum error. This 

number gives the maximum range over which the correct value for a measurement might vary from that 

recorded, and represents the precision of the measuring instrument. 

Propagation of errors 

In many experiments the desired quantity, call it Z, is not measured directly, but is computed from one 

or more directly-measured quantities A,B,C,... with a mathematical formula. In this experiment, the 

directly-measured quantities are T, y and b, and the desired quantity g is calculated from g = 4π 2 L/T 2 , 

with L = y + 1 2b. The following rules give a quick (but not exact) estimate of σ(Z) if σ(A), σ(B) etc. 

are known Always use the absolute value of an error in a calculation . Error rules are tabulated in the 

Appendix. 

1. If Z = cA, where c is a constant, then σ(Z) = |c|σ(A). This is used only if A is a single term. For 

example, it can be used for Z = 3y, so that σ(Z) = 3σ(y), but not for Z = 3xy. 

2. If Z = A+B +C +···, then σ(Z) = σ(A)+σ(B)+σ(C)+···. For example, if 

L = y + 1 2 b 

then σ(L) = σ(y)+σ 

( 1 

2 b ) 

(See 2. above.) 

σ(L) = σ(y)+ 1 σ(b) (See 1. above.) 

2

13 

3. To derive an error equation for any relation, rewrite that relation as a series of multiplications, then 

apply the change of variables method as shown in the Appendix to evaluate the error terms: 

g = 4π2 L 

T 2 −→ g = 4π 2 LT −2 −→ g = ABCD, ( letting A = 4, B = B = π 2 , C = L, D = T −2 ) 

Then 

and σ(A) = σ(4), 

σ(g) 

|g| 

σ(B) 

|B| 

= σ(A) + σ(B) 

|A| |B| 

( σ(π) 

= |2| 

|π| 

+ σ(C) 

|C| 

) 

, σ(C) = σ(L), 

+ σ(D) , (Appendix, Rule 4) 

|D| 

σ(D) 

|D| 

= |−2| 

( ) σ(T) 

. (Rules 1,6) 

|T| 

Thequantities 4 and π are constants and have no error (strictly speaking, σ(4) = σ(π) = 0), therefore 

these terms do not contribute to the overall error. The error equation for g then simplifies to 

σ(g) 

= σ(4) ( ) σ(π) 

+2 + σ(L) ( ) σ(T) 

+|−2| −→ 

σ(g) = σ(L) ( ) σ(T) 

+2 . 

|g| |4| |π| |L| |T| |g| |L| |T| 

The right hand side of the above equation, called the “relative error” of g, results in a fraction that 

describes how large σ(g) is with respect to g. The desired quantity, σ(g), is obtained by multiplying 

both sides of the equation by g: 

[ ( )] σ(L) σ(T) 

σ(g) = |g| +2 . 

|L| |T| 

Standard deviation of a series of measurements 

When the same measurement or calculation is repeated several times, an error estimate can be calculated 

form the variation in this set of values. This is convenient because on other error values, measured or 

calculated, need to be known. For example, the five results for g can be used to estimate an error σ(g). 

To calculate the standard deviation σ(g) for a sample of N values of g: 

1. determine the average 〈g〉 of N values g i , where i = 1,2,···,N by summing the values and then 

dividing by the number of values: 

〈g〉 = 1 N (g 1 +g 2 +···+g N ); 

2. for each g i , calculate the deviation ∆g i = g i −〈g〉 from the average 〈g〉; 

3. for each g i , calculate the square of the deviation (∆g i ) 2 to make all the values positive; 

4. calculate the variance σ 2 (g) of the sample by summing the squares of the deviations (∆g i ) 2 and 

dividing by the number of values minus one, N −1 

σ 2 (g) = 1 

N −1 (∆g2 1 +∆g 2 2 +···+∆g 2 N); 

5. undo the previous squaring operation by taking the square root of the variance. This is the root 

average squared deviation, or standard deviation σ(g), of b: 

√ 

σ(g) = σ 2 (g) 

A large σ value indicates a result which is not very precise. The theory of statistics shows that the 

probability that a further measurement of g falls within the range of σ(g) is 68% and that σ is proportional 

to 1/ √ N, so σ will decrease as the number of samples N is increased.


Rounding a final result: X ±σ(X) 

The value of σ(x) is rounded to one significant digit whether it represents a maximum error estimate, 

calculated error, or standard deviation of a sample. The result correspondingto this error must be rounded 

off and expressed to the same decimal place as the error. For example, 〈x〉 = 25.344 mm and σ(x) = 

0.0427 mm. Rounded to one digit, σ(x) = 0.04 mm. Rounded to the same decimal place, 〈x〉 = 25.34 mm. 

The final result is expresses as 〈x〉±σ(x) = (25.34 ± 0.04) mm. 

Do not use a rounded off value in further calculations. Use the original unrounded value. Use of a 

truncated value will decrease the quality of your result. 

Powers of 10 

Express both the result and its error to the same power of 10. This allows the reader to immediately judge 

how large the error is relative to the result: 

1. 2.68 × 10 −2 ± 5 × 10 −4 should be written as 0.0268 ± 0.0005 or (2.68 ± 0.05) × 10 −2 . Note the 

parentheses, indicating that both the result and the error are to be multiplied by 10 −2 . 

2. 1.634±3×10 −3 m should be written as 1.634±0.003 m 

Format of calculations 

Record all calculations, in the appropriate space if provided or on a separate sheet of paper. A calculation 

is performed in three lines. The first line displays the formula used. In the second line, the variables in the 

formula are replaced with the actual values used in the calculation. The third line shows the final answer 

formatted according to the section on Rounding above and if any, the units associated with the result. 

Review questions 

Using a textbook, the Physics Handbook, the Internet or some other source, find the accepted value of g. 

Cite your reference so that a reader can find this information. Be specific; do not cite ’my physics book’, 

’my buddy’ or ’i remember it from high school’ as your source. Include specific info such as the uncertainty, 

or error, in the value, and the condition (i.e. sea level) under which it was measured. 

g = .............................................................................. 

An error value is only meaningful when expressed with ..... significant digits. 

The measurement error σ of a scale graduated in minutes and seconds is .................... 

My Lab dates: Exp.1:...... Exp.3:...... Exp.4:...... Exp.5:...... Exp.6...... 

I have read and understand the contents of the Lab Outline (sign) ................ 

CONGRATULATIONS! YOU ARE NOW READY TO PROCEED WITH THE EXPERIMENT! 

Procedure and analysis 

You will determine theperiodT of a pendulumof length L by monitoringthe distance of thependulumbob 

from a computer controlled rangefinder. This device determines the distance to an object by transmitting 

an untrasonic ping and measuring the time required to receive the echo of the original signal after it is 

reflected from the object.

15 

The transmitted signal propagates in a narrow conical beam and as it is the echo from the nearest 

object that is accepted, you will need to be sure that you are pinging the pendulum bob and not the 

pendulum arm or some other structure in the cone of the beam. As the pendulum bob swings toward and 

away from the rangefinder, the variation in distance over time can be approximated by a sine wave. By 

analysing the properties of this sine wave, we can calculate the period T of the pendulum. 

The pendulum is an aluminum ball suspended by a light nylon string from a sliding arm. Assume the 

mass of the string is negligibly small compared to the mass of the ball. To calibrate the pendulum: 

1. Release the string to set the ball on the table. Align the bottom of the sliding arm with the zero 

mark on the scale. Adjust the string length so that the top of the ball lightly contacts the bottom of 

the arm, ensuring that the string is not stretched. Secure the string with the clamping nut. 

2. Raise the arm away from the ball and carefully reposition the arm until it once again just contacts 

the top of the ball. If the bottom of the arm is no longer at the zero mark, repeat the sequence of 

adjustments until the bottom of the arm is in line with the zero mark of the scale and the arm lightly 

contacts the top of the pendulum ball. All members of the group must check the calibration. 

The scale is now calibrated to display the length of the string y from the bottom of the sliding arm, 

the pivot point of the pendulum, to the top of the ball, to a precision of one millimetre (mm). This 

number is not exactly equal to L since the ball is not a point-mass. The length of the pendulum L is the 

sum of the length of the string y and half the ball’s diameter b given in Table 1.1: 

L = y + 1 b. (1.2) 

2 

Mount the larger ball m 1 and calibrate the pendulum. Adjust the sliding arm so that the string length 

y is approximately 0.3 m. Record the actual length to a precision of 1 mm. 

• Set the pendulum swinging in a straight line, keeping α less than approximately 15 ◦ . Wait several 

seconds to allow for any stray oscillations present in the bob to dissipate. 

• Shift focus to the Physicalab software. Check the Dig1 box and choose to collect 50 points at 0.1 

s/point. Click Get data to acquire a data set. 

• Click Draw to generate a graph of your data. Your points should display a nice smooth sine 

wave, without peaks, stray points, or flat spots. If any of these are noted, adjust the position of the 

rangefinder and acquire a new data set. 

• Select fit to: y= and enter A*sin(B*x+C)+D in the fitting equation box. Click Draw . If 

you get an error message the initial guesses for the fitting parameters may be too distant from the 

required values for the fitting program to properly converge. Look at your graph and enter some 

reasonable initial values for the amplitude A of the wave and the average distance D of the wave 

from the detector. The value of B (in radians/s) is given by B = 2π/T since the period of a sine 

wave is 2π = B∗x radians, where x = T is the period in seconds. Estimate the time T between two 

adjacent minima of the sine wave. Estimate and enter an initial guess for B. (See Appendix A) 

• Label the axes and identify the graph with your name and the string length y used. Click Print 

to print your graph. Every student should print a copy of each graph. 

• Record in Table 1.1 the trial length y and the computer calculated value of the fitting parameter B 

(radians/second). Do a quick calculation for g to make sure that the value is reasonable. 

• Repeat the above steps for m 1 with y = 0.45, 0.60, 0.75 and 0.90 m. 

• Mount the second ball m 2 , recalibrate the pendulum and verify the calibration, set the string 

length to approximately y = 0.5 m, and record this one measurement in Table 1.1.


Run, i m (kg) b (m) y (m) L (m) B (rad/s) T (s) T 2 (s 2 ) g i (m/s 2 ) 

1,m 1 0.0225 0.02540 

2,m 1 0.0225 0.02540 

3,m 1 0.0225 0.02540 

4,m 1 0.0225 0.02540 

5,m 1 0.0225 0.02540 

1,m 2 0.0095 0.01904 

Table 1.1: Table of experimental results 

Part I: Determining g from a single measurement 

• Use Equation 1.2 and the single measurement of m 2 to calculate L and g. Show the calculation 

below, in three steps as previously outlined and recalling the proper application of significant figures 

and physical units. 

L = ............................... g = ............................... 

............................... ............................... 

............................... ............................... 

• The measurement errors in y, b, and T, represented by σ(y), σ(b), and σ(T), respectively, are 

determined from the scales of the measuring instruments. The micrometer used to measure the ball 

has a resolution, or scale increment of 0.00001 m. The data logger can measure time with a precision 

of 0.00002 s. Summarize these measurement errors below: 

σ(y) = ±.................. σ(b) = ±.................. σ(T) = ±.................. 

• Calculate the error σ(L) in the length L and the error σ(g) in g. Express the final values as L±σ(L) 

and as g±σ(g), with the correct units. 

σ(L) = ............................... σ(g) = ............................... 

............................... ............................... 

............................... ............................... 

L = ..............±............... g = ..............±...............

17 

Part II: Determining g from a series of measurements 

• Use Equation 1.1 to calculate the five measurements of g for m 1 . Record the results in Table 1.1 and 

Table 1.2. Include all the calculations for m 1 as part of your Lab Report. 

• Use Table 1.2 to calculate the average and standard deviation of the sample of g values, then summarize 

the result in the proper format below: 

i g i g i −〈g〉 (g i −〈g〉) 2 

1 

2 

3 

4 

5 

〈g〉 = variance = 

σ(g) = 

Table 1.2: Calculation template for the standard deviation of g. 

g = .............±............. 

Part III: Determining g from the slope of a graph 

Equation 1.1 can be written as: 

L = 

( g 

4π 2 ) 

T 2 (1.3) 

This is the equation of a straight line Y = mX + b, where in this case Y = L, X = T 2 , b = 0, and the 

slope m = rise/run = g/(4π 2 ). 

• Using the Physica Online software, enter the five coordinates (T 2 ,L) in the data window. Select 

scatter plot. Click Draw to generate a graph of your data. Select fit to: y= and enter A*x+B 

in the fitting equation box. Click Draw . The computer will generate a line of best fit through the 

data points. Print the graph, then record below the slope A and error σ(A) 

• Calculate below values for g and σ(g). 

A = .............±............. 

g = ....................... = ....................... = ....................... 

σ(g) = ....................... = ....................... = ....................... 

g = .............±.............


Summary of results 

Nowthatyouhavequantitatively estimatedthereliability ofyourvariousresultsforg byapplicationoferror 

analysis, you can make a meaningful comparison of these results with the accepted value of g = 9.80 m/s 2 . 

• In the space below, plot and label your three results for g and the error bars representing the 

uncertainty in each of these results. Begin by drawing on the given line a scale that is representative 

of the range in your results. The range of agreement in two results is determined by the region of 

overlap of their error bars. For clarity, plot your data and error bars above one another. 

• Which of the error bars overlap the accepted value of g? ............................... 

• Is there a region where all the error bars overlap? ..................................... 

• Do any of your error bars overlap the accepted value of g? .............................. 

IMPORTANT: BEFORE LEAVING THE LAB, HAVE A T.A. INITIAL YOUR WORKBOOK! 

Discussion 

Your Discussion should summarize the essence of this experiment in a clear and organized way, so that 

someone reading (or marking) this report will obtain a good understanding of what was done, how it was 

done, and of the results that were obtained. Here are some ideas on the issues you should address in your 

discussion: 

Begin by tabulating your three experimental values and the accepted value of g. Identify the source of 

the data and include appropriate units. Give the table a descriptive title. 

Compare your three results for g. Do your values agree with the accepted value of g. Explain your 

reasoning. 

Consider which method is likely to provide the most reliable value for g. Which method is the least 

reliable? Explain your conclusions. 

From your graph, determine whether T varies with L. Does T vary with the mass of the ball? Explain. 

Do your conclusions agree with the predictions of Equation 1.1? 

Both the averaging method and the line of best fit use the same five experimental values for L and T. 

What similarities do you think there may be in these two approaches to determining g? 

If drawing a line through the plotted experimental points, how would you choose where to place the 

line? How might this choice affect your resulting value of g? 

What is the resolution, or scale, of the time measurement and how did you determine this? How might 

the resolution be improved? 

A Final Note: Have you printed and included the complete email that Turnitin sent you 

containing the full content of your Discussion? If not, you will lose 40% of your grade. Printouts of the 

receipt from the Turnitin webpage or your wordprocessor will not be considered for marking.

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