UNIVERSITI TUNKU ABDUL RAHMAN

CENTRE FOR FOUNDATION STUDIES

PHYSICS LABORATORY

PRACTICAL REPORT GUIDELINE

FOUNDATION IN SCIENCE (P, S)

1

Errors and Uncertainties

All measured quantities have uncertainties associated with them. The purpose of error analysis is to

determine how such uncertainties influence the interpretation of the experimental results

1. Systematic Error - Results from consistent bias in observation (ie. Instrument-calibration error, natural errors or

personal error).

- Can be eliminated by pre-calibrating against a known, trusted standard. - Affects accuracy

2. Random Errors - Results from fluctuations in the readings of a measurement apparatus, experimenter's

interpretation of the instrumental reading or randomly changing conditions (weather,

humidity, etc.).

- Can be reduced by averaging multiple measurements. - Unbiased - Affects precision

Uncertainties in Measuring Devices

General rule of thumb used to determine the uncertainty in a single measurement when using a scale

or digital measuring device.

1. Uncertainty in a Scale Measuring Device is equal to the smallest increment divided by 2.

2. Uncertainty in a Digital Measuring Device is equal to the smallest increment.

In general, any measurement can be stated in the following preferred form:

The measured value is just an estimate and thus it cannot be more precise than the uncertainty of the

device. (i.e. The number of decimal places for the measured value must match the number of decimal

places for the uncertainty, and in multiples of the uncertainty)

Example:

The smallest increment in a meter rule (scale measuring device) is 0.1 cm. Following the general rule

of thumb to determine the uncertainty in a scale measuring device, it would therefore be half of the

smallest increment ( l = 0.1/2 = 0.05 cm).

The uncertainty of a meter rule is 0.05 cm, thus for the length measured,

l = 31.225 0.05 cm (incorrect)

l = 31.23 0.05 cm (incorrect)

l = 31.25 0.05 cm (correct)

2

Calculations

Significant Figures

1. Measurements

Number of digits recorded is a direct indicator of the nature of measuring device and process, and hence, how precise the measurement is.

Left-most non-zero digit in any number = most significant digit

Right-most digit (zero or non-zero) = least significant digit

Zeroes are insignificant if used to hold the place of decimals {0.004785 (4 s.f.)}

Trailing zeroes are significant if followed after a decimal point because it indicates precision of measuring device {42.0 (3 s.f.)}

To avoid ambiguity in significant figures, use scientific notation

2. Computation

Addition and subtraction Round to same number of decimal places as element with the least decimal places.

Multiplication and division Round to the same number of significant figures as the factor with the fewest

significant figures

Note: results should not be quoted with a precision higher than the absolute uncertainty associated with it.

Error of a Derived Quantity

If the measured values are used to calculate or to derive another quantity, the value of the derived

quantity will bear an error (and not an uncertainty).

The error of the derived quantity can be calculated using the following rules:

Error Propagation Rules

Relation Error

1. (Use only if A is a single term, i.e. Z = 3x)

2.

3.

4.

5.

a, b, c, ..., z represent constants.

A, B, C, ..., Z represent measured or calculated quantities

, , , , Z represent the errors in A, B, C, ..., Z respectively.

3

Constructing Tables

1. Follow instructions in the practical manual to construct the table, taking into consideration all values that need to be tabulated and any multiple readings if necessary.

2. State the physical quantity (e.g. Mass), the symbol (e.g. m), followed by the uncertainty (if applicable) and units (e.g. 0.01 g) at the headings of the table.

3. Uncertainties should be stated for all measured readings. 4. When recording data, a minimum of 6 readings are required for the plotting of straight line graphs

and 8 readings for curved graphs.

Guidelines for Plotting a Linear Graph

1. Provide a title of the graph and label the axes with correct symbols and units. 2. Choose and specify an appropriate scale on the graph paper and plot the data points. Ensure all

data points are contained within the largest values in the x and y axes. Use as much of the graph

paper as possible. Your graph should be more than half the size of your graph paper.

3. Identify outlier(s) (if any) and label them clearly to differentiate it from other data points. Remember to exclude the data point(s) of the outlier(s) from calculations of the centroid; and the

error box or linear least square fit computation.

4. Calculate the centroid. Add the centroid to the graph and circle the centroid to differentiate it from other points.

5. Draw a best fit linear graph passing through the centroid that best represents the average behavior of data. (Balance the points on the graph.)

6. Calculate the gradient and error of the gradient of the linear graph using either the error box method or linear least square fit method.

Estimating the minimum and maximum slope

Method 1: Using Error Box Method

1. Plot best fit line. 2. Draw a line parallel to the best fit line using the furthest point away from the top of the best fit

line and another line using the point furthest from the bottom of the best fit line.

3. Draw the box to enclose all the points. 4. The minimum and maximum slopes can be estimated using the opposite corners of the box as

guidelines.

5. Draw a triangle to identify points used for calculating the gradient. Make the triangle as large as possible using two widely separated points on the line. Avoid using data points to calculate the

gradient. The slope that you are determining is of the line drawn (the data plotted no longer plays

a role).

6. Calculate gradient of best fit line, minimum and maximum slopes (s, smin and smax). 7. Determine uncertainty of the slope.

8. State the values of the best-fit line, minimum and maximum y-intercept (if applicable). 9. Determine uncertainty of the y-intercept (if applicable).

4

Drawing the error box

0

20

40

60

80

100

120

140

160

180

0.0 2.0 4.0 6.0 8.0 10.0 12.0

y-a

xis

x-axis

Graph of Y against X

X

Maximum

slope, smax

Best fit line

slope, sbest

Minimum

slope, smin

Box to enclose

all points

Furthest point away

from the bottom of

the best fit line

Furthest point away

from the top of

the best fit line

Centroid

Maximum

y-intercept,

ymax

Best-fit line

y-intercept,

ybest

Minimum

y-intercept,

ymin

5

Method 2: Linear Least Square Fit Template

Linear Least Squares Fit

x y xy x^2 y^2 mx mx + c y - (mx + c) [y - (mx + c)]^2 1

Fill

in x

-axi

s va

lues

her

e

Fill

in y

-axi

s va

lues

her

e

2 3 4 5

The programme will perform the calculation here (obtain calculated values below) 6 7 8 9 10 11 12 0.000 0.000 0.0000 0.0000 0.0000 0.0000

n = number of data

Std deviation of y : = average uncertainty of y-axis values

m = gradient

Uncertainty in m : m= uncertainty of gradient

c = y-intercept

Uncertainty in c : c= uncertainty of y-intercept

Linear correlation coefficient (Worst =0, Best = 1) : r = How well x-axis values relate to y-axis values

6

Discussion

1. Settings of experiment a. Highlight the usage of certain apparatus and significant methods implemented for

measurements (if any).

Example: Plumb line as reference position for measuring distance, x (Mechanics, P3) b. Do not repeat the whole setup and procedure. c. Explain how certain parameters were fixed or varied for the setup of the experiment.

(Example: Controlled/Independent variables, Dependent variables)

2. Observation and results a. Observations worth highlighting while running the experiment and your

responses/evaluations with rationales/justifications.

b. What physical quantities were calculated? c. State the results and comments on the results. (e.g. Do your observations match your

predictions (theory)? Is it high or low? Why is it so?)

3. Precautions, limitations and modifications a. State the precautions taken and explain the reason they were taken. b. State the limitations (if any) of the experiment and how they affected the outcome of the

experiment

c. Propose modifications (if any) to the experiment based on the previous mentioned limitations

4. Applications (if any) of the experiment

5. Additional info about the experiment

7

Sample Report Results, Data Analysis and Discussion

1. The values of m, W, L, and L were tabulated.

Note that all readings are recorded according to the precision of the measuring device. (i.e. The number of decimal places in the measured readings matches the number of decimal

places of the uncertainty of the measuring device.)

For calculated values, follow rules of significant figures for computations. All calculations and labeling have to be consistently presented.

Data:

Frequency of function generator, f = 30.00 0.01 Hz.

Mass of string, M = 0.48 0.01 g

Total length of string, l = 1.5480 0.0005 m

Mass of slotted

masses, m (g)

Tension on string,

W = mg (N)

Length, L

( 0.0005 m)

L2 (m

2)

1 10 0.10 0.3350 0.1122

2 20 0.20 0.3920 0.1537

3 30 0.29 0.4710 0.2218

4 40 0.39 0.5450 0.2970

5 50 0.49 0.6200 0.3844

6 60 0.59 0.6840 0.4679

7 70 0.69 0.7300 0.5329

8 80 0.78 0.7900 0.6241

Centroid = 44.0,3493.0,2

n

w

n

L

Drawing Graphs

Graph:

Title: Graph of tension, w against L2.

Scale: x-axis 2 cm: 0.050 m2

y-axis 2 cm: 0.10 N

Labels: x-axis: L2 (m

2)

y-axis: W (N)

More than one

reading can be

taken and

averaged out

Since W = mg,

it has same

number of

significant

figures as m.

L2 has same

number of sig.

fig. as L

The number of significant figures or

decimal places should be the same as the

readings used to calculate it.

If using Linear Least Square Fits method: Draw the best-fit line by using the gradient from the

LLSF calculation and passing the line through the

centroid or draw the best fit line to pass through the

centroid and the y-intercept obtained from LLSF if

your axis begins at (0,0).

Ensure that your graph contains the following:

Title, scale, labels with units, centroid, best-fit line, and error box

unless using LLSF method. Circle and identify outliers if any.

Every measured reading must

contain a value, uncertainty

and unit.

8

Alternative 1: Using Error-Box method of calculating error:

Data points for gradients

1. Best fit (0.125, 0.15), (0.610, 0.78) (Black line)

2. Max slope (0.150, 0.15), (0.610, 0.80) (Red line)

3. Min slope (0.110, 0.15), (0.610, 0.74) (Blue line)

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

-0.050 0.000 0.050 0.100 0.150 0.200 0.250 0.300 0.350 0.400 0.450 0.500 0.550 0.600 0.650 0.700

Ten

sio

n, W

(N

)

L2, (m2)

Graph of Tension, W against L2 Scale: x-axis: 2 cm to 0.050 m

2

y-axis: 2 cm to 0.10 N

X

9

Data Analysis

1. Calculating gradients

2N/m 3.1485.0

63.0

0.125 - 0.610

0.15 - 0.78s

2

max N/m 4.1460.0

64.0

0.150 - 0.610

0.15 - 0.79 s

2

min N/m 2.1500.0

59.0

0.110 - 0.610

0.15 - 0.74 s

2

minmax

N/m 1.0

2

2.14.1

2

||

sss

2N/m 0.1 3.1 s

2. Calculate experimental value for linear density,

kg/m105.3

)00.30(4

3.1

44

4

2exp

2

2

f

sfs

kg/m10 3

)105.3()08.0(

)105.3()0007.008.0(

)105.3(00.30

01.02

3.1

1.0

2

5

4

4

4

exp

f

f

s

s

kg/m103.03.5 4exp

Use the same format and exponent.

Dont write like this: 3.510

4 310

5 kg/m

Present your calculations

in this form xbest x

Intermediate values in calculations

(ie. values that are not the final answer)

can take an additional significant figure.

Example:

2

max N/m 93.1460.0

64.0

0.150 - 0.610

0.15 - 0.79 s

10

3. Calculate reference value for linear density,

kg/m101.3

5480.1

48.0

4

l

mref

kg/m100.06

kg/m10 6

)101.3()02.0(

)101.3()0003.002.0(

)101.3(5480.1

0005.0

48.0

01.0

4

6

4

4

4

l

l

m

mref

kg/m101.03.1 4 ref

4. Percentage error

%13

%1001.3

1.35.3

%100%exp

ref

referror

Results

The linear density of the string is kg/m103.03.5 4exp with a percentage error of 13%.

In general, it should have no more than

1 2 significant figures.

The error should have the same

number of decimal places as the

obtained value.

If a standard value is available

(i.e. g = 9.8 m/s2), the percentage error

can be calculated as follows:

%100%standard

standardexp

x

xxerror

11

Alternative 2: Using the Linear Least Square Fit Method for calculating error:

Linear Least Squares Fit

x y xy x^2 y^2 mx mx + c y - (mx + c) [y - (mx + c)]^2

1 0.1122 0.10 0.0112 0.0126 0.0100 0.1453 0.1343 -0.0343 0.0012

2 0.1537 0.20 0.0307 0.0236 0.0400 0.1990 0.1880 0.0120 0.0001

3 0.2218 0.29 0.0643 0.0492 0.0841 0.2872 0.2762 0.0138 0.0002

4 0.2970 0.39 0.1158 0.0882 0.1521 0.3846 0.3736 0.0164 0.0003

5 0.3844 0.49 0.1884 0.1478 0.2401 0.4978 0.4868 0.0032 0.0000

6 0.4679 0.59 0.2761 0.2189 0.3481 0.6059 0.5949 -0.0049 0.0000

7 0.5329 0.69 0.3677 0.2840 0.4761 0.6901 0.6791 0.0109 0.0001

8 0.6241 0.78 0.4868 0.3895 0.6084 0.8082 0.7972 -0.0172 0.0003

9

10

11

12

2.7940 3.53 1.5410 1.2138 1.9589 0.0022

n = 8

Std deviation of y : = 0.0193

m = 1.2949

Uncertainty in m : m = 0.0395

c = -0.0110

Uncertainty in c : c = 0.0154

Linear correlation coefficient (Worst =0, Best = 1) : r = 0.9972

12

Draw the best-fit line by using the gradient from the LLSF calculation and passing the line through the centroid

or draw the best fit line to pass through the centroid and the y-intercept obtained from LLSF.

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

-0.050 0.000 0.050 0.100 0.150 0.200 0.250 0.300 0.350 0.400 0.450 0.500 0.550 0.600 0.650 0.700

Ten

sio

n, W

(N

)

L2, (m2)

Graph of Tension, W against L2

X

Scale:

x-axis: 2 cm to 0.050 m2

y-axis: 2 cm to 0.10 N

13

Data Analysis

From the linear least square fit computation, the following values can be obtained to perform

the calculations.

1. Gradient

s = 1.295 N/m2

s = 0.0395 N/m2

2N/m 0.040 295.1 s

2. Calculate experimental value of linear density

kg/m10597.3

4(30.00)

1.295

4

4

2

2exp

f

s

kg/m10 1.2

)10597.3()032.0(

)10597.3()0007.0031.0(

)10597.3(00.30

01.02

295.1

040.0

2

5

4

4

4

exp

f

f

s

s

kg/m1012.03.60 4exp

Use 4 significant figures at most as the measured values have

a maximum of 4 significant figures and include units.

The uncertainty should have the same number

of decimal places as the obtained value.

The error should have the same

number of decimal places as the

obtained value.

14

3. Calculate reference value for linear density,

kg/m101.3

5480.1

48.0

4

l

mref

kg/m100.06

kg/m10 6

)101.3()02.0(

)101.3()0003.002.0(

)101.3(5480.1

0005.0

48.0

01.0

4

6

4

4

4

l

l

m

mref

kg/m101.03.1 4 ref

4. Percentage error

%16

%1001.3

1.360.3

%100%exp

ref

referror

Results

The linear density of the string is kg/m1012.03.60 4exp with a percentage error of 16%.

In general, it should have no more than

1 2 significant figures.

The error should have the same

number of decimal places as the

obtained value.

15

Discussion: (sample)

1. Settings of the experiment

As the apparatus was being set up, it was ensured that the string was aligned with the groove of the

pulley to ensure that the tension in the string was only exerted by the slotted mass. While moving the

wooden wedge to obtain the standing waves, it was found that the rough surface of the wooden wedge

had caused additional friction, resulting in a change in the tension of the string. To overcome this

problem, the string was lifted and released back onto the wooden wedge (using fingers) for each

reading to ensure the readings obtained were accurate. The standing waves formed were left

undisturbed for 1 to 2 minutes to ensure that they were stable before taking readings. The

measurements were done carefully. Extra care was taken to avoid contact with the standing waves

while taking measurements.

2. Results

The graph shows that the tension in the string is linearly proportional to the square of the length

between the two nodes of a standing wave, as illustrated in the equation W = 4f 2L

2. The gradient

obtained from the graph (or Linear Least Square Fit method) is 1.295 0.040 N/m2. From the

gradient, the linear density of the string is calculated, bearing the value of (3.60 0.12) 104

kg/m.

On the other hand, the linear density of the string determined from direct measurements is

(3.1 0.1) 104

kg/m. Thus, the error of the experimental value is 16% as compared to the value

obtained from direct measurement.

3. Sources of Error

The main source of error is in identifying and measuring the length between the two nodes. A major

contributor to error is the accuracy of judgment as to when the standing wave actually occurs.

Secondly, because of the dynamic nature of the experiment, the string being vibrating all the time, it is

not possible to accurately measure the length between the two nodes to the accuracy inherent in the

ruler used as the measuring instrument, hence the expected uncertainty in measurement is likely to be

more than 1 mm. The frequency or function generator has a small inherent uncertainty of 0.01 Hz and

although the frequency often varies slightly during the experiment 0.05 Hz, this variation is unlikely

to contribute much to the overall error of the experiment.

The third and possible source of uncertainty is the mass of the slotted masses, which we assume to be

accurate to its stated value although it was not measured using a weighing machine.

4. Precautions taken

Care is taken to determine when the standing wave occurs as this is identified to be a major

variability. For this, instead of merely taking the distance from the vibration generator to the wedge as

the length between two nodes, the distance between two actual nodes were identified instead, where

possible, moving the wedge away from the vibration generator to obtain at least 3 static nodes

whenever possible.

In measuring the length between two nodes, several readings were taken for each measurement to

minimize the errors in the readings.

16

5. Modification to the experiment

The following modifications to the experiments are suggested to improve the results further:

First, a vibration generator that could generate clearer and stronger waves is recommended to produce

clear nodes and antinodes.

Second, the weighted masses can be weighed using a weighing scale to ensure we have a more

accurate value of the tension. The measuring ruler can also be held up using a set of retort stands so

that readings can be taken in a stable condition.

Another suggestion is to fix the length between the two nodes and vary the frequency as the tension

varies, as the variation in the frequency is more accurately determined than the measurement of the

length between the nodes.

6. Applications

A large number of musical instruments work on the principle of resonance of standing waves in

strings that are fixed between two fixed ends. Examples of these are the guitar, violin, ukulele, piano,

and many other instruments. Resonance of standing waves also play an important role in the design of

cables used and the safety mechanisms incorporated into, for instance, suspension bridges, as such

resonance can result in standing waves with vibration large enough to cause the collapse of the bridge,

as had happened during one stormy gale in 1940 when the Narrows Bridge in USA collapsed as it

galloped under resonance due to the strong winds, and became known as the Galloping Gertie.

Conclusion:

The experimental data and graph showed that the theoretical formula for standing waves on a string

during resonance,

T

Lf

2

1 was valid with small variation in the readings obtained and the linear

density of the string was found to be kg/m103.03.5 4exp with a percentage error of 13%.

Bibliography and references: if any.