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Joule Physics Laboratory

International Foundation Year: Physics

Laboratory Handbook

Joule Physics Laboratory International Foundation Year (IFY)

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Introduction to the Laboratory

The purpose of the IFY laboratory is to:

allow students to become familiar with and competent in the use of basic measuring equipment and techniques,

allow students to measure physical constants and common material properties that often arise in their theory classes,

allow students to test the validity of physical laws,

train students in good practice in the conduct of experiments, the collection of data and data analysis,

introduce students to the estimation of experimental error,

impress upon students the need to write a report on each experiment that they complete and to train students in the writing of a short formal report.

Experiments are best performed if you know what you have to do and you understand the

underlying principles of the experiment. So, before starting any experiment, read the laboratory

script and make sure you understand what the experiment requires. Make sure that you are

aware of the safety hazards for each experiment before you start. Ensure that the equipment is

arranged sensibly and within easy reach so that you may perform the experiment comfortably

and in safety. If there is any doubt concerning any aspect of an experiment then ask your

laboratory supervisors for assistance.

To increase accuracy in experiments you should:

Measure the same quantity as many times as is practical given the time you have to carry

out the experiment,

Change the values of set variables as many times as is practical

When you have a choice, ensure measured values are as large as possible compared to the

error in those values.

CONDUCT OF EXPERIMENTS

Students are allowed entry to the laboratory only during timetabled laboratory classes and whilst

under the supervision of authorised physics staff.


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Students may perform only one experiment per laboratory session. Students must perform only

the particular experiment that is scheduled for a given laboratory session. Students are allowed

to commence a new experiment only once they have submitted a completed report for the

previously conducted experiment. If a student has not fully completed a laboratory report and

wishes to commence a new experiment, they may do so on the understanding that the

uncompleted work will not be considered for assessment.

If any student fails to observe all necessary health and safety protocols, or displays disruptive or

unruly behaviour, they will be banned from entering the laboratory and may have their student

status revoked.

LABORATORY REPORTS A short report must be written for each experiment, each report is marked out of ten. Each

experiment is performed by a pair of students, but each student of the pair must produce their

own, individual work. The University of Salford operates a very strict plagiarism policy; the

penalties for copied and plagiarised work are detailed in your Student Handbook.

You may not start a new experiment unless the report for the previous experiment has been

handed in.

In Semester 1 the experimental reports shall be handwritten, and the five highest scoring reports

will contribute to the final laboratory module mark. In Semester 2 the experimental reports must

be word processed and include a full error-analysis; the five highest scoring reports will

contribute to the final overall laboratory mark.

The combined Semester 1 and Semester 2 laboratory marks contribute 25 % to the final Physics

Module mark.


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FORMAT OF THE LABORATORY REPORT All laboratory reports, whether hand written or word processed, must follow the same general

format:

General Information

The title and number of the experiment, and the date the experiment was performed must be

included at the start of the report. The aim of the experiment should follow; be aware that this

may not always be the same as the title of the experiment. Each experiment has an associated

risk assessment form that must be consulted prior to any practical work. The safety information

associated with each experiment must be stated in the report, and the report itself should follow a

logical structure. For reports written in Semester 1 marks are awarded for including this general

information; in Semester 2 marks will be deducted for omitting any of this general information.

Theory and Method

This section describes the basic physical principles under investigation, and exactly how the

experiment was performed. The theory and method must be presented in sufficient detail such

that the reader may fully understand the purpose and procedure of the experiment. This section

should be written in the past tense and third person, and should be accurate.

Diagram

A diagram of the experimental apparatus should be included to aid description of the

experimental procedure. Diagrams must be clear, actual and fully labelled. In Semester 1

diagrams may be hand drawn, in Semester 2 diagrams must be drawn using a computer.

Diagrams electronically scanned from the laboratory handbook will not be accepted.

Results

All raw data generated throughout the experiment should be presented in a clear table. To

increase accuracy all measurements should be repeated where practical, and should yield

appropriate values taken over an appropriate range. All columns in the table should be

labelled, with units and measurement errors included.


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Graph

For experiments requiring a graphical method of analysis, all necessary data must be clearly and

accurately plotted on a suitable graph. The graph should include a relevant title, the axes must

be labelled, and appropriate units clearly indicated. Graphs must be presented as large is

practically possible, featuring maximised scaling and an appropriate line of best fit. In Semester

1 graphs must be drawn by hand, in Semester 2 graphs must be plotted using appropriate

computer software.

Calculations

The working equation for the experiment must be included. If necessary the working equation

must be successfully rearranged into a usable form, and any subsequent calculations

successfully completed. Calculations should be presented clearly and follow a logical structure.

In Semester 2 a full error analysis is required.

Conclusion

The conclusion provides a summary of the experiment. The result must be stated, with

appropriate units and then compared to the standard value if relevant. Reasons for any possible

deviation between the experimental value and the standard value should be given.

Copies of the marking schemes for both Semester 1 and Semester 2 laboratory sessions are given

overleaf.


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Semester One Marks Title Date and Expt. No. Safety Info Aims

General Info

Logical structure

10%

Sufficient detail Past tense Third person

Theory and Method

Accurate

10%

Clear Actual Diagram

Fully labeled

10%

Measurements repeated App. Values App. Range Colums labeled Units Measurement errors

Results

Clear

20%

Title Axes labeled Graph

Units (no graph, results and Maximised scaling calc times 1.5)

Appropriate best line fit

20%

Working equation given

Rearranged (if necessary)

Calculations

S.C.C.

20%

state result with units

compare to expected value Conclusion

Explain any possible reason for deviation from the expected value.

10%


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Semester Two Marks Title No marks awarded but

Date and Expt. No. a maximum of 10% General Info

Safety Info deducted for Aims missing information Logical structure

Introduction 4%

Sufficient detail Past tense Third person

Theory and Method

Accurate

10%

Clear Actual Diagram

Fully labeled

13%

Measurements repeated App. Values App. Range Colums labeled Units Measurement errors

Results

Clear

20%

Title Axes labeled Graph

Units (no graph, results and Maximised scaling calc times 1.5)

Appropriate best line fit

20%

Working equation given

Rearranged (if necessary) S.C.C.

Calculations

Error Analysis

23%

State result with units

Compare to expected value

Conclusion Explain any possible reason for deviation from the expected value.

10%


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Micrometers and Verniers

Vernier callipers and scales

The object to be measured is placed in the jaws of the callipers which are then closed so that the jaws touch the two surfaces the separation of which is required.

The main scale is marked in cms and gives the reading up to and including the first decimal place (in cms).

The short, moveable scale, of 10 principal divisions, is the vernier scale. This gives us the second decimal place (in cms) without having to estimate a fraction of a division.

The first scale is read at the point of '0' on the vernier scale. From the diagram this is 2.0. We then look for the vernier scale marking that is exactly opposite any main scale marking. This is 4 in the diagram. Therefore the reading is 2.04 cm.

With most verniers we can measure 'outside' dimensions, 'inside' dimensions and depths.


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Micrometer screw gauge

This is 10 × more accurate than the vernier callipers, but generally will only measure small outside dimensions.

The object to be measured is placed in the gap between the jaws and the jaws are closed using the ratchet until a clicking sound is heard. One full rotation of the circular scale moves the sleeve up the main scale by 0.5 mm (or 0.05cm). There are 50 divisions on the circular scale, therefore each division is

0.5 0.0150

= mm.

The first two figures are read from the main scale at the sleeve to the nearest 0.5 mm. In the diagram this is 5.5 mm. Then the circular scale is read. This is 22 i.e. 22 0.01 0.22× = mm. The reading is thus 5.5 + 0.22 = 5.72 mm.

Make sure that the jaws are clean before use. Close the jaws and check that the reading is 0.00 mm. Any other reading here is called the zero error and must be accounted for in your measurement.


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Digital Multimeters

Digital multimeters (DMMs) are now almost universally used instead of analogue devices for making electrical measurements. As the name suggests, the circuitry inside a DMM is configured such that a single meter can measure, for example, AC and DC current and voltage, resistance, temperature and signal frequency. This versatility is one reason why digital meters are often preferred over analogue meters. The second reason is that digital meters are direct read, that is that the meter automatically does the scale conversions and a measurement can be read directly from the meter, without the need for additional calculations. DMMs exist in two main types: those that require a mains supply, and those that operate off batteries. The use of both types will be discussed in this handout. Hand-held multimeters Figure 1(a) shows the layout of a typical hand-held DMM that you will encounter in the laboratory. The main features of the meter are: 1. The LCD display In general, battery-powered meters have liquid crystal displays as this display type has a sufficiently low power consumption to be used with a battery power source. One disadvantage of such a display is that it uses ambient light, and so in a dark room you will not be able to read it, although some meters have back-lighting to overcome this problem. Figure 1(b) shows an example of all of the symbols that can be shown by the LCD display, although, clearly, only a subset of these will be displayed at any one time when the meter is in use. The symbols on the left-hand side of the display are: AC This indicates when the meter is being used to measure AC current or voltage. If the

time period of the alternating signal is short compared to the response time of the meter then the reading will be the rms value of the signal.

- This indicates that the current or voltage being measured is negative, i.e. that the

signal going into the positive input of the meter is at a lower potential than the signal going into the common input of the meter. Note that this does not necessarily mean that the input signal is less than 0 V.

This indicates that the continuity option has been selected. With this option the

meter will sound a note when the potential difference between the positive and common input signals is small, i.e. when they are electrically connected through a small resistance.

hFE This indicates that the transistor testing option has been chosen. If this symbol shows, then the battery is low and needs changing. + −+ −


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Appearance of a hand-held DMM

The symbols on the LCD display of the meter shown above


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The symbols on the right-hand side of the display are: pF nF µF Indicates that a capacitance measurement is being made, along with the units of

measurement.

mV V Indicates that a voltage measurement is being made, along with the units of measurement.

µA mA A Indicates that a current measurement is being made, along with the units of measurement.

Ω kΩ MΩ Indicates that a resistance measurement is being made, along with the units of measurement.

°C Indicates that a temperature measurement is being made, along with the units of measurement.

kHz Indicates that a frequency measurement is being made, along with the units of measurement.

Note that not all meters have these symbols. In that case you have to determine the units of measurement (for example milli-, micro-) from the type and range chosen using the rotating dial. The symbols in the centre of the display show the numerical value of the measurement being made. 2. The rotating dial The rotating dial is used to select: (i) the type of measurement being made, for example voltage; (ii) the measurement range, for example 20 V. Measurement type Both current and voltage measurements can be made in AC or DC form. The AC measurement ranges are indicated by ‘∼’ and the DC measurement ranges are indicated by ‘− − − ’. Measurement range For the greatest precision a measurement should always be measured on the most sensitive scale possible. The appropriate range can easily be determined by reducing the range until only the ‘1’ digit on the display is shown. In order to make a measurement you will then need to change the range, using the rotating dial, back to the one that is next most sensitive. WHEN THE INPUT SIGNAL BECOMES OVER RANGE DURING MEASUREMENT, FOR EXAMPLE YOU ARE TRYING TO MEASURE 2.1 V ON THE 2 V RANGE, THEN AGAIN ONLY THE ‘1’ DIGIT WILL BE DISPLAYED AND YOU WILL NEED TO CHANGE THE RANGE.


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3. The banana plug sockets Signals are put into the DMM using two wires terminated in 4 mm banana plugs. In order to make measurements it is important to use the correct terminals of the meter. COM One of the two input wires should always be plugged into the COM or ‘common’

socket. The signal on the second wire is then compared to that on the COM wire. If you have one signal wire which is at 0 V, in other words is at electrical ground, then it is usual practice to make this the COM input, although it is not required that you do this. Since the meter will measure the input from the second wire relative to that on the COM input you do not blindly accept the sign of the eventual meter reading; think carefully about what you are doing.

20A On the meter show, if you are expecting to measure currents (ac or dc) bigger than

200 mA then you must plug the second wire into this socket. Putting large currents through another input could damage the meter.

µmA On the meter shown, if you are expecting to measure currents (ac or dc) of up to

200 mA then plug the second wire into this socket. VΩ If you are measuring voltage or resistance (any range) then plug the second wire

into this socket. BEFORE YOU PUT ANY SIGNALS INTO THE METER, DETERMINE THE MAXIMUM SIGNAL THAT IT CAN TAKE FOR THE TYPE OF MEASUREMENT YOU ARE MAKING. PUTTING A SIGNAL GREATER THAN THE MAXIMUM ALLOWED INTO THE METER WILL DAMAGE IT. For example, the meter shown can only take voltages of up to 600 V, and currents of up to 20 A without damage. Excessively large resistances will not damage the meter. However, as the size of the resistance being measured approaches the inherent input resistance of the meter voltage, divided effects will mean that the resistance value displayed by the meter becomes increasingly inaccurate. You should always measure resistance with a meter which has an input resistance much greater than that which you are measuring.


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Bench-top Battery/Mains powered multimeters The layout and operation of bench-top multimeters is very varied, and they may be dual powered (battery or mains) or mains powered only. Shown in Figure 2 is the front face of one of the meters you may encounter in the laboratory. This meter is on experiment Q7. Operation of this type of meter is still relatively straightforward, but does require a little more attention to detail than the hand-held meters you will encounter. As previously, the front face of the meter is composed of the display, the banana plug sockets for inputting the signal, and buttons to select the type and range of measurement. Figure 2 shows the front face of the meter used on experiment Q7. 4. The LCD display

Figure 2 Front face of the multimeter on experiment Q7

With this meter the LCD display only shows the numerical value of the parameter being measured. The units of measurement (for example milli-, micro-) are taken from the range chosen using the buttons. 5. The buttons Use of the buttons on this meter requires a little thought. Starting from the left-most button: ON/OFF This button (button 1) is depressed to turn the meter on. AC or DC Ω When this button (button 2) is depressed the meter will make AC measurements

(voltage or current). When it is released the meter will make DC (voltage or current) or resistance measurements.

V or A When this button (button 3) is depressed the meter will measure currents of up to

10 A. When this button is released the meter will measure AC voltages of up to 750 V or DC voltages up to 1200 V. SAFETY NOTE: When measuring voltages above 50 V fully insulated test leads should be used.


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Ω or A When this button (button 4) is depressed the meter will measure currents of up to 800 mA. When this button is released the meter will measure resistances up to 32 MΩ.

AC and DC measurements of current and voltage are selected by the appropriate combination of buttons 2, 3 and 4. Buttons 5 to 10 are used to select the range of a measurement. Remember that this also will give you the units of the number displayed. As with the hand-held meters you should use the most sensitive range that you can. If the input signal is over range then the display will show on the digit ‘1’: 6. The banana plug sockets Signals are put into the DMM using two wires terminated in banana plugs. As previously, in order to make measurements it is important to use the correct terminals of the meter. kHz If you wish to measure the frequency of a signal, one of your wires should be

plugged into this socket. Note that the result displayed is in kHz. 10A If you wish to measure currents that are between 800 mA and 10 A then one of

your wires should be plugged into this socket. Using the other current input socket will damage the meter.

500V max/ This is the common input of the meter and you should always plug your second

signal wire into this socket. The signal of the first wire is then compared to that on the common wire. If you have one signal wire which is at 0 V, in other words is at electrical ground, then it is usual practice to make this the common input, although it is not required that you do this. Since the meter will measure the input from the first wire relative to that on the COM input you do not blindly accept the sign of the eventual meter reading: think carefully about what you are doing.

It is important to note that the common input does not have to be at 0 V: you can

measure the size of a signal relative to any other signal. HOWEVER because this is potentially a mains-powered unit it will have a connection to 0 V through the earth plug. This means that, in practice for such meters, there is a maximum amount by which the common input can ‘float’. For this meter it is 500 V.

V Ω A The input to be used for voltage or resistance measurements, and low current

measurements. Note that the maximum allowed input signal sizes are quoted next to the socket.


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Straight line graphs

In many experiments you are asked to plot a graph. These are very useful for displaying information, illustrating and proving functional relationships and for calculating physical quantities. Often, the graph that you are asked to plot is a straight line graph. Before describing the uses of such graphs, we lay down the rules for plotting them. 1. Hand plotted graphs must be produced on real graph paper. 2. Graphs are most accurate when as large as possible. So, arrange the scales on the axes of your graph so that the plotted line covers as much of the page as possible, e.g. suppose that we are investigating the variation of a distance, y in metres with time, t, in seconds. The data to be plotted shows y increasing from near to 2 m to around 12 m as t goes from 0 to 5 s.

Graph of y versus t Graph of distance y versus time t

3. Label axes with well spaced and regular scale points as above. Do not mark data points on the axes. 4. Label both axes with the quantity plotted and the units, as on the 'DO' graph above. 5. Give the graph a title, as in the 'DO' graph above. The title should name the quantities plotted, not just give their symbols. 6. Mark data points boldly. Lightly marked data points can often be obscured by the drawn line.


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Do not illustrations

Features of straight line graphs

There are two basic equations of straight line graphs:

y = mx + c and y = mx

m and c are constants which can be +ve or −ve. m is the gradient (or slope) of the graph and c is the intercept (the value of y when x = 0). Both often represent physical quantities.

Gradient (or slope)

The gradient m is found from both of the above equations by differentiating y with respect to x:

dy mdx

=

For a straight line, this is a change in y divided by the corresponding change in x. To find this


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we:

1. Construct the largest possible right angled triangle as shown. To do so we use points on the drawn line and not data points as triangle corners.

2. Measure a and b with due regard to the scales on the axes.

Then:

Gradient ab

=

The larger the triangle used to find the gradient, the more accurate is the gradient found. When m is +ve, the line slopes upwards from left to right as in all the above illustrations. When m is −ve, the line slopes downwards from left to right:


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Intercept

The value of y when x = 0 is called the y-intercept of the straight line graph . It can be either +ve or −ve and is found by inspection of the graph. For y = mx + c, y = c: Note that, for the intercept to be revealed on the graph, the x-scale must start at x = 0. There is a less used quantity called the x-intercept. This is the value of x when y = 0. Putting y = 0 into y = mx + c gives

cxm

= −

i.e. the x-intercept is related to both the y-intercept (c) and the gradient (m). The simplest straight line graph, y = mx has zero intercepts i.e. it passes through the origin. The use of graphs 1. Graphs give us a visual indication of how one quantity varies as another is changed. This visual indication usually allows much more critical a judgement to be made from the data than is possible by looking at lists of figures. For example


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In both cases we could have seen, by looking at a list of the data from which these graphs were plotted, that y increases as x increases. But we would need to plot the graphs to see if the variation is linear or not 2. We can use graphs to test the proportionality (linearity) of physical laws. For instance, Ohm's law says that the voltage across a resistor is proportional to the current passing through that resistor i.e.

V ∝ I or V = RI where the constant of proportionality, R, is called the resistance of the resistor. To test Ohm's law we would measure the voltage across a resistor for various values of current passing through that resistor, and then plot a graph of V versus I. If the graph is linear, then the proportionality of V and I is proven.

I

3. Graphs can be used to obtain physically significant quantities. For example, in the above case of V = RI, a plot of V versus I will have a gradient equal to R. From the graph

aRb

=

We can measure a and b from our graph and thus obtain R. 4. We can also use graphs to look at physical relationships and find significant physical quantities when the relationship is not linear. For instance, suppose that a car starts from zero velocity and has a constant acceleration – a in a straight line drive. The relationship between the car's speed, v, and the distance travelled, s, is

v2 = 2as

This is a non-linear relationship. If we read the speedometer of the car to get v at a series of


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known distances (each time we passed a street lamp, say) we would have a set of data of v against s. If we used this data to plot v versus s, we would get a non-linear graph (see next page). But if we now plot v2 versus s, we obtain a straight line graph. From the equation v2 = 2as, we can see that a plot of v2 (y-axis) against s (x-axis) will have a slope of 2a. Thus

2 PaQ

=

which we can measure from the graph and hence get a value for a.


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Graph exercise In electricity theory. Ohm's law states that the voltage across a resistor, of resistance R, is given by:

V = RI

where I is the current passing through the resistor. In an experiment, the value of V is measured for various values of I. The following data is obtained:

I (mA)

V (volts)

0 0

3 5.1

6 9.9

9 15.2

12 19.8

15 25.0 Plot a graph of this data to test Ohm's law. Is the graph linear? Does the graph suggest that Ohm's law is valid? Calculate the gradient of the graph. What physical quantity does the gradient represent and what is the value of this quantity?


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EXPERIMENTAL ERRORS

It is impossible to measure any physical quantity exactly. Every measurement is subject to a range of uncertainty which we call the error in the measurement. Here, 'error' does not mean mistake, as its occurrence is unavoidable. Mistakes made by experimenters do lead to errors e.g. if an instrument is wrongly used or mis read, or the measured quantity is mis-recorded (all experimenter mistakes), the measurement will have an error due to this. But even if the measurement is properly made and recorded, there will still be an error in it. We all should know that a result is meaningless if the units of the measurement are not given and dangerous and/or expensive if they are wrong (the 1999 Mars Climate Orbiter crashed immediately after arriving at the planet because someone used Imperial units instead of SI, causing erroneous data to be fed to a computer calculating weight – the monetary cost was £76,000,000). It can be equally meaningless if the range of uncertainty – the error – is not given along with a result. Thus, to develop as skilled experimentalists, we must train ourselves to record error as well as recording measurements. This is the first step in error calculation and we take it immediately. In this first laboratory module we record both measurements and the error in measurements TOGETHER. Types of error 1. Systematic error If a systematic error exists in a measuring situation, it occurs every time the measurement is made. Repeating the measurement therefore gives the same result. Systematic errors are thus difficult to detect. The error may occur because something is wrong with the measuring instrument, e.g. the spring in an analogue voltmeter may have weakened in time; so that when a voltage is present at the terminals, the pointer swings too far across the scale, thereby consistently indicating a high result. Nobody knows that the error is occurring unless the instrument calibration is checked or unless the result is so illogical as to be recognised as such. Another example is that of a timer (e.g. stopwatch) that runs fast or slow. Expert experimentalists can almost always eliminate systematic error. We ignore all but the most obvious systematic errors e.g. a non-zeroed instrument. We assume that all our instruments are properly calibrated and working well. But, be aware that systematic errors exist.


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2. Random errors Random, errors can always occur and cannot usually be eliminated, but they are easy to detect. If the result of a measurement is subject to significant random error then repeating the measurement several times will produce several different results. Such errors are caused, for example, by variation in the ambient conditions under which the measurement is made. Vibration, air temperature and pressure fluctuations, draughts, light level changes etc can all cause random errors as can minor, unnoticed variation in experimental technique. The accuracy of a result that is subject to random error is greatly improved by repeating the measurement several times and taking the average result. In all but a few simple cases we ignore random error, but again, be aware that such errors exist. 3. Reading errors No instrument or scale can be read exactly: Is this reading 4.27, or 4.26, or 4.28? We cannot tell for certain and opinions will differ, so we record our best estimate and the uncertainty:

4.27 ± 0.01 units The ±0.01 is the reading error. It is our estimate of our own uncertainty of the reading. Always estimate on the high side. If uncertain of an estimate for a reading from an analogue scale, take ¼ of the smallest marked division as the reading error. These days, people tend to have more faith in digital meters than in analogue. This is because the numbers are presented to you. But there is still a reading error: Suppose a 3 digit meter reads 4.23. The next possible higher and lower readings are 4.24 and


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4.22, even if the digital instrument itself is perfectly accurate (unlikely!). The instrument will read 4.23 for any true value between 4.225 and 4.234, so the result should be recorded as 4.23 ± 0.005 units. If unsure, in the digital instrument case, then take half the smallest possible change that can be read on the instrument. Reading errors are the errors with which you will be most concerned. Every time you record a measurement, you should record the reading error with it. We use reading error because it is the simplest error to estimate and we wish to induce good error practice in the taking of results. But, be aware that it is not usually the dominant error in a measurement. Expression of errors Errors can be expressed as absolute errors, relative errors or percentage errors. The symbol that we use for an absolute error is ∆. Thus the absolute error in a measured quantity, x, is ∆x. Absolute error: This is the actual error in a measurement or result. It has the same units as the quantity that is being measured. Examples:

10.0 0.5R = ± Ω

The 0.5 Ω is ∆R, the absolute error in R. Note that this should not be written as 10.0 0.5R = Ω± .

3.26 0.01v = ± m s−1

here, the 0.01 m s−1 is ∆v, the absolute error in v. It should not be written v = 3.26 m s−1 ± 0.01. When noting down the absolute errors of individual measurements, we generally use the absolute error as above. Relative error: This is the ratio of the absolute error to be measured i.e. an expression of the absolute error as a fraction of the measured value. Relative errors have no units. Examples (following the above):


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0.5 0.0510

0.01 0.0033.26

RR

vv

∆= =

∆= =

It is not unusual to express the error in an individual measurement or in the final result of an experiment as a relative error. The usefulness of relative error expressions is their frequent use in the calculation of the error in a final result from the errors in individual measurements made to obtain that final result, and in the calculation of percentage error. Percentage error:

Percentage error = relative error × 100

Examples (following the above):

% error in 100 0.05 100 5%RRR∆

= = × =

Thus, we write

10.0 5%R = Ω±

not R = 10.0 ± 5% Ω

% error in 100 0.003 100 0.3%vvv∆

= = × =

Thus, we write

v = 3.26 m s−1 ± 0.3%

not v = 3.26 ± 0.3% m s−1. The error in the final result of an experiment is usually expressed as an absolute error or a percentage error. Graphs and errors in graphs In many experiments you are asked to plot a straight line graph. In doing so the following guidelines should be observed: (a) The quantities plotted should be clearly indicated in a title.


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(b) Suitable scales should be chosen so that the relevant part of the graph fills most of the page. The scale is indicated by marking straightforward values of the quantity plotted at sensible, regular intervals along the axes. Do not mark axes with irregular or measured values of the plotted quantity.

(c) It is not necessary to start a scale from zero unless it is required to find an intercept or is

convenient. It is better that the data points are spread as widely over the page as possible. (d) Axes should be labelled with the quantity plotted and its units. (e) Points should be boldly marked. (f) The gradient of a graph should be found by drawing what you judge to be the best straight

line through the data points and then constructing the largest triangle (ABC in diagram below) and finding the ratio AB/BC. Note that the magnitude of AB and BC is taken with due regard to the scale on the axes and not just by counting squares or measuring lengths on the graph paper. Suppose that we call the best gradient, m.

Graph of velocity (v) versus time (t)

After drawing the best line, find the gradient as above. Then draw the lines of maximum possible gradient and minimum possible gradient (see below). These lines are judged sensibly and not by the extremes of single points. Measure the maximum and minimum gradients (mmax and mmin respectively) using the triangle method above and take half of the difference between these gradients as the error in the best gradient.


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Graph of velocity (v) versus time (t) Error in the final result of an experiment In general, an individual measurement made in an experiment, or a single gradient or intercept of a straight line that is plotted, does not give the result of the experiment on its own. Usually, the measurements are substituted into an equation — the working equation — to obtain the final result R. The absolute error in the final result, ∆R, therefore depends on the errors in the individual measurements AND on the form of the working equation. There are, however, a few simple cases when the result of the experiment depends upon a single measured quantity, x. This x might be the value of a single measurement, the average of several measurements of the same quantity, the gradient or intercept of a graph that you plot etc. In Semester 1 we calculate the error in the final result of an experiment only when the result is such a function of a single variable. Below we show how this is achieved. In Semester 2 we shall progress to calculating the error in the result of any and all experiments. Errors in results that are functions of a single measured quantity only Let R depend only on some measured quantity, x (e.g. R = x2, R = sin x etc). This can be mathematically expressed as

( )R R x= From calculus we can write

dRR xdx

∆ = ∆


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Here, ∆R is a small change in R due to a small change in x of ∆x. If we think of this small change in x as the absolute error in x then ∆R is the corresponding absolute error in R. The above equation allows us to derive an ‘error equation’ that gives the absolute error in the result. All that we have to be able to do is to differentiate the working equation i.e. find dR/dx. How this is done is best illustrated by example: Suppose that the working equation is R = x then

1dRdx

=

and therefore R x∆ = ∆

the absolute error in the result equals the absolute error in the measured quantity. Suppose that the working equation is now R = 3x then

3dRdx

=

and so 3R x∆ = ∆

i.e. the absolute error in the measured value must be multiplied by 3 to give the absolute error in the result. Suppose that the working equation is R = x2

2dR xdx

= and so 2R x x∆ = ∆

The relative error here is

2

2 2R x x xR x x∆ ∆ ∆

= =

which is the relative error in the measurement multiplied by the power to which the measured quantity is raised in the working equation. This is always so for a quantity raised to a power and it can be useful e.g. if R x= , then the relative error is

12

R xR x∆ ∆

=

Now look at a trig function. Suppose that R = sin x is the working equation. Then

cosdR xdx

= and thus (cos )R x x∆ = ∆ or cos (cot )sin

R x x x xR x∆

= ∆ = ∆


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so that there is no real advantage in using either relative error or absolute error. Now suppose that R = ln x

1dRdx x

= thus xRx∆

∆ =

i.e. the absolute error in the result is equal to the relative error in the measurement! The inverse is true if R = ex as then

xdR edx

= and xR e x∆ = ∆

Therefore

x

xR e x x

R e∆ ∆

= = ∆

i.e. the relative error in the result is equal to the absolute error in the measurement! Negatives

Suppose that R = −x, then 1dRdx

= − and ∆R = −∆x.

But ∆x, being an absolute error, can be either +ve or −ve. We do not know which. So the −ve sign from the differential is meaningless and we ignore it. This means that the formula that we really use to calculate ∆R is

dRR xdx

∆ = ∆

where /dR dx is the modulus of dR/dx i.e. its value when ignoring its sign. Summary of Semester 1 requirements in errors 1. When a measurement is recorded, the reading error is also recorded. 2. When a straight line graph is plotted and the slope (or intercept) of the graph is needed to

find the result, the error in the slope (or intercept) is found. 3. When the result of an experiment is a function of a single measurement, the error in the

result is calculated.


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Combination of errors In many of the experiments that you do, the final result will be calculated by combining several (sometimes many) pieces of data. In the same way, the final error value will be obtained by combining several different errors. In the second semester you will be expected to combine your reading errors and calculate an error in the final overall error. An additional handout will be provided at the start of the second semester detailing how this is done.


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Health and Safety and the Law

The Health and Safety at Work etc. Act 1974 places on employers the statutory duty to ensure, so far as is reasonably practicable, the safety and health and welfare of their employees at work. The Act is a broad, generalised piece of legislation, not going into a great deal of detail. Section 2 of the Act requires employers to provide safe systems of work, which includes disseminating and revising as necessary written statements of their general policy. The statements should ensure that all who are at risk are aware of the hazards, the reasons for control in working practices and the part they as individuals must play in maintaining a safe and health working environment. Whilst the overall responsibility for health and safety rests at the highest management levels, each and every individual at all levels has to accept responsibility for carrying out that policy. The University periodically reviews its policy statements, and as new statements become available these are posted on safety notice boards, in laboratories, etc. Students presently come within a category of lawful visitors in University premises, rather than employees, and as specified by the Occupier’s Liability Act, 1957, the University must take care to see that they are safe. Sections 3 and 4 of the 1974 Act further reinforce this duty. Conversely, all persons owe a ‘duty of care’ and must take reasonable care to avoid acts or omissions which it can be reasonably foreseen may lead to injuries to their fellows. Recently more stringent requirements have been introduced at national level. Since 1 October 1989 the use of chemicals and other hazardous subjects has been subject to the Control of Substances Hazardous to Health Regulations (the COSHH Regulations). Since 1 April 1990 the Electricity at Work Regulations have been in force. In response to these regulations and to anticipate further national regulations the University now requires that all experimental work must be subject to a detailed assessment for all hazards using the University Hazard Assessment Form. All current undergraduate laboratory experiments have been assessed in this way. New experiments are assessed at the time of their introduction. STUDENTS SHOULD NOTE PARTICULARLY THAT THEY AND THEIR SUPERVISORS MUST MAKE AN ASSESSMENT FOR ANY ADDITIONALWORK CARRIED OUT IN THE UNIVERSITY, FOR EXAMPLE, FINAL YEAR PROJECTS. Electrical Safety Most electrical accidents are associated with inadequate or non-existent insulation for current-carrying wires, which may result in shocks, burns, fire or explosions. The susceptibility of individuals to shock varies, but currents of a few mA and above are dangerous, and correspondingly contact with voltages above 50 B must definitely be avoided. Even slight shock might result in involuntary movement, causing other accidents. Students are likely to meet: (i) Permanently wired-in, fixed equipment such as machine tools;


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(ii) Equipment connected to the mains by a plug and socket, either heavy and not normally moved, such as an oven, or smaller portable items such as lamps, power supplies or soldering irons. The frequent reconnections of these leads to deterioration of the mains cable;

(iii) Experimental circuits in various laboratories. Insulation of equipment must correspond to one of the following categories: (i) Relies upon the insulation of the supply cable and the connection of the metal casing to

the mains earth lead; (ii) Extra insulation is provided and there is no earth connection via the mains. There are two

types:

(a) equipment totally enclosed by insulation materials; (b) metal casing is isolated from the supply lead by an extra layer of insulation.

This should be marked with a double square sign. (iii) Equipment in which the voltage between any two parts in the circuit must be less than

50 V (rms, ac or dc). In all circuits one should avoid bare sections of the wire as far as possible. WORK ON EXPOSED, LIVE CIRCUITS CARRYING ABOVE 50 V MAY NOT BE CARRIED OUT BY UNDERGRADUATES WITHOUT APPROVAL BY THE DEPARTMENTAL SAFETY OFFICER Before commencing work, check for signs of damage to the equipment, especially to mains voltage supply cables. Avoid unnecessary strains in cables and avoid also having long lengths trailing about the floor or over equipment. Check that protective covers are in place and secured. Use the mains switch to isolate sockets when inserting or removing plugs, but in any case insert or remove plugs carefully. Hold the main body of the plug and avoid touching the contacts. When making connections even in low voltage circuits, turn the power off and do not touch the circuit when live if it can be avoided. Do not work on wet surfaces. Avoid liquid spillages and generally keep liquid away from the circuit where possible. If in any doubt about the safety of the equipment, seek immediate help from the staff. If this is not possible for any reason, attach a label to the equipment indicating what is faulty, preferably giving some indication of the fault. Code of Practice for the Use and the Disposal of Chemicals Chemicals are divided into four groups using the nomenclature of the Classification, Packaging and Labelling Regulations 1984 (the CPL Regulations), namely: Extreme Hazard High Hazard Medium Hazard Low Hazard


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All chemicals purchased now in the UK should have a mandatory hazard classification on the label, e.g. harmful, irritant, toxic or very toxic. The label should provide additional information on the specific nature of the risk, e.g. may cause cancer, very toxic by inhalation. These help to determine the classification in Table 1. Hazard Category CPL classification (or equivalent) or CPL risk indication

Extreme Hazard Explosive risk; Violent reactions; Very toxic; Reactions liberate very toxic gases; Risk of cancer, genetic effects or birth defects; Risk of cumulative health effects or very serious irreversible health effects.

High Hazard Toxic; Corrosive; Extremely flammable or highly flammable.

Medium Hazard Harmful; Irritant; Flammable.

Low Hazard Substances not reaching criteria for CPL classification of ‘Harmful’, ‘Irritant’ or ‘Flammable’.

A copy of Hazard Data Sheets is available, providing a useful source of information for chemicals without adequate labels. The University Hazard Assessment Form requires a statement of Means of Containment (e.g. fume cupboard, total enclosure), Personal Protection used (e.g. lab coat, gloves, eye protection) and Waste Disposal Procedures. The University Code of Practice discusses storage requirements for chemicals. The common solvents must be sorted in labelled, lockable fire-proof metal cupboards containing not more than 50 litres in total. Acids must be stored in a separate cupboard, with strict segregation of organic (e.g. acetic) from other acids, especially nitric acid. Toxic and Very Toxic chemicals must be stored in a labelled lockable poison cupboard. A register of the poisons must be kept. A further lockable cupboard should be provided for carcinogens etc. Provided these are compatible with the toxic material they may be stored in the poisons cupboard. Disposal of Chemicals and Glass Certain chemicals may be disposed of via the sink: (i) All water soluble/miscible liquids except (a) heavy metals and their compounds, particularly mercury and cadmium; (b) flammable substances with flash points below 55°C, including common solvents. (ii) Acids and alkalis should be buffered to about neutral pH prior to sink disposal.

(Experimenters using caustic chemicals should protect themselves with goggles, rubber gloves and overalls.)


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All sink disposals should be heavily diluted. Solvents should be returned to the stores in marked bottles. Other unwanted chemicals should be listed and transferred to Mr T Meachin, Occupational Hygiene and Safety Services.

(iii) Glass should not be put in the waste bins as it constitutes a hazard to cleaners etc. Small

quantities may be disposed of by Technical Support Staff in room Newton 171. The portering staff should be requested to remove larger quantities.

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