Measurements Techniques

Aslevels Physics Notes

Topic 2: Measurements and Techniques

Reading is a single determination of the value of an unknown quantity. It is the actual reading during an experiment.

Measurement is the final result of the analysis of a series of readings. A measurement is only accurate up to a certain degree depending on the instrument used and the physical constraints of the observer.

Methods of measuring length

Meter rule

Reading with a metre rule can be taken with an uncertainty of 0.5 mm. There are three possible sources of error in using a metre rule.

The first may arise if the end of the rule is worn, giving rise to a zero error. For this reason it is bad practice to place the zero end of the rule against one end of the object to be measured and to take the reading at the other end. You should place the object against the rule so that a reading is made at each end of the object. The length of the object is then obtained by subtraction of the two readings. A zero error like this is a systematic error, because is involved every time a reading is taken from the zero end of the rule.

Another systematic error may arise because marking is incorrect. Compare a 30 cm rule with another 30 cm rule there may be a discrepancy of one or two mm. A one mete steel rule is many time expensive than a plastic rule because it is more accurate

Another source of error with the metre rule is parallax error. The angle at which the scale is viewed affects the result. This is random error because the angle of view may be different for different readings

Micrometer screw gauge

The screw of the micrometer screw gauge advances exactly 1 mm for two revolutions. That is, the pitch of the screw is 0.5 mm or 500 𝜇m. If you look at the graduations on the barrel of the screw bearing you will see that there are divisions every 0.5mm. The reading on the barrel corresponds to the position of the edge of the thimble. When taking a reading it is important to check which half of the mm the edge of the barrel is in. the graduations round the circumference of the thimble run from 0 to 50. Each division corresponds to one – hundredth of a mm, or 10 𝜇𝑚. The reading on the thimble is added to the reading on the barrel. Figure shows a reading of 9.5 mm on the barrel plus 0.36 mm on the thimble, 9.86 mm in total. You can easily read to the nearest division on the thimble, that is, to the nearest 0.01 mm.

Vernier caliper

A Vernier caliper is a versatile instrument for measuring the dimensions of an object, the diameter of a hole, or the depth of a hole. Its range is up to about 100 mm, and it can be read to 0.1 mm or 0.05 mm depending on the type of Vernier with which it is fitted. The object to be measured is placed between the jaws and the sliding part is moved along until the object is gripped tightly. A reading to the nearest mm is taken on the fixed scale, at the zero end of the Vernier scale. The reading to a tenth of a mm is obtained by finding where a graduation of the Vernier scale coincides with a graduation of the fixed scale. Figure shows the scale of a Vernier caliper giving a reading of 25.4 mm.

Method of measuring mass

The top - pan balance

The top – pan balance is a direct-reading instrument based on a pressure sensor, or sometimes a spring. The unknown mass is placed on the pan, and its weight applies a force to the sensor. The mass corresponding to this force is displayed on a digital read-out. When using the balance, ensure that the initial (unloaded) reading is zero. There is a control for adjusting the zero reading. The uncertainty in the reading of a particular top-pan balance will be quoted in the manufacturer’s manual likely to be expressed as a percentage.

The spring balance

Spring balances are based on Hooke’s law, the extension of a loaded spring is proportional to the load. The extension is measured directly, by a marker moving along a straight scale, or by a pointer moving over a circular scale. You should take care to avoid parallax error when you take readings.

The lever balance

Lever balances are based on the principle of moments. In one common type, the unknown mass is placed on a pan, and balance is achieved by sliding a mass along a bar, calibrated in mass units, until the bar is horizontal. This represents the condition in which the moment of the load is equal and opposite to the moment of the sliding mass and the bar. A reading is taken from the edge of the sliding mass on the divisions marked on the bar. In this case, parallax error is less likely to be serious. Again check for zero error before taking a reading.

Measuring an angle

angles are measured using an instrument called a protractor. To measure the angle between two lines, the centre of the circle of the protractor is placed exactly over the point of intersection of the lines and one lin is aligned with the 0⁰ direction of the protractor. The angle between the lines is then given by the reading on the scale at which the second line passes through the circumference of the circle.

Method of measuring time

Stop watch

In physics practical you often need to measure time intervals. The basic method of measuring time interval is with a stopwatch or digital timer. The instrument has a digital display. It is based on the oscillations of a quartz crystal. The read out is nearest onehundredth of a second. The instrument is started and stopped by pressing a button, and re-set by pressing another control. It is important that you should not fumble a start or stop signal. You should familiarize yourself with the way of operating the instrument before you start a timing experiment. Remember that the reaction time of the experimenter (a few tenths of a second) is likely to be much greater than the uncertainty of the instrument itself. If you do not reduce the effects of reaction time, an unacceptable systematic error may be built in to the experiment. One way of reducing the effect of reaction time is to time enough events (e.g. swings of a pendulum). Count a large number of events (swings) so that the time of events is very much larger than the reaction time of the experimenter. Wherever possible work with about 20 oscillations and repeat each set of events three times. (Sometimes, when carrying out experiments on damped oscillations, you will have to be satisfied with fewer swings, but try not to go below 10 events).

Measurement of frequency using c.r.o.

A cathode-ray oscilloscope (c.r.o.) has a calibrated time-base, so that measurements from the screen of the c.r.o. can be used to give values of time intervals. One application is to measure the frequency of a periodic signal, for example the sine-wave output of a signal generator. The signal is connected to the Y-input and the Y-amplifier and time-base controls are adjusted until a trace of at least one, but fewer than about five, complete cycles of the signal is obtained on the screen. The distance L on the graticule (the scale on the screen) corresponding to one complete cycle is measured. It is good practice to measure the length of, say four cycles, and then divide by four so as to obtain an average value of L. the graticule will probably be divided into cm or mm or 2 mm divisions. If the time-base setting is x (which will be in units of seconds, milliseconds or microseconds per centimeter), the time T for one cycle is given by T=Lx. The frequency f of the signal is then f=1/T. The uncertainty of the determination will depend on how well you can estimate the measurement of the length of the cycle from the graticule. Remembering that the trace has a finite width, you can probably measure this length to an uncertainty of about 2 mm.

Method of measuring current and potential difference

Multimeters

Multimeters are available in digital forms. Such meters may include switched options of direct and alternating currents and voltages, and of resistance, with several ranges for each quantity being measured. If you use a multimeter, make sure that you are familiar with the controls, so that you can set the instrument to measure the quantity you require. Remember that, to measure a current in a component is a circuit, an ammeter should be connected in series with the component. To measure the potential difference across the component, a voltmeter should be connected in parallel with the component. The arrangement is shown in figure.

Error And Uncertainties

Significant figures

The number of significant figures in an expression indicates the confidence or precision with which a quantity can be stated i.e. more significant figures indicate more precision.
The presence of non-decimal, trailing zeros could be ambiguous, e.g. a distance reported as 1200 m, as it does not indicate whether the last two zeros are significant or not.
Possible ways to avoid such ambiguities include

Explicitly specify the number of significant figures e.g. 1200 m (2s.f.).
Use scientific notations e.g. 1.2 × 103 m

Generalized guidelines for calculations include;

For multiplications and divisions, the final result should be reported with the same number of significant figures as the least precise quantity used in the calculations.
e.g. if two lengths are reported as 1.23 m and 2.1 m
their combined length =1.23+2.1=3.33=3.3 m (1dp)
you are advised to apply the guidelines with common sense, e.g. for lengths reported as 99 m or 101 m, a simple criterion of 2 or 3 significant figures could be misleading.
The context of calculation should also be examined.
In examination papers or books, you may be asked to give you answer to 3 s.f.
e.g. a room with length and width given as 1.23 m and 2.3 m respectively.
Area of room = 1.23 × 2.3 = 2.829 = 2.83 𝑚2 (3 s.f.)

Error

An error is the difference between the measured value and the expected value of something. (An error is unavoidable).

Uncertainty

An uncertainty is a way of expressing or summarizing the error. (Uncertainty is unavoidable)
If a measured length is reported as 1.23 m, the observer is indicating that the last digit “3” has some uncertainty, but the digits “1” and “2” are certain. The real length could be between 1.225 m and 1.235 m (i.e. ±0.005 m)
Thus, an error in not the same as an uncertainty, though both are unavoidable. They are often used in common language alternatively to mean the same thing, but they are not!

Mistake

A mistake is simply not doing something correctly through carelessness (avoidable)

Precision and accuracy

When we make measurement, we generally assume that some exact or true value exists. While we may never know this true value exactly, we attempt to find this ideal quantity to the best of our ability with the time and resources available to the maximum of accuracy and precision.

Accuracy

Accuracy is the closeness of agreement between a measured value and a true or accepted value (measurement error reveals the amount of inaccuracy). Smaller the measurement error, higher the accuracy)

Precision

Precision is a measure of the degree of consistency and agreement among independent measurements of the same quantity (also reliability or reproducibility of the result).

Example

Consider the following two sets of measurements for the diameter of wire

Set A Mean diameter = 0.40 Mean deviation =0.12/5 = 0.024

Set B Mean diameter = 0.40 Mean deviation = 0.06/5 = 0.012

Since the mean deviation of the measurements in set B is smaller, the measurements in set B are more precise than the measurements in set A

The precision in the measurement can be improved by using a magnifying glass.

Think of it, while you are playing darts, like this;

Random and systematic errors

There are two types of errors in measured data. It is important to understand which you are dealing, with, and how to handle them.

Random errors

The characteristic of random errors is that it can be positive or negative and its magnitude is not constant. Thus, the reading obtained may sometimes be greater than the actual value and at other times smaller than the actual value.

Random errors refer to random fluctuations in the measured data due to;

The readability of the instrument
The effects of something changing in the surroundings between measurements
The observer being less than perfect  Parallax when reading a scale.
Using a micrometer screw gauge to measure the diameter of a wire if different pressures are applied when closing the gap of the micrometer screw gauge.
Changes in temperature during an experiment can result in measurements being sometimes bigger or smaller than the actual value.

Random errors can by averaging. A precise experiment has small random error.

Systematic errors

Systematic Errors are uncertainties in the measurement of physical quantities due to instruments, faults in the surrounding conditions or mistakes made by the observer.

Systematic errors refer to reproducible fluctuations consistently in the same direction due to;

An instrument being wrongly calibrated (e.g. a slow running stopwatch)
An instrument with zero error (it does not read zero when it should – to correct this, the value should be subtracted from every reading)
The observer being less than perfect in the same way during each measurement (e.g. in starting and stopping stopwatch)

One important characteristic of systematic errors is that the size of the error is roughly constant and the measurement obtained is always greater or less than the actual value.

Such as zero error.

Systematic errors cannot be detected or reduced by taking more measurements. An accurate experiment has small systematic error.

When graphing experimental data, you can see immediately if you are dealing with random or systematic errors (if you can compare with theoretical or expected results).

Error due to instruments.

A watch is fast. An ammeter which is used under different conditions from which it had been calibrated. An ammeter manufactured in Japan had been calibrated under different temperatures and earth’s magnetic fields from Singapore where the ammeter is being used.

Error due to wrong assumption

let’s say the value of g is assumed as 9.81 ms-² where the actual value of g is not 9.81 ms-² but say 9.87 ms-² .

Compound Errors;

Most experiments in physics involve measurement of various physical quantities which may then be used to evaluate a particular quantity. Errors in the measurement of the various physical quantities are then compounded, resulting in a larger error.

Addition or Subtraction

If there are two values ( 𝑥 ± 𝛿𝑥) and (𝑦 ± 𝛿𝑦)