### Collection Physical quantities guidance notes

## Physical quantities guidance notes

for 14-16

Physical quantities guidance notes.

The verb to weigh

and the noun weight

are used in conflicting ways, even in science.

You cannot hope to clear up the great distinction between mass and weight by narrowing down the use of those common words. Nor can you exclude them or replace them in science. Students have to learn to live with their sloppy complexity.

Therefore, at an introductory stage you might use weigh it

to mean put it on the balance and see what the balance says. Others will want to drop the verb to weigh

and ask students to find the mass

of the object.

Even that is not very helpful, because what the balance feels, and what I feel if I put the object in my hand, is a force that arises from the pull of gravity on the object. This force is the object's weight. With a balance we are comparing the weight of one object with the weight of some other standard thing, a standard kilogram. That comparison will give the same ratio if we can transport the whole experiment to the Moon, or anywhere else where the gravitational field is strong enough to make the machine work at all.

Whatever you decide to do about the distinction between mass and weight, it is important to use these terms consistently. The student should insensibly acquire the right idea by the teacher's care in their use. Actually, the distinction between mass and weight comes fairly easily to some students in this space age, so be prepared to make a short, general comment if a good occasion arises.

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### Introducing the concept of density

There is a strong tradition of beginning physics with careful measurements of volumes and masses and calculations of densities, and with considerable care over arithmetic.

Generally, students find the measuring a fairly interesting routine which does not require much thought. The calculation of density appears an unnecessary interruption of interesting experiments.

Many teachers find the calculation simple and so are tempted to rush to arrive at a characteristic physical quantity. Later, when they discover their students are having difficulties, they may revisit the concept of density with greater care. By that time, it's likely that damage has been done to the picture of science which students are forming.

#### Another approach

Solid blocks of material, which are the same size, aid the comparison of density of different materials. Rectangular shaped containers for liquids and gases help in the measurement and calculation of volume. Using spreadsheets to cope with arithmetic problems enables you to emphasize the concept of density and handle materials of different densities. Once the concept has a secure foundation, the arithmetic skill can be introduced quickly, at a later stage.

We suggest, with an average group, that you get students to compare blocks of the same dimensions. Then ask them how they can bring other blocks into comparison. If they are interested (if you've succeeded in making it an intriguing problem), you can coax the class into a discussion of ways and means. If necessary, offer the suggestion of finding out the mass of one little block of 1 cm x 1 cm x 1 cm.

*"Yes. If you had them, you could weigh little blocks like that little cube. But you have not got them. No, we cannot cut them up with a saw. That would take too long and spoil the big blocks. Can you count the cubes in a big block without cutting it up?"*

At this point, draw pictures on the blackboard/whiteboard in a progression of problems, or give problems for homework, such as the following:

*"Here is a block of wood 2 cm long, 1 cm wide and 1 cm high. Here is a little cube of Plasticene 1 cm by 1 cm by 1 cm. How many little cubes are there in this block?"*

*"Here is a block 2 cm long, 3 cm wide and 1 cm high. How many little cubes would fit along the 2 cm by 3 cm face? How many layers of cubes from front to back? How many cubes altogether will fit into the block?"*

*"So it is all a matter of counting cubes by multiplying length x breadth x height."*

Of course, students will have learnt this in mathematics but you hope to have produced some practical meaning.

A well-packed box of sugar cubes helps this discussion. There may also be a set of Tillich's bricks, either in the science preparation room or in the mathematics department. These bricks are 1 cm^{3} and have a density of 1 g/cm^{3}.

Density is just the name that scientists use for the mass of a unit cube.

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### Introducing pressure

Begin with some simple questions in order to find out what students know about pressure and so determine a starting point for the topic.

This series of questions describes four pairs of events. The items in each pair are similar, but with a difference. For example, in pair **A , the shoes are different.**

#### A

A girl stands on soft sand in flat shoes

A girl stands on soft sand in high-heeled shoes

How would the marks in the sand be different?

#### B

A boy presses his thumb on the flat top of a drawing pin

A boy presses his thumb on the point of the drawing pin

What difference would his thumb feel?

#### C

The flat side of a knife is pressed against butter

The sharp edge of a knife is pressed against butter

What difference would you see?

#### D

A saucer is carefully placed flat onto water in a bowl

A saucer is lowered edge down into water

What difference would you see?

Illustrate your answers with diagrams if you want.

*"You stand with your bare feet on a smooth concrete floor. Then someone sprinkles gravel around you so that you have to walk across the gravel. Why does the gravel hurt while the concrete does not? (Because the gravel sticks to your feet. Is that the whole answer?)"*This should lead to a discussion of load and area comparisons leading on to a need for a new concept: pressure.- Other examples might include pressing on the bench and then a corner of the bench with the flat of your hand; holding a ruler by squeezing the edges and the flat surface; leaning against a wall with the flat of the palm of your hand and then a finger.

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### Managing open ended practical investigations

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### A language for measurements

#### What is a measurement?

A measurement tells you about a property of something you are investigating, giving it a number and a unit. Measurements are always made using an instrument of some kind. Rulers, stopclocks, chemical balances and thermometers are all measuring instruments.

Some processes seem to be measuring, but are not, e.g. comparing two lengths of string to see which one is longer. Tests that lead to a simple yes/no or pass/fail result do not always involve measuring.

#### The quality of measurements

Evaluating the quality of measurements is an essential step on the way to sensible conclusions. Scientists use a special vocabulary that helps them think clearly about their data. Key terms that describe the quality of measurements are:

- Validity
- Accuracy
- Precision (repeatability or reproducibility)
- Measurement uncertainty

Validity: A measurement is ‘valid’ if it measures what it is supposed to be measuring. What is measured must also be relevant to the question being investigated.

If a factor is uncontrolled, the measurements may not be valid. For example, if you were investigating the heating effect of a current ( *P* = *I*^{ 2} *R*) by increasing the current, the resistance of the wire may change as it is heated by the current to different temperatures. This would skew the results.

Correct conclusions can only be drawn from valid data.

Accuracy: This describes how closely a measurement comes to the true value of a physical quantity. The ‘true’ value of a measurement is the value that would be obtained by a perfect measurement, i.e. in an ideal world. As the true value is not known, accuracy is a qualitative term only.

Many measured quantities have a range of values rather than one ‘true’ value. For example, a collection of resistors all marked 1 kΩ. will have a range of values, but the mean value should be 1 kΩ.. You can have more confidence in a number of measurements of a sample rather than an individual measurement. The variation enables you to identify a mean, a range and the distribution of values across the range.

Precision: The closeness of agreement between replicate measurements on the same or similar objects under specified conditions.

Repeatability or reproducibility

(precision): The extent to which a measurement replicated under the same conditions gives a consistent result. Repeatability refers to data collected by the same operator, in the same lab, over a short timescale. Reproducibility refers to data collected by different operators, in different laboratories. You can have more confidence in conclusions and explanations if they are based on consistent data.

Measurement uncertainty: The uncertainty of a measurement is the doubt that exists about its value. For any measurement – even the most careful – there is always a margin of doubt. In everyday speech, this might be expressed as ‘give or take…’, e.g. a stick might be two metres long ‘give or take a centimetre’.

The doubt about a measurement has two aspects:

- the width of the margin, or ‘interval’. This is the range of values one expects the true value to lie within. (Note this is not necessarily the range of values one might obtain when taking measurements of the value, which may include outliers.)
- confidence level’, i.e. how sure the experimenter is that the true value lies within that margin. Discussion of confidence levels is generally appropriate only in advanced level science courses.

Uncertainty in measurements can be reduced by using an instrument that has a scale with smaller scale divisions. For example, if you use a ruler with a centimetre scale then the uncertainty in a measured length is likely to be ‘give or take a centimetre’. A ruler with a millimetre scale would reduce the uncertainty in length to ‘give or take a millimetre’.

#### Measurement errors

It is important not to confuse the terms ‘error’ and ‘uncertainty’. Error refers to the difference between a measured value and the true value of a physical quantity being measured. Whenever possible we try to correct for any known errors: for example, by applying corrections from calibration certificates. But any error whose value we do **not** know is a source of uncertainty.

Measurement errors can arise from two sources:

- a random component, where repeating the measurement gives an unpredictably different result;
- a systematic component, where the same influence affects the result for each of the repeated measurements.

Every time a measurement is taken under what seem to be the same conditions, random effects can influence the measured value. A series of measurements therefore produces a scatter of values about a mean value. The influence of variable factors may change with each measurement, changing the mean value. Increasing the number of observations generally reduces the uncertainty in the mean value.

Systematic errors (measurements that are either consistently too large, or too small) can result from:

- poor technique (e.g. carelessness with parallax when sighting onto a scale);
- zero error of an instrument (e.g. a ruler that has been shortened by wear at the zero end, or a newtonmeter that reads a value when nothing is hung from it);
- poor calibration of an instrument (e.g. every volt is measured too large).

Whenever possible, a good experimenter will try and correct for systematic errors, thus improving accuracy. For example, if it is known that a balance always reads 2 g greater than the true reading it is perfectly possible to compensate for that error by simply subtracting 2 g from all readings taken.

Sometimes you can only find a systematic error by measuring the same value by a different method.

Errors that are not recognized contribute to measurement uncertainty.

#### ASE/Nuffield booklet: The Language of Measurement

In 2010, following a series of meetings with Awarding Organisations, the ASE and Nuffield Foundation jointly published a booklet to enable teachers, publishers, awarding bodies and others in England and Wales to achieve a common understanding of key terms that arise from practical work in secondary science. Order a copy or see extracts from the booklet

#### Acknowledgement

This webpage is based on the National Physical Laboratory's *Good Practice Guide: A Beginner's Guide to Uncertainty of Measurements* written by Stephanie Bell.

A Beginner's Guide to Uncertainty of Measurements

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### Variables

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### Rough and ready measurements

To many students, the image of science is one of exactness and perfection. And yet, good scientists make rough estimates again and again, sometimes without ever making a precise measurement. It is important to teach students that rough measurements are respectable.

Of course, high precision is of the essence in many cases. A modern mass spectrograph must yield measurements of high precision if tiny mass-differences between one atomic nucleus and another are to be interpreted as energy-differences using *E = m c^{ 2}*.

Yet when Chadwick measured the nuclear charges of copper, silver and platinum, by alpha scattering in 1920, relatively rough measurements showed Rutherford's atomic model was correct. Chadwick showed that the nuclear charge (in electron units) is just equal to the atomic number

, the number of the element in the periodic table, a series arranged in order of atomic masses. Those answers were suspected from the general pattern of theory and had to be whole numbers since a complete atom (of nucleus plus outside electrons) is neutral. Much more precise measurements were neither needed, nor at the time, possible. Even before that, the first hint of atomic number measurements came in 1906, from Barkla's attempt to measure the number of electrons in a carbon atom by scattering X-rays. His measurements suggested a number of about 6 electrons per atom, in fact somewhere between 5 and 7, yet this rough estimate enabled the founding of atomic theory to proceed.

Galileo made the roughest measurements for his test of constant acceleration down an incline. He knew he was right in his simple summary of natural behaviour. He just wanted to convince some people by quoting an experiment.

Rough estimates are not just a misfortune peculiar to early, clumsy experimenters. They are the right thing in some parts of a growing science. Nuclear physicists and some cosmic ray physicists make very precise measurements. In other cases, they seek only a rough estimate to settle an essential point in the progress of their knowledge.

You cannot give the above examples to students if they do not know the science. In that case, the following may be some help.

*"An invading army is about to go into a foreign land and the general wants to know the size of the enemy's forces. He learns that it is 18 000. Does it matter much to his plans if it is 19 000 or 15 000? What he wants to know is that it is about 18 000 and not 30 000. If he waits for his staff to carefully sift through reports and add up the guesses and check them and find that the enemy really has 18 473 men, then the general may set out too late to win the battle."*

Other examples include:

- estimating how many snow ploughs are needed to clear a snowfall in the middle of the night;
- the Chancellor of the Exchequer makes a clever guess on the number of road vehicle licences which will be paid in the next year;
- a rough guess that the Sun is 300 000 times as massive as the Earth suffices to tell astronomers that the Earth is not massive enough to affect the orbit of the planet Venus, significantly.

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### Straight line graphs

#### Drawing straight line graphs

Once you have plotted the points of a graph, checked for any anomalies and decided that the best fit will be a straight line:

- To select the best fit straight line, take a weighted average of your measurements giving less weight to points that seem out of line with the rest.
- Use a ruler to draw the line.

#### Interpreting straight line graphs

Proportionality:
A straight line through the origin represents direct proportionality between the two variables plotted, *y = mx*. If the plotted points (expressing your experimental results) lie close to such a line, then they show the behaviour of your experiment is close to that proportionality.

Linear relationships:
In many experiments the best straight line fails to go through the origin. In that case, there is a simple linear relationship, *y = mx + c*. Historically, one of the most far-reaching examples is the graph of pressure of gas in a flask (constant volume) against temperature. The intersect on the temperature axis gives an absolute zero of temperature, and an estimate of its value.

Identifying systematic errors:
In some experiments, all measurements of one quantity are wrong by a constant amount. This is called a ‘systematic error’. (For example, in a pendulum investigation of *T* against *l* all the lengths may be too small because you forgot to add the radius of the bob. Plotting T^{2} against *l* will still give a straight line if every value of *l* is too short by the radius but the line does not pass through the origin.) In such cases, the intersect can give valuable information.

Checking for constancy:
Consider the acceleration of a trolley. If you plot *s* against t^{2}, where *s* is the distance and *t* is the total time of travel from rest, then you hope to get a straight line through the origin. [A straight line through the origin shows that s = constant t^{2}]

In fact we know that *s* is proportional to t^{2} for any case of constant acceleration from rest. Simple mathematics lead from the statement that Δv / Δt = acceleration, giving s = 1/2at^{2} providing *a* is constant. [Δv = change of velocity, Δt = time taken.]

IF *a* is constant, THEN *s* = 1/2at^{2} because logic does that. So why might you plot the graph? To find out whether the trolley moved with constant acceleration.

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