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### Measuring mass, length and time

- A simple balance 1
- A simple balance 2
- Making a microbalance
- Measuring lengths in centimetres
- Measuring the thickness of a coin
- Measuring paper
- Measuring the thickness of aluminium leaf
- Measuring the radius of a marble
- Measuring time intervals
- Time intervals using a heavy pendulum
- Investigating the time of oscillation of a pendulum
- Mass and weight
- Rough and ready measurements

## Measuring mass, length and time

for 14-16

It can be easy to use a modern measuring device to make measurements without thinking about the underlying ideas. A serious look at measurement techniques will also reveal the meaning of these three fundamental quantities – mass, length and time.

**Demonstration**

A brief introduction to measuring mass by balancing.

Apparatus and Materials

- Crude balance (see technical notes)
- Retort stand, boss, and clamp
- Hanger with slotted masses, 10 g
- Hanger with slotted masses, 100 g

Health & Safety and Technical Notes

Read our standard health & safety guidance

This simple lever balance is made out of a lath of wood or a metre rule (with the graduations ignored) drilled with holes at the ends and centre, together with three hooks. It is not meant to produce accurate results nor is it meant to be an accurate part of teaching the lever law, though it could be an introduction. Suspend the simple lever by one of the hooks positioned centrally. It is convenient to support the hook from a clamp attached to a retort stand by a boss. Position the other two hooks near the ends of the lever at equal distances from the centre.

The sensitivity of the balance can be changed by adding a small load (e.g. Blu-Tac or a screw) on the lower edge of the beam, at the centre, to bring down the centre of gravity. Better still, drill another hole nearer to the top of the lath so that the lath is suspended with a lot of it below the suspension point.

Procedure

- Weigh a small parcel (about 500 g), using a 100 g hanger and 100 g slotted masses on one side, the parcel on the other. Since this is only a demonstration to introduce a problem do not labour this experiment. Do not add smaller masses to make the weighing more
accurate

. It is only a rough measurement. - Weigh a letter to the nearest 10 g by the same method, using the 10 g slotted masses.
- Ask whether the beam will weigh a single hair. It fails, suggesting the need for a more sensitive balance. This could lead on to:

Teaching Notes

- You could introduce this experiment to students in the form of a problem-solving task, to be designed and perhaps built at home. One context which has been used successfully is that of finding out how much a parcel and a letter cost to send by post. The current postal rate leaflets, obtainable from a Post Office or related website, make this task more realistic.
- A parcel and a letter can be weighed relative to the mass of laboratory masses or substances found at home such as a bag of sweets, bag of sugar, slab of butter and so on. All the masses are marked on these products. However, you might find yourself debating whether to sort out the mass/weight problem.
- Many teachers tend to try to use the correct words without penalizing students for getting the wrong ones at the introductory stage. That is, mass is measured in kilograms relating it to
massive

objects. A discussion of units and the standard kilogram in Sèvres may also come into the discussion when comparing slabs of butter, bags of sugar and bags of sweets as fractions of a standard kilogram. How would a slab of butter stand up to reliability over time?

*This experiment was safety-tested in July 2007*

### Up next

### A simple balance 2

**Demonstration**

A balance which introduces the principle of reading masses against a scale.

Apparatus and Materials

- Crude balance (see technical notes)
- Retort stand, boss, and clamp
- Mass with hook (mass approx 25 g)
- Hanger with slotted masses, 10 g
- Another mass, approx 25 g
- Long strip of card (to act as scale)
- Triangle of card (to act as pointer)
- Sticky tape

Health & Safety and Technical Notes

Read our standard health & safety guidance

This simple lever balance is made out of a lath of wood or a metre rule (with the graduations ignored) drilled with holes at the ends and centre, together with three hooks. Suspend the simple lever by one of the hooks positioned centrally. It is convenient to support the hook from a clamp attached to a retort stand by a boss. Position the other two hooks near the ends of the lever at equal distances from the centre.

The sensitivity of the balance can be changed by adding a small load (e.g. Blu-Tac or screw) on the lower edge of the beam, at the centre, to bring down the centre of gravity. Or, drill another hole nearer to the top of the lath so that the lath is suspended with a lot of its mass below the suspension point.

Procedure

- Set up the simple lath balance (see the experiment A simple balance 1 below) and discuss how it measures the mass of a small parcel...
- Position the scale so that the pointer moves up and down next to it. Introduce a new balance which uses a counterpoise and the angle of the beam to measure mass.
- Attach the
anonymous

counterpoise mass on the left. - Try 10 g, 20 g, 30 g on the right, and mark the pointer positions on the card scale. The third weighing shows excess mass by the tilt of the beam
- Hang another mass, approx 25 g, on the right. Read its value from the scale.

Teaching Notes

- This balance is a lever balance using a counterpoise weight with the suspension point in the middle or off-centre depending on the sensitivity required. This is a useful design for finding the mass of lighter things such as a letter.
- To make a success of this experiment, you will need to prepare the beam quite carefully so that it is rather insensitive, bringing the 10-, 20-, 30-g scale into a reasonable range. That simply needs a move of the central pivot to a higher hole, or the adding of a considerable load under the beam at the centre. A large bulldog clip with a piece of metal anchored to it will make the latter change easy.
- Ask students if one could weigh still smaller things on this balance: a ring, a pin, a hair? Try hanging a hair alone on one end of the balance. Do this demonstration very quickly and lightly to suggest the need for a sensitive
microbalance

. - Students given a
design and build

task for homework can come up with some very ingenious designs, perhaps made with Lego. Top-pan balances of varying mechanical advantage can be built and some very small masses measured. - Students should have some practice at guessing the masses of various objects. They should then check their guesses by using a balance to find the object's mass more accurately. This can be treated as a game around the class.

*This experiment was safety-tested in July 2007*

### Up next

### Making a microbalance

**Class practical**

Students can make an amazing instrument which is able to measure the mass of a hair or a grain of sand.

Apparatus and Materials

*For each student group*

- Wooden block (4 cm x 4 cm x 2 cm)
- Wooden strip (15 cm x 2 cm, for example, a medical tongue depressor)
- Elastic bands (5 cm), 2
- Needle (fine)
- Metal screw (1.2 cm)
- Aluminium support (see Technical notes)
- Drinking straw (waxed paper or plastic)
- Graph paper - at least 100 sheets
- Scissors
- Sensitive laboratory balance

Health & Safety and Technical Notes

Take care with the needles. Issue them in a little plastic box so they don't become lost.

Read our standard health & safety guidance

The aluminium support is made from a bent sheet of aluminium. An alternative support could be made from two plastic credit-card-sized cards, glued to a wooden block.

The screw should be the correct size to screw into the end of the drinking straw.

Have plenty of spare straws available.

A pile of graph paper can be used. It needs to have no margins on the paper and to have 10 x 10 large squares marked on it and each large square divided up into 10 x 10 smaller squares. The pile of graph paper should have a mass of 1 kilogram and this may be conveniently near to 1000 sheets. If this is not possible, then suitable values should be used in order to make the arithmetic easier, all the time discussing order of magnitude calculations.

Once the mass of 1000 sheets is known then the mass of one sheet can be calculated, the mass of one big square and the mass of one small square. This is 10 ^{ -7 } of the original mass of the pile of paper. Grains of sand and short hairs have a mass in this range. So the vertical scale can now be calibrated by using 1, 2, 3. etc., small squares. Less mathematically capable students can measure the mass in terms of squares of paper or be told the real mass once they have begun to grasp the principle of finding the mass of many sheets in order to calculate the mass of a small square.

*NB:* Paper changes in mass depending on the moisture content.

Procedure

- Fix the wooden strip to the wood block as illustrated using the elastic bands twisted twice round the block.
- Insert the small metal screw in one end of the drinking straw.
- Cut away the other end of the straw with scissors so that it acts both as a pointer and as a little scoop into which items to be weighed can be placed.
- Find the approximate centre of gravity by balancing the straw on the needle. Then push the needle through the straw to act as the rolling axle. It should be put through just above the long axis of the straw.
- Use standard masses (paper squares) to calibrate the microbalance. Mark the zero position of the pointer high up on the vertical scale. Put the 1 mm
^{2}weight into the scoop and mark the pointer position. Continue adding weights until a uniform scale is constructed. - Other things that might be weighed include a flake of mica, a small piece of thread, a small piece of iron wire before and after rusting, a drop of olive oil, and the loss in mass as a drop of ether evaporates.
- Order of magnitude calculations can be fun. Once the mass of a grain of sand is known estimate and then measure the mass of a beaker of sand.

Teaching Notes

- This amazing instrument is able to measure the mass of a hair or a grain of sand. It is sensitive, even if not overly precise, but it will give excellent order of magnitude values if it is built carefully.
- It is helpful for you to have made a microbalance before the lesson so that students are able to suggest improvements when making their own. Their smaller fingers may find the task easier than you do.
- Once students have a microbalance they will want to know where the weights are. Offer a small piece of paper and students will realize that they do not know its mass. The ensuing discussion may lead to using a known mass of paper, which can be subdivided. Mark the zero of the scale on a piece of paper attached to the upright wooden strip.
- Students could make a microbalance at home. Use a cutaway matchbox as the support, a straw and suitable screw and any suitable scale rigged up from a piece of card.
- The position of the needle (how high up on the cross-section of the straw) determines the sensitivity of the balance, but students can be left to find this out for themselves. However, some students may need help. When help is given, take away the
right

straw you have set up and give the student a new straw for a fresh attempt. - Some students will be able to give the mass of the grain of sand in grams and others only in the number of (mm) squares.

*This experiment was safety-tested in July 2007*

### Up next

### Measuring lengths in centimetres

**Class practical**

Practice at measuring and estimating lengths.

Apparatus and Materials

*For each student group*

- Paper scale
- Sticky tape or glue

Health & Safety and Technical Notes

Read our standard health & safety guidance

Prepare a paper scale by photocopying a metre rule in sections of 25 cm. (Check that the photocopier is giving true 100% reproduction.) From this, make a master sheet with sections 0-25 cm, 25-50 cm, 50-75 cm, 75-100 cm. You should be able to fit two of these on an A4 sheet of paper.

Students can then cut strips, glue them end-to-end, and make a 2 m rule.

Procedure

- The sheet gives a paper ruler marked in centimetres. Cut out the sections and stick them end-to-end. (Make sure that the ends meet accurately.)
- Make some measurements with your tape and record them. Here are some suggestions:

- Length of your foot
- Width of your foot where it is widest
- Height of a neighbouring boy or girl (record their age)
- Height of a parent or other adult
- Width and height of your front door
- Length and width and thickness of a book
- Length of an egg
- The length, width and height of a room in your house (or your classroom): write each of them down in centimetres, then in metres. Work out the volume of the room in cubic metres.
- Diameter of a clock face
- Diameter of a watch face
- Diameter of a drinking glass and its circumference. Then divide the circumference by the diameter to find a rough value for π.
- Your own diameter (at waist)
- The difference in height between two people (such as father and mother, two sisters, or two other students in your class)
- The circumference of your upper arm, round your biceps muscle, first when your arm is slack, then when you are holding a heavy load, with your arm bent up at the elbow.

Teaching Notes

- Students will be familiar with the use of the metric system of length measurements. This is a time to measure varying lengths, first of all with a ruler and then with many other kinds of length-measuring devices, such as vernier calipers, micrometers and sonic distance measurers.
- Students should be able to estimate lengths and then check to see how accurate their estimates have been. Encourage them to make an estimate before measuring each length. Checking the table of estimates and measurements may reveal whether their ability to estimate is improved by this exercise. The table can also be checked to see if students find it easier to estimate large or small lengths.
- Students can make a ruler from paper. Download an example and photocopy it so that the scale is correct.

*This experiment was safety-tested in July 2007*

### Up next

### Measuring the thickness of a coin

**Class practical**

Introducing the idea of measuring multiple objects and finding an average.

Apparatus and Materials

*For each student group*

- Large supply of coins (all of one denomination)
- Ruler with graduations in mm (e.g. metre rule)
- Micrometer (OPTIONAL)

Health & Safety and Technical Notes

Read our standard health & safety guidance

There is a significant risk that the coins will go missing! You could use steel washers as an alternative if you are concerned about this.

Procedure

- Estimate the thickness of a single coin.
- Measure one with a ruler.
- Measure a pile of coins stacked on each other. Calculate the average thickness.

Teaching Notes

- Each student should try to measure the thickness of a single coin with a ruler with millimetres marked on it. The teacher should ask for results and also ask how reliable students think they are. Then ask for suggestions of improvement. Some may suggest measuring the height of a pile of coins and then calculate the thickness of one.
- Discuss the general idea of accuracy behind the method by saying:
*"Suppose you have just one good coin and this ruler marked in millimetres, how thick would you find the coin if you could measure it very carefully? Yes, we do now know that the thickness is, say, 1.3 mm but could you really see that if you had just one coin to measure? Even if you thought you could see it, would that be a safe and fair answer to give? With just one coin what would be the fairest thing to say? If you wanted to be quite safe, what would you say? Yes I agree; all we can say is somewhere between 1 and 2 millimetres. Now suppose you have 10 coins in a pile and you measure the pile. Even if you make a mistake of 1 millimetre in that measurement, how much of a mistake is that in the thickness of one coin? So if you measure 10 coins you could say that you think each coin is 1.3 millimetres thick. What would you say if you measured 100 coins in a pile?"*- At an introductory level, you might leave this problem there and come back to it later. Big numbers and small decimals are not easy, and the problem of accuracy is not a particularly interesting one yet.
- If a student points out that worn coins are thinner than new coins, then it might be worthwhile sorting them into two stacks, using the faces on the heads and the date on the coins. Compare the heights of the stacks. Of course, this kind of experiment is of far greater value if pupils suggest it themselves, or even if the teacher can coax it out of them in a way that makes them feel it is their own suggestion. Then they are doing science. Measuring many atoms in order to find the size of one atom is a skill which scientists have.
**How Science Works extensions:**Point out that it is worthwhile changing and improving an experimental method as you carry out an experiment and that deciding on a method does not preclude subsequent changes.- Collecting the data from the class for the three different measurements of the thickness can be used to prompt discussion about data spread, variation and accuracy.
- You could ask students to use the micrometer on a single coin and compare the value measured this way with the value obtained from the stack of coins. If students have not used a micrometer before, allow time for teaching them how it works and have them take sample readings before expecting them to use it confidently.

*This experiment was safety-checked in July 2007*

### Up next

### Measuring paper

**Class practical**

Measuring many sheets to find the thickness of one.

Apparatus and Materials

*For each student group*

- Pack of paper containing a known number of sheets (e.g. 500)
- Sheet of the same paper
- Ruler with millimetre scale (e.g. a metre rule)
- Chemical balance
- Micrometer (OPTIONAL)
- Selection of rulers/tape measures with different scale divisions (OPTIONAL)

Health & Safety and Technical Notes

Read our standard health & safety guidance

Procedure

- Attempt to measure the thickness of a single sheet of paper. Fold the paper in half, in half again, and so on, to obtain multiple thicknesses. Measure the thickness and calculate the thickness of a single sheet.
- Measure the thickness of a pile of a known number of sheets (say, 100 or 200). Calculate the thickness of a single sheet.
- Compare these two methods for measuring the thickness of a single sheet. Which is better?

Teaching Notes

- A book could be used instead of a pile of sheets of paper. Students should make a rough measurement of a pile of paper or the thickness of the book (remember the book is numbered on both sides of the paper).
- As well as the length, width and thickness of the paper, some students may attempt to measure its density. Often paper is sold with a g/m
^{ 2 }value on the packet (which is**not**a density) and so this may provide a value against which they could check their measured value. - Alternatively, a textbook can be used, but beware of the factor 1/2 since a book has half as many leaves as pages.
- To encourage order of magnitude estimates, you could go on to give pupils a rough value for the diameter of an atom so that they can calculate how many atoms would sit next to each other in the thickness of a piece of paper. For paper made of cellulose (which contains carbon, hydrogen and oxygen) then the average atomic diameter is probably only about 1.5 x 10
^{ -9}cm. Students who do not delight in using large numbers should not be dragged through these calculations. - There are excellent books, videos and web sites on Powers of ten: Classroom displays of large and small distance measurements with pictures of the objects measured will create a good background to this work.
- You could follow this up with the activity
Practice in using large and small numbers

, which can be downloaded from: **How Science Works extensions:**Although the measurement methods here are relatively straightforward, this provides an opportunity for students to design and carry out their own experiment. Emphasise that they should measure as accurately as possible and clearly explain how their method improves the quality of data collected. Get them to try and estimate the uncertainties in their results and to identify which measurement they feel has the greatest uncertainty.- You could encourage students to write a full plan/method before they begin. If they are to do this, make relevant teaching points about the importance of clear and unambiguous instructions. Each group/student could write a procedure, which is then passed on to a different group/student to carry out, exactly as written. The second group/student can then evaluate the information that they have been given.

*This experiment was safety-checked in October 2007*

### Up next

### Measuring the thickness of aluminium leaf

**Class practical**

Using mass, density and area to deduce thickness.

Apparatus and Materials

*For each student group*

- Aluminium leaf, 1 sheet approx 5 cm square
- Microbalance (see Technical notes)
- Ruler

Health & Safety and Technical Notes

Read our standard health & safety guidance

If students have made microbalances (see the experiment below) the microbalance used in this experiment should be their own. Alternatively, they will require access to a sensitive laboratory balance.

Aluminium leaf is the thinnest sheet of aluminium you can get, beaten out until it is so thin that it flutters in air and you can almost see through it.

Procedure

- Measure the area of the aluminium leaf.
- Weigh the aluminium leaf to find its mass. (It may be necessary to screw the leaf into a ball.)
- The density of aluminium is 2710 kg /m
^{ 3 }(or 2.710 g /cm^{ 3 }). Calculate the volume of the leaf (= mass/density). Calculate its thickness (= volume/area).

Teaching Notes

- Put to your students the problem of how the thickness of aluminium leaf can be measured. It is not possible to adopt the procedure of... ...by measuring a pile of aluminium leaves.
- Aluminium leaf is much thinner than a sheet of paper and if two pieces are placed together then they stick.
- The best procedure is to use a microbalance to find the mass of a known area of aluminium leaf (5 cm
^{ 2 }), screwed up into a ball. Given the density of aluminium, the volume of the leaf can be calculated. Since the area is known, the thickness can be estimated. This is an indirect way of measuring the thickness of materials by knowing some other property such as density. - The comparison between the thickness of paper and aluminium leaf will be useful when it is found that paper will stop alpha particles and alumunium leaf does not.

*This experiment was safety-tested in July 2007*

### Up next

### Measuring the radius of a marble

**Class practical**

Controlling the object you are measuring.

Apparatus and Materials

*For each student group*

- Metre rules, 2
- Ruler, 15 or 30 cm
- Set squares, 2
- Marbles, a supply of
- Supply of ball bearings of different sizes (at least 5 of each size) OPTIONAL

Health & Safety and Technical Notes

Care needs to be taken as there is a trip hazard if marbles fall and roll around the floor.

Read our standard health & safety guidance

Procedure

- Give students a single marble and ask them to measure the radius using a ruler.
- Discuss with the class the difficulties with this method, in particular the shape of the marble.
- Give the students the opportunity to devise an improvement before suggesting the method illustrated above. Two metre rules provide a trough which the marbles sit in so that they do not roll away. The rules are held next to each other and the pair of set squares makes the ends of the line of marbles clear.
- Students then can carry out this revised and improved measurement of the marbles and work out the radius of one.

Teaching Notes

- In step 1 students should be able to get a reading but the variation across the class may be considerable. Check that they have not given the marble diameter as their result, instead of its radius.
- This experiment is designed to support the development of practical skills and confidence, not simply finding the radius itself. Students may ask
*'how many marbles should we use?'*, to which a non-committal reply such as*'how many do you think?"*or*'enough is appropriate'*. Using increasing numbers of marbles reduces the uncertainty in the diameter measured, but there is no right number. It is worth mentioning that errors may be introduced when more marbles are used, e.g. not all the marbles may be in a perfectly straight line. **How Science Works extension:**This experiment can support the skills involved in devising experimental methods which improve the accuracy of measurements. If students have carried out the experiment Measuring paper or Measuring the thickness of a coin, then they will already have encountered some of these ideas. In both of those experiments, a micrometer can be used to collect an accurate answer from a single coin or piece of paper.

*This experiment was safety-checked in December 2006*

### Up next

### Measuring time intervals

**Class practical**

Practice in estimating time and measuring time.

Apparatus and Materials

*As many different time measuring devices as possible, including:*

- Wall clock
- Wrist watch (to allow the use of the human pulse as a timer)
- Laboratory timers, electronic (ideally ones with different scale intervals, e.g. 0.1 s , and 0.01 s )
- Sand hourglass / egg timer
- Kitchen timer (either wind-up or digital)
- Electronic timers or datalogging package (OPTIONAL)

Health & Safety and Technical Notes

Read our standard health & safety guidance

Some mobile phones also have in-built timers which can be used.

Procedure

- Ask students to make some rough measurements of time, to gain familiarity both with time intervals, e.g. of the time between hand claps, the time for a book to drop, the time to walk or run a given distance and stopwatches and other measuring devices.
- Have them measure (or attempt to measure) the same time interval with a variety of devices.

Teaching Notes

- This activity can begin with students estimating time intervals between various activities and then by measuring time intervals to check how good their estimates are. Young students will find five minutes of silence to be a life sentence!
- Some real tasks will improve motivation, e.g. measuring the time to run a 100 m race. Average speed measurements might also be included, e.g. the time taken for a ball to fall from a high window.
- Discuss the difference between timing an unexpected time period (e.g. between hand claps hidden under a desk) compared to something that they can predict (e.g. a falling object).
- There are many electronic timers and datalogging packages available for measuring time, which will make a change from stopwatches and provide an introduction to electronic time measurement.
**How Science Works extension:**The selection of appropriate equipment to carry out a particular measurement is an important skill that is often overlooked, particularly if the ‘correct’ equipment for any experiment is generally made available for students. Students need practice choosing equipment and justifying their choice if they are to become confident experimenters.- This activity allows students to look at different ways of measuring time and to compare uncertainties in measurements. They could also discuss measurement errors. Point out that no single timing device will suit all situations.

*This experiment was safety-checked in July 2007*

### Up next

### Time intervals using a heavy pendulum

**Demonstration**

Measurement of regular time intervals.

Apparatus and Materials

- Broom handle
- Lengths of wood, about 5 cm x 5 cm x 50 cm, 2
- Wooden board, about 20 cm x 8 cm x 2 cm
- Nails, 15 cm, 2
- Bricks, 6
- Cross-beams of wood, 2
- Card, about 8 cm x 4 cm
- Wooden block, about 6 cm x 4 cm x 12 cm

Health & Safety and Technical Notes

Read our standard health & safety guidance

Drill a broom handle about 3 cm from the end to take a 15 cm nail. Push a wooden platform with a hole in it to take the broom handle over the handle. Secure it in place by a second nail pushed through a hole drilled in the lower end of the handle. This second hole should be about 110 cm below the top hole.

Screw a 25 mm round-headed screw into the bottom of the broom handle so that the head projects about 1 cm. Place two lengths of wood horizontally across two table tops. (Bricks resting on the ends will hold them rigid.) The pendulum is supported from these wood lengths, the 15 cm nail will roll on them. When stable, load the pendulum platform with two bricks, which can be tied into place.

Fasten a piece of thin card to a small block of wood. Position it so that, in swinging through the mid-position, the screw head which protrudes from the pendulum bob strikes the broad side of the card, making an audible click. Alternatively, amplify the sound this way: stick the card to part of a balloon which has been stretched tight and tied over the top of a 1000 ml beaker.

If the pendulum does not make enough swings, add more bricks to the platform.

Procedure

- Set the pendulum swinging. Listen to its regular sound.
- Time a number of time intervals – say, 10\. (Remember to start counting from zero.)
- Calculate the average time interval.
- The time interval may change slightly as the pendulum swing decreases. Ask:
*"Can this change in time interval be measured?"*

Teaching Notes

- A rigid pendulum of this type has a period of about two seconds so that the audible clicks occur at about one-second intervals. Rigidity is essential, otherwise there will be an unnecessary loss of energy.
- It is good for students to know that a lack of beautiful shiny apparatus need not stop the progress of science. Where apparatus for modern physics is needed and expensive, an economy like this may be wise.
- The crude brick pendulum hits a card as it passes through the lowest point each time to give audible signals. The piece of card that is to mark the swings audibly must be placed so that the pendulum hits it sharply and does not slide along its surface. Thus the card must be perpendicular to the plane of the swing. Simple schemes for mounting the card on a beaker or even on a rubber drum to increase the sound may be devised. Even a small bell could be rigged up.
- The hits which signal the time will, of course, make the amplitude of the pendulum decrease fairly rapidly. However, that will not change the timing appreciably so you can let the amplitude die down without worrying. Since the impacts which lessen the motion occur at the midpoint of the swing, the phase of the motion is not changed in that abrupt decrease of motion. Thus the pendulum continues to keep its true period. Those who have tried this pendulum confirm that students enjoy the ingenuity of it and find it less clumsy and more satisfying than adults do.

*This experiment was safety-tested in July 2007*

### Up next

### Investigating the time of oscillation of a pendulum

## Investigating the time of oscillation of a pendulum

Practical Activity for 14-16

**Class practical**

Students make a timer using a simple pendulum.

Apparatus and Materials

*For each student group*

- Pendulum (e.g. Plasticine bob on string/thread)
- Stopclock
- Chemical balance (0-100 g)
- Stand, clamp and boss
- Protractor

Health & Safety and Technical Notes

If large masses are used then the stands may need to be clamped to the bench.

Read our standard health & safety guidance

Procedure

- Show a demonstration pendulum and ask students to think about the variables that may affect the time period for one oscillation.
- Ask students to select one independent variable, collecting a set of data to investigate its effect on the oscillation time.
- After students have completed an initial investigation and drawn conclusions, ask them to evaluate their method in terms of its accuracy and improve on it.

Teaching Notes

- You may need to explain what one oscillation for a pendulum is (motion
). Variables to investigate include the mass of the pendulum bob, length of the pendulum (best measured to centre of bob), and initial amplitude (angle or displacement).there and back again

- A discussion following students' first attempts might lead to the following ideas for improving their measured value.
- Reduce the uncertainty in a measurement of periodic time by:
- measuring many oscillations to calculate the average time for one oscillation
- increasing the total time measured for multiple swings
- There is some uncertainty when measuring both the start time and also the stopping time, resulting from the experimenter's reflex time (as much as 0.2 s each, i.e. totalling 0.4 s ). The percentage uncertainty which this 0.4 s represents decreases as the total time measured increases. Students could carry out simple error calculations to discover, for example, the effect of a human reaction time of 0.2 s econds on timings of 2 s 20 s and 200 s.
- You may wish to get them to estimate the human reaction time or measure it as a separate activity. There are many web-based activities freely available.
- Improve the accuracy of a measurement of periodic time by:
- making timings by sighting the bob past a fixed reference point (called a
fiducial point

) - sighting the bob as it moves fastest past a reference point. The pendulum swings fastest at its lowest point and slowest at the top of each swing.
- The periodic time for a swinging pendulum is constant only when amplitudes are small. Its period of oscillation is then
*T*=2π √ _(l /g)_where *T*= Time period for one oscillation (s)*l*= Length of pendulum (m)*g*= acceleration due to gravity (m s^{-2})- Students investigating the effect of bob mass or pendulum length should keep the maximum angle of swing under 5 °.
**How Science Works Extension:**This provides an excellent opportunity for planning, carrying out and evaluating an investigation using multiple skills. The number of variables is limited but there is enough scope to allow students to develop an approach and select appropriate ranges and intervals.- Students often assume that any sensibly selected independent variable will always have an effect upon the dependent variable. Many may decide to investigate the effect of the mass of the bob, which yields an unexpected (counter-intuitive) result: the mass has no effect on periodic time. Proving that there is no link between two variables can be just as significant as finding one.
- The introductory discussion can put the pendulum into a scientific and historical context by describing the development of timing devices. Start with the hours of a day as one of the simplest units of time, easily measured with a sundial. Use this to introduce Galileo Galilei (1564-1642) and the (possibly apocryphal) story that his understanding of the behaviour of pendulums was spurred by observing the bronze chandelier or incense burner in the cathedral at Pisa. Galileo's pendulum introduced a method of measuring short periods of time that improved on the use of the human pulse. You could extend this timeline by describing further developments in timing devices, right up to the atomic clock (usually containing caesium) which is accurate to within 10-9 seconds per day.
- For students who take a particular interest in the measurement of time, suggest the book Longitude by Dava Sobel (ISBN 0007214227), which provides further background about the development of clocks and their use in navigation. It also has some examples of the struggles that can happen in the development of science and technology.

*This experiment was safety-checked in January 2007*

### Up next

### Mass and weight

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.

### Up next

### 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.