Measuring length, volume and time
Physics starts with measurement: choosing the instrument, reading it properly, and making a small quantity big enough to measure well.
Measuring length with a ruler
A metre rule or 30 cm ruler reads to the nearest millimetre, its smallest division.
- Line one end of the object up with the zero mark. If the ruler's end is worn, start from another mark, such as 1.0 cm, and subtract it.
- Keep your eye directly above the mark. Looking from an angle shifts the reading, an error called parallax.
Measuring volume with a measuring cylinder
A measuring cylinder measures the volume of a liquid, in cm³ (1 cm³ = 1 ml).
Stand it on a level surface.
The water surface curves where it meets the glass; this curve is the meniscus.
Read the scale at the bottom of the meniscus, with your eye level with it.
The smallest cylinder that holds the liquid has the finest scale.
Measuring time
- A clock or watch suits long intervals, such as minutes or hours.
- A stopwatch (a hand-operated digital timer) suits a few seconds or more. It may display 0.01 s, but your reaction time, about 0.2 s to 0.3 s, is far bigger.
- A light gate connected to a digital timer suits very short intervals, such as a trolley passing through. It starts and stops automatically, with no reaction time.
Measuring many to find one
A sheet of paper is about 0.1 mm thick, less than one ruler division, and a pendulum swing lasts about a second, not much longer than your reaction time.
The answer is to measure many together and divide: measure 100 sheets and divide by 100, or time 20 swings and divide by 20.
The reading error is now shared out over them all, so its effect on each one is tiny.
The period of a pendulum is the time for one complete oscillation: from one side, across to the other side, and back again.
To time it well:
1. Put a marker (a fiducial mark) behind the centre of the swing and count each pass of the bob in the same direction.
2. Start the stopwatch as the bob passes the marker and count that moment as "zero", not "one".
3. Time 20 oscillations, then repeat twice more and find the mean.
4. Divide the mean time by 20.
The idea behind it
No measurement is exact.
Every instrument has a smallest step it can show, its resolution, and every reading is really "this value, give or take a bit".
A ruler marked in millimetres cannot tell 12.3 mm from 12.4 mm.
The skill in this section is not reading scales for its own sake: it is choosing a method so that the "give or take" is small compared with the thing you are measuring.
That is why measuring many and dividing works so well.
Your reaction time makes a stopwatch reading wrong by about 0.2 s, and that error comes from starting and stopping the watch, so it is the same however long you time for.
Time one swing of 1.6 s and 0.2 s is more than a tenth of the answer.
Time 20 swings (32 s) and the same 0.2 s is less than one hundredth.
The error has not shrunk; the quantity has grown, so the error matters less.
Notice what the answer means, too.
"One sheet is 0.12 mm thick" is the average sheet.
Some sheets will be a little thicker and some thinner; the method cannot tell you which.
You will meet the same idea, stated as percentage uncertainty, in practical work and at A Level.
- Parallax error: a reading error caused by not having your eye directly in line with the scale mark.
- Meniscus: the curved surface of a liquid in a narrow tube; read the bottom of it for water.
- Period: the time taken for one complete oscillation.
- Oscillation: one complete swing, from one side to the other and back again.
A stack of 200 identical sheets of paper is 2.4 cm thick.
Calculate the thickness of one sheet in mm.
1. Convert to mm: 2.4 cm = 24 mm.
2. Divide by the number of sheets: thickness = 24 ÷ 200.
Answer: 0.12 mm.
A student times 20 oscillations of a pendulum three times and gets 31.8 s, 32.2 s and 32.0 s.
Determine the period.
1. Mean time for 20 oscillations = (31.8 + 32.2 + 32.0) ÷ 3 = 96.0 ÷ 3 = 32.0 s.
2. Period = mean time ÷ number of oscillations = 32.0 ÷ 20.
Answer: period = 1.60 s.
A full answer on finding the period accurately has three parts: time many oscillations (such as 20), divide by that number, and repeat and take the mean.
Repeating a single swing does not remove the reaction-time problem.
- Reading the top of the meniscus, or looking down at it from above. Read the bottom, with your eye level with it.
- Counting "one" as the stopwatch starts. That gives 20 counts for only 19 oscillations.
- Thinking a stopwatch reading to 0.01 s is accurate to 0.01 s. Human reaction time is much larger.
Check your understanding
1. State where on the meniscus you should read the volume of water in a measuring cylinder. [1]
2. Suggest why a light gate gives a more accurate time than a stopwatch for a trolley passing a point. [2]
3. A stack of 50 coins is 9.5 cm tall. Calculate the thickness of one coin in mm. [2]
4. Describe how to find the period of a pendulum as accurately as possible with a stopwatch. [3]
Answers
1. At the bottom of the meniscus, with the eye level with it. [1]
2. The timer starts and stops automatically [1], so there is no human reaction time in the measurement [1].
3. 9.5 cm = 95 mm [1]; 95 ÷ 50 = 1.9 mm [1].
4. Time a large number of oscillations, such as 20 [1]; divide the time by the number of oscillations [1]; repeat and find the mean (or use a marker at the centre of the swing) [1].