School Science Lessons
Physics - Measurement
Updated: 2008-06-25
Please send comments to: J.Elfick@uq.edu.au
See also: Interesting websites
Table of contents
7.0 International system of units (SI), the 7 base units
2.0.1 Right-angled triangle
2.0.2 Golden mean, golden ratio
2.0.3 Greek alphabet, phonetic alphabet
2.0.5 Conic sections, parabola. ellipse,hyperbola
2.0.6 Parabola equation
2.0.7 Change scale of a map
2.0.8 Mathematics for science teachers
2.0.9 Handspans, trundle wheel
3.2.2 Rank scaling tables
2.1.6 Volume of liquid
3.2.2 Rank scaling tables
3.3.0 Accuracy and error
3.1: Measurement for primary science, estimation (Primary)
6.22 Pendulum tells the time (Primary)
3.3.7 Table of numerals adding vertically or horizontally or diagonally to 33
3.3.0 Accuracy and error
3.3.1 Significant digits and standard form, scientific notation
3.3.2 Order of magnitude (nearest power of ten, a factor or factors of ten)
3.3.3 Factors that affect readings, obtain data from the equipment
3.3.4 Record measurements in tables
3.3.5 Graph measurements
3.3.6 Graph the speed of a car
3.1.0 Counting, numerals, Roman numerals
3.2.0 Mass and weight
3.3.0 Length (l), the kilometre (km), Angstrom unit, nanometre, Astronomical unit, AU
3.4.0 Area, square metre (m2), hectare
3.5.0 Volume (vol.), cubic metre (m3),
3.6.0 Estimating
3.7.0 Ratio and proportion, concentration, degrees proof
3.8.0 Angle, degree, arc minute, arc second, radian
3.9.0 CGS units (centimetre, gram, second)
3.10.0 The m.k.s. units
3.11.0 Imperial units used in land surveying (1 hectare = 10,000 m2, 1 kilometre = 1,000 m)
3.12.0 SI, CGS and FPS conversion, metric conversion
3.13.0 Energy conversion KJ, MJ, KWh, therm, BTU, calorie, horsepower
3.14.0 Oven temperatures
3.15.0 Measurement lessons for primary schools
3.1 Measurement for primary science
1.14 Mark our height
1.19 Length game
1.20 Pace distances
1.22 Compare different shapes
1.23 Make new shapes
1.25 Pouring water game
1.44 Area game
2.14 Measure in hand spans
2.16 Compare our weights
2.22 Copy with a rubber band
3.21 Trundle wheel
2.23 See-saw balance
2.24 Steelyard balance
2.25 Make a ruler balance
2.26 Balance bottle tops
2.27 Nail balance
2.28 Beam balance
2.29 drinking straw balance
3.18 Volume of our fist
3.22 Throw up and fall down
3.23 Volume of a liquid
4.14 Length game
4.15 Pace distances
4.17 Shapes game
4.18 Diameter of a thread
6.21 Estimating
6.22 Pendulum tells the time
3.3.0 Measurement and errors, error and accuracy
3.3.1 Significant figures and standard form, scientific notation
1. Significant figures are all the figures that can be read with meaning from an instrument. Significant figures of a number are the digits that contribute to its value. For measurement, the significant figures are those you know with certainty plus the digit that is uncertain. A "2 tonne truck" could weigh between 1.5 and 2.5 tonnes. A reading of 25 cm could have a value between 24.5 and 25.5 cm. So you say that the last digit is uncertain. You count zeros between integers and zeros to the right of the decimal point following non-zero integers. You do not count other zeros. The following examples each have four significant figures:
0.01234
0.1023
0.1230
In the last case you are saying that the reading is closer to 0.1230 than 0.1229 or 0.1231. So be careful about zeros, especially the last zero.
2. If rounding off to 3 significant figures:
4.657 becomes 4.66 because 7 > 5.
4.655 becomes 4.66 because last digit is 5 and digit behind it is odd.
4.645 becomes 4.64 because last digit is 5 and digit behind it is even.
4.654 becomes 4.64 because 4 < 5.
3. When adding or subtracting, all numbers must have the same number of digits after the decimal point. This is equal to the least number of digits after the decimal point of any number in the addition or subtraction.
19.43 + 6.456 + 101.9 becomes 19.4 + 6.5 + 101.9 =127.8
4. When multiplying or dividing numbers, the answer can have only as many significant figures as the number with the least number of significant figures. 17.9 X 4.3 = 76.97 Answer = 77 (4.3 has only 2 significant figures)
5. Standard form or scientific notation expresses a number as a product of a number between 1 and 10 and a power of 10. It is a convenient way to express large and small numbers for easy comparison and it can show the number of significant figures. So you can write 18,000 as 1.8 X 104 (2 significant figures, i.e. the value is between 1.7 and 1.9 X 104), or 1.80 X 104 (3 significant figures, i.e. the value is between 1.79 and 1.81 X 104). Express decimal fractions in standard form: 0.1 = 1 X 10-1, 0.019 = 1.9 X 10-2
Standard form (scientific notation), e.g. 8.04 X 102, shows the significant figures expressed unambiguously. The coefficient, 8.04, must be greater than or equal to 1 and less than 10. The base number 10 is written in exponent form, so in 8.04 X 102, the number 2 is the exponent or power of ten.
Express decimal fractions in standard form, e.g.
0.1 = 1 X 10-1
0.2 = 1 X 10-2
0.019 = 1.9 X 10-2
0.00087 = 8.7 X 10-4
3.3.2 Order of magnitude (nearest power of ten, a factor or factors of ten) +
Order of magnitude is a value expressed to the nearest power of ten. Sometimes you are interested in knowing the approximate rather than the precise values, so you just use the nearest power of ten, e.g. speed of light: 3.0 X 108 ms-1 = (approx.) 108 ms-1, the radius of the Earth: 6.38 X 106 m = (approx.) 10 X 106 m= 107 m, the radius of the Moon = 3.8 X 108 = (approx.) 109 m! (3.8 is closer to 100 (1) than to 101 (10), 3.8 is greater than 10 ½ = 3.14).
3.3.3 Factors that affect readings, obtain data from the equipment
3.3.3.1 Relative positions between measured object and equipment +
When you read on a scale with a measured object directly touching with the equipment, you must be careful as their relative position will probably affect precision of your readings. For example, if you measure temperature of liquid by a thermometer, you must immerse completely the measuring bulb in the liquid as you take readings.
3.3.3.2 Reaction time of the equipment
Some equipment reacts to measured quantities very quickly, such as meters for measuring electricity. However, some equipment needs a certain reacting time, such as a mercury thermometer. So you must take readings after the equipment stabilizes. Even with equipment that reacts quickly you need to pay attention to such problems, e.g. when measuring electric potential, be certain that the pointer no longer moves before you read from the scale.
3.3.3.3 Line of vision
The angle between your line of vision and the object referred to can cause errors. Your eye should be at right angles to the scale and directly opposite the part of the scale you are reading. Reading a scale from the left side or the right side or above or below are all wrong because they result in parallax error.
3.3.4 Record measurements in tables
Set up a table vertically if there is a possibility of additional requiring some extra space. Include a title and table number on the top of a table to state what data the table contains. The first column should contain data for the independent variable rather than the dependent variable. The weight is the independent variable because you decide its values, usually before doing the experiment. The increase in length of spring is the dependent variable because it depends on the weight added. Express all data in standard form.
Increase in length of spring. (Original length = 28.0 cm)
Weight
(N)		 Length of spring
(cm)		 Increase in length
(cm)
0.49 (0.5 kg) 32.8 4.8
0.98 (1 kg) 36.3 8.3
1.47 (1.5 kg) 39.4 11.4
1.96 (2.0 kg) 41.9 13.9
3.3.5 Graph measurements
A graph is a drawing that shows the relationship between variables.
Terminology: Axes of a graph, co-ordinates of a position on a graph, independent variable, dependent variable, line of best fit, area "under" a curve, interpreting graphs, linear graphs, gradient (slope of a graph), intercepts on a graph.
The graph of Y varies directly with X, e.g. weight on spring.
The graph of Y varies inversely with X, e.g. PV of gas.
The graph of Y varies directly with X2, e.g. acceleration.
3.3.5.1 Select scales
Select the scale of the axes to make the shape of the graph display the relation between data. The starting a point of the co-ordinate axis does not have to begin with zero and the scales of the two axes need not be the same. The variable you set up is the independent variable and is placed on the horizontal axis, the x axis. The variable that results from the independent variable is the dependent variable and is placed on the vertical axis, the y axis. If you investigate the cooling of a bucket of water, time is the independent variable and temperature of the water is the dependent variable. When you say "Graph speed against time" or "Draw a velocity time graph", then "time" is the independent variable because you have mentioned it after the dependent variable "speed". You show the position of any plotted point as (XY).
3.3.5.2 Plot points and draw by hand
See diagram 2.0.4.1: Drawing a graph
When plotting points in the co-ordinate system by hand the symbols may be small dots surrounded by a circle or a thin cross shape. In the diagram, the computer using "brush" from the Windows XP Paint program has generated the symbols. When you have two graphs in one co-ordinate system, different symbols should express the points in different graphs. Do not graph if less than 6 points. Draw the graph by using the inner drawing method so that your wrist that is a centre to turn around in forming a smooth graph. Check the points that are far from the graph because measuring them again may be necessary. A dotted line should express a graph that you have deduced to distinguish from the graph obtained from experiment.
3.3.6 Graph the speed of a two cars
See diagram 2.0.4.2: Speed of two cars
Suppose you mark a straight road every 10 metres and can use a stopwatch to record when a car reaches each mark. The following table shows your data for 2 cars, car A and car B. In the graph the points for Car A are almost in a straight line. You can say that the line of best fit is a straight line. However, the graph line does not go exactly through each point because some experimental error can occur when reading the stopwatch, recording the data and plotting the graph. However, if you assume that the graph line is properly straight then you can say that each quantity is proportional to the other, distance = speed (velocity) X time, d = vt. Car A was moving with constant speed 4.2 m / s. Estimate how far Car A had moved after 8 seconds, by interpolation = 33 m. See the P on the graph. Estimate how far Car A had moved after 8 seconds, by calculation, d = vt, d =4.2v X 8 t = 33.6 m.
Car A
Distance
(m)		 Elapsed time
(seconds)		 Speed
(m / s)
0 0 0
10 2.3 4.3
20 4.9 4.1
30 7.1 4.2
40 9.7 4.1
50 12.0 4.2
- - Average speed = 4.18 = 4.2
In the graph the points for Car B are not in a straight line. The line of best fit is a curve so the speed is constantly changing. The graph shows the method of calculating the instantaneous speed at two distances 15 m and 35 metres. Draw a tangent to the point on the graph corresponding to the distance. Construct a right angle triangle with the tangent as hypotenuse then read the corresponding values for for distance and time from the two sides of the triangle then calculate the speed, v = d / t. At 15 m, the instantaneous speed was s / t, 10 d / 3.2 t = 3.125 m / sec. = 3.2 m / sec. At 35 m, the instantaneous speed was s / t, 10 d / 7.8 t = 6.25 = 6.2 m / sec.
Car B
Distance
(m)		 Elapsed time
(seconds)		 Instantaneous speed
(m / sec.)
0 0 -
10 4.0 (15 m, 3.2 m / sec
20 7.0 -
30 8.9 (35 m. 6.2 m / sec)
40 10.5 -
50 12.0 -

3.3.7 Table of numerals adding vertically or horizontally or diagonally to 33
1
 14
 14
 4
11
 7
 6
 9
8
 10
 10
 5
13
 2
 3
 15