Chapter 1 Physical Quantities, Units and Measurement · 2011-02-10 · Chapter 1 Physical Quantities, Units and Measurement 1.3 Prefixes pageS 1. Discuss with students the convenience

Chapter 1 Physical Quantities, Units and Measurement

Chapter 1 Physical Quantities, Units and Measurement Learning Outcomes

After completing this chapter, students should be able to:

1. show understanding that all physical quantities consist of a numerical magnitude and a unit 2. recall the following base quantities and their units : mass (kg), length (m), time (s), current (A),

temperature (K), amount of substance (mol) 3. use the following prefixes and their symbols to indicate decimal sub-multiples and multiples of the SI

units: nano (n), micro (11), milli (m), centi (c), deci (d), kilo (k), mega (M) 4. show an understanding of the orders of magnitude of the sizes of common objects ranging from a

typical atom to the Earth 5. state what is meant by scalar and vector quantities and give common examples of each 6. add two vectors to determine a resultant by a graphical method 7. describe how to measure a variety of lengths with appropriate accuracy by means of tapes, rules,

micrometers and calipers, using a vernier scale as necessary 8. describe how to measure a short interval of time including the period of a simple pendulum with

appropriate accuracy using stopwatches or appropriate instruments

1.1 Physical Quantities page 3 1. One way to introduce this topic to students is to ask them how they describe the physical world

around them. Take, for example, if the students want to describe the size of their school compound and their answer is 'My school is big ', then the answer is not precise. Exactly how big is big? If it is compared to the compound of a local university, do they still consider it big?

2. To give a precise description of their school, students need to specify the area of the school as a quantity and use a standard unit like square metre.

3. Ask students to name some examples of physical quantities. Classify these quantities into base quantities and derived quantities.

Answer to Think Time question page 4

Base quantities are needed so that all the other quantities can be derived from them. Units are defined to ensure the result of a measurement is meaningful.

1.2 SI Units page 4 1. Emphasise the importance of units to students. For example, if we state that the distance between two

points is 50, then the statement is meaningless when no unit is specified after the numerical quantity. Does it mean 50 mm, 50 cm, 50 m, 50 km or other distances? Students are reminded throughout the course the importance of stating units for the measurements taken.

2. Discuss with students how other quantities are related to the base quantities like mass, length, and time. Examples include speed, acceleration, area, volume and density. As shown in Table 1.1 (page 5), we only need seven main base quantities, five of which - mass, length, time, electric current, temperature and amount of substance - are commonly used.

3. Discuss with students the advantages of using SI units when we are talking about globalisation.


We can have a different system of base units. The Imperial system of foot , pound and second is an example. If there is a widespread use of different units for measurement, it wi ll be very difficult for us to understand each other. We need to have a conversion table wherever we go.

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1.3 Prefixes pageS 1. Discuss with students the convenience and advantages of using prefixes and standard scientific

expressions. To write 45000000 m as 45 Mm, or 4 .5 x 107 m, and 0.000 000 0056 mas 5.6 nm, or 5 .6 x 10 m-9

, is much more convenient and results in less chances of making mistakes because of missing zeros.

2. Ask students to list the prefixes denoting quantities larger than giga (G) and smaller than nano (n).

Factor Name Symbol Factor Name Symbol 1024 yotta Y 10-1 deci d lOLl zetta Z 10-" centi c 10 1~ exa E lO-J milli m 101) peta P 10-0 micro Il lOlL tera T 1O-~ nano n 109 giga G 10-11 pico P 100 mega M 10-15 femto f lOJ kilo k 1O-1~ atto a 102 hecto h 10-21 zepto z 101 deka da

3. Discuss the roles of the Singapore Standard, Productivity and Innovation Board (SPRING Singapore).

Answers to Section Review questions page 7

1. Refer to Table 1.1, page 5 for base 3. (a) 0.760 m (b) 0.0000045 s quantities and their SI units. Use of prefixes (c) 3.2km (d) 1 mm and the standard form (e) 7200000 mm (f) 2500 Ils

2. (a) 5 MJ (b) 48 kg 4. 80 x 109 bytes (c) 0.9 rns (d) 485 kN (e) 7 fls

1.4 Scalars and Vectors page 7 1. Introduce the topic by asking students to perform the following additions:

(a) Water of volumes 200 ml and 300 ml (b) Forces of magnitudes 6 N and 8 N In the case of adding volumes which is straight forward , the answer is 500 mi. In the case of adding forces , however, it is more complicated. Consider the following cases: (i) If the forces are parallel and in the same direction , the answer is 14 N. (ii) If the forces are parallel and in the opposite direction , the answer is 2 N. (ii i) If the forces are in directions other than (i) and (ii), the answer is between 2 Nand 14 N,

depending on the direction of forces. In particular, if the forces are perpendicular to each other, the answer is 10 N.

Addition of vectors perpendicular to one another can be performed using Pythagoras Theorem. In this case, the resultant, R, is given by

R2 = 62 + 82 = 36 + 64 = 100 :.R = 10 r

2. Addition of vectors in other orientation can be done by the graphical method using either the parallelogram or the 'start to end point' method (page 9). For students taking Additional Mathematics, yo u may also like to show them the alternative way to find the resultant using the sine rule and the cosine rule.

3. What is the boy's resultant displacement from P if he walks 40 m due west from P and then another 30 m due south and finally 40 m due east? [Answer: 30 m due south)

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Teachers may wish to illustrate this particular example using a chess board and a counter. Using the counter to represent the boy, place it on one of the boxes of the chess board. Take each square box to represent 10 m and move the counter accordingly. For example, a displacement of 40 m due east corresponds to a movement of 4 squares to the right and so on.


When the forces 4 N and 6 N are not parallel, the resultant force will be between 2 Nand 10 N.


l. (a) Resultant force = 5 N - 3 N 2. Maximum force = 8 N + 5 N = 13 N = 2 N to the east Minimum force = 8 N - 5 N = 3 N

(b) Resultant force = 5 N + 3 N = 8 N to the east 3. P + Q = 13 N ... (1)

(c) Resultant force = J5 2 +52 P - Q = 7 N ... (2) (1) + (2): p= 10 N

= 7.1 N to the northeast .". Q = 3 N (d) The two forces to the right and to the

left cancel each other. Resultant force = 5 N to the north

1.5 Measurement of Length and Time page 11 1. Discuss with students that while we would like to take accurate measurements, errors are inevitable

due to the precision of the instrument.

2. Accuracy is the degree of conformity of a measured quantity to its actual (true) value.

3. Precision is the degree to which further measurements will show the same or similar results . The precision of a measuring instrument is taken as the smallest division of its scale. For example, the metre rule is said to give a precision of 0.1 cm. This means that repeated readings taken will have a difference of ±0.1 cm.

4. The results of a measurement can be accurate but not precise, precise but not accurate, neither precise nor accurate or both precise and accurate. If a result is both accurate and precise, it is a valid measurement.

5. Random error and systematic errors are introduced briefly to convince students that no practical measurement is perfectly accurate. However, these are not required by the syllabus.

6. Discuss with students how a measurement of 2 cm using a metre rule should be written. Should it be written as 2 cm, 2.0 cm, 2.00 cm or 2.000 cm? The correct answer should be 2.0 cm because of the precision of the instrument used. If we are using a pair of vernier calipers or a micrometer screw gauge, then the respective answer should be 2.00 cm and 2.000 cm. Emphasise to students the importance of writing the zeros, if necessary, to indicate that the last figure written is significant with respect to the precision of the instrument (i.e. the instrument can measure the length up to that precision.).

7. Introduce parallax error and show students how to avoid them. To avoid parallax. error, a plane mirror can be used. When the observer's eye is positioned vertically above the object being measured, the image of the object should not be seen, i.e. it should be just out of sight.

8. Ammeters or vo ltmeters with reflecting surfaces below their pointers may be shown to the students . If possible, connect some simple circuits so that students can practice reading the scale with reduced parallax error.

Vernier Calipers Page 12

1. Visit the website: http://www.upscale.utoronto.ca!PVBlHarrisonlVernierNernier.html. The java applet shows how to read a vernier.

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2. Measure a few different objects and show students how to take readings from the vernier calipers.

3. Demonstrate how to measure external diameter, internal diameter and depth using vernier calipers.

4. Demonstrate how to deal with zero error in vernier calipers. Particular attention should be given to dealing with negative zero errors.

5. illustrate how to eliminate zero error by adding or subtracting the zero error from the reading shown on a metre rule (see Table l.7 on page 13).

Micrometer Screw Gauge Page 14

1. Micrometer screw gauge is an important instrument to measure short lengths. Demonstrate how to use the micrometer screw gauge properly.

2. Go through with students the precautions they should take when using a micrometer. Practice measuring the lengths of a few objects using the micrometer.

Answer to Think Time question Page 13

Zero error = -0.02 mm Micrometer reading = l.50 nun Thickness of coil = l.52 mm

Time Page 15

l. Ask students what they understand by the·term "Greenwich Mean Time" (GMT). Answer: GMT is the time at Greenwich, London. Times in the rest of the world are compared to this and said to be a number of hours earlier or later.

2. Ask students, also, how many seconds there are in one year. Answer: Taking one year to have 365 days, there are 31 536 000 seconds in a year.

3. In the experiment on page 16, the amplitude of oscillation must be small so that the motion is simple harmonic. Its period will then be constant. Instead of taking directly the time for just 1 oscillation (i.e. the period), take the time for 20 or more oscillations. This is to reduce the percentage error made when measuring a small quantity. Repeating the measurements allows us to check the reading and also to get a more accurate average value.

Pendulum Clock page 16

Discuss with students how temperature changes in the day and night affect the pendulum of a grandfather clock which, in tum, determines whether the clock runs faster or slower. [The metallic chain of the pendulum expands during the day, when the temperature is higher, causing its length to be longer. During the night, when the temperature is lower, the metallic chain contracts, and the pendulum's length is shorter.]

Answer to Think Time question age 16

For the investigation of the relationship between period and length of a simple pendulum, plot a graph of time against length. The length of pendulum that gives a period of oscillation 1 s can be determined from the graph.

Stopwatch age 17

1. Ask students about the precision of analogue stopwatch (0. 1 s) and digital stopwatch (0.0 I s).

2. Time measured using a stopwatch is recorded to the nearest 0.1 s, although the precision of analogue stopwatch is 0.01 s. This is because we need to consider the systematic error due to reaction time, which is about 0.3 s.

3. To estimate the reaction time Hold a ruler vertically above the fingers of a student. Say 'now' and drop the ruler at the same time. The faster the student catches the ruler, the shorter is his reaction time. The reaction time can be

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Chapter 1 Physical Quantities , Units and Measurement

calculated using the formula s = Y2 gP where s is the distance the ruler falls through, g is the acceleration due to gravity = 9.8 m S- 2 and t the reaction time.

Ticker-tape Timer Page 18

1. The ticker-tape timer is a rather primitive timing device by today's standards. Visit the website http://en.wikipedia.orglwiki/Clock to check out other timing devices.


1. The sundial depended on the constant period of rotation of the Earth. It was quite accurate. The hourglass/sandglass or water clock depended on the rate of the falling sand or water, which is not constant. Hence it is not accurate. However, the time taken for all the sand to fall through the sandglass is constant.

2. This is not a reasonable answer. He is most likely to be using a metre rule or measuring tape. If the measuring tape is marked in millimetres, the precision of the measuring tape is 1 rnm, so he should give the measurement as 1.515 ill.

Physics in Society: The Scientific Method page 20 Answers to Q

1. The scientific method is the process by which scientists study and describe the physical universe through observation, hypothesis, formulation, experimentation, review and revision.

2. The scientific method provides mankind with a logical and systematic way of investigating how nature works. Scientists who use this method learnt to be objective. As they constantly experiment and revise their hypothesis and theories, they expand their boundaries of knowledge. This helps scientists to understand nature better and enable them to make predictions or to control the environment in order to improve life in general.

Answers to Misconception Analysis page 22

1. True 2. False. Base quantities and base units are not

the same. 3. False. Both the base quantities and derived

quantities are physical quantities. 4. False. The SI unit for mass is the kilogram. 5. False. Prefixes are used to express both big

and small numbers.

6. True 7. True 8. False.

9. True 10. True

Zero error can only be eliminated if we know its value.

Answers to Multiple Choice Questions page 22,23

1. B Weight is a derived quantity. 6. B 2. D Change all values to metre before 7. B

comparing 8. C 3. A Maximum force = 7 N + 3 N = 10 N; 9. D Time for one oscillation is from X to Y

minimum force = 7 - 3N = 4N and back to X. 4. A Use the starting point to ending point 10. C The time taken is 5 intervals

method _ I

5. C A pair of vernier calipers has inside =) X 50s = 0.10 s

jaws to measure internal diameter.

Answers to Structured Questions age 23, 24

1. (a) 10 17 s 73600724 7365 = 3 170979 198 yr = 3 x 109 yr (b) 0.08 cm per day x 10-2 x 109 7 3600724 = 9 nm S- I

(c) 70 km h- I = 70 km = 70x 1000 m = 19.4 m S-I

1 h 36005

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(d) 1 g cm-3 = _l_g_ = _---;;-_ _ l X_lO--:-;-3 _k"'-g_-,-_ = 1000 kg m-3

lcm3 (1O-2 m)x(lO-2 m)x(1O 2m)

2. (a) Force and acceleration are vector quantities. (b) (i) The two forces are parallel and in opposite direction.

(ii) The two forces are parallel and in the same direction. (iii) The two forces are acting at an angle 1200 to each other.

3. (a) (6.2 - 1.8) cm = 4.4 cm (b) Height of Mount Everest, 10 000 m

Thickness of paper, 1 x 10-4 m Height of a person, 1.8 m

Radius of Earth, 6000 km Distance from Earth to Moon, 4 x 108 m

4. (a) P, metre rule; Q, vernier calipers; R, micrometer (b) (i) 2.8 cm (ii) 0.14 cm (iii) micrometer

Answers to Critical Thinking Questions page 24

1. (a) 50 cm x 75 cm (b) 8 m x 10 m (c) 15 m x 29 m (d) The minimum length of the football field is 100 m and the width has to be between 64 and 75 m for international matches. The stadium is larger than the football field.

2. (a) 2 g (b) 300 g (c) 600 g (d) 5 kg (e) 1500 kg

3. (a) 105 heartbeats per day (b) 3 x 109 heartbeats in an average lifetime (assuming the average life expectancy is 75 years)

4. This is a bad way to define a unit because the palm size of different people is different, so there is no standardisation.

Extension page 24

l. kilogram The kilogram is the unit of mass; it is equal to the mass of the international prototype of the kilogram.

metre The metres the length of the path travelled by light in vacuum during a time interval of 11299 792 458 of a second.

second The second is the duration of 9 192 631 770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium 133 atom.

ampere The ampere is that constant current which, if maintained in two straight parallel conductors of infinite length, of negligible circular cross-section, and placed I metre apart in vacuum, would produce between these conductors a force equal to 2 x 10-7 newton per metre of length.

kelvin The kelvin, unit of thermodynamic temperature, is the fraction 11273.16 of the thermodynamic temperature of the triple point of water.

mole The mole is the amount of substance of a system which contains as many elementary entities as there are atoms in 0.012 kilogram of carbon 12.

2. Refer to the following websites for estimating the Moon's distance: http://www-spof.gsfc.nasa.gov/stargaze/Shipprc2.htm http://www.uh.edu/engines/epiI457.htm

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Chapter 1 Physical Quantities, Units and Measurement · 2011-02-10 · Chapter 1 Physical Quantities, Units and Measurement 1.3 Prefixes pageS 1. Discuss with students the convenience