By The End Of The Subtopic Learners Should Be Able To;
  1. measure physical quantities
  2. read an instrument scale to the nearest fraction of a division
  3. express quantities in terms of S.I. units and derive other units from base units
  4. determine density of regular and irregular objects

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  • It is the continued scrutiny of everything we know about the world around us that leads people to the lifelong study of physics.
  • This section is the foundation that will equip you with the basic tools required in the communication of physics and the world in general.

1.1 Measurements

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Physical quantities

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4. Fig.1.1.1_forcemeter.jpg (57 KB)
4. Fig.1.1.1_volumeter.jpg (63 KB)
4. Fig.1.1.1_thermometer.jpg (52 KB)
4. Fig.1.1.1_micrometer.jpg (58 KB)
4. Fig.1.1.1_ammeter.jpg (62 KB)
4. Fig.1.1.1_digital balance.jpg (71 KB)
4. Fig.1.1.1_vernier caliper.jpg (54 KB)


  • The kilogram, the second and the metre are three of the seven base units. Other units are derived from these base units
  • S.I. units (Systeme Internationale d’Unites) are the internationally agreed units used in science.
Table 1.1.1 Example of S.I. units  
Physical quantity Name of S.I. unit Symbol for unit
Length Metre M
Mass Kilogram Kg
Time Second S

When writing units note the following:

  1. Always use the agreed abbreviation or write the words in full, for example, second or s.
  2. The symbol is not to be a capital letter unless it is named after a person, for example, 10N (N named after Sir Isaac Newton but when writing in words do not use a capital letter- 10 newtons.)
  3. Do not add an 's' to the symbol to make it a plural, for example: 8 kilograms = 8kg

Prefixes of the metric units

  1. A prefix represents the multiple of a unit.

Example 1.1.1:

1000W = 1kW

k is the prefix (kilo-); k = 1000

Example 1.1.2:

0.001g = 1mg

m is the prefix (milli-); m = 11000 = 0.001

Name symbol Factor Scientific notation
tera- T 1 000 000 000 000 1012
giga- G 1 000 000 000 109
mega- M 1 000 000 106
kilo- K 1 000 103
hecto- H 100 102
deca- Da 10 101
deci- d 0,1 10-1
centi- c 0,01 10-2
milli- m 0,001 10-3
micro- µ 0,000 001 10-6
nano- n 0,000 000 001 10-9
pico- p 0,000 000 000 001 10-12

    • Scientific notation or standard form is a system of writing numbers using the powers of ten.
    • This system makes it easy to read a number which has a lot of zeros for example:

3 450 000 000 =3.4×109 m

Using measuring instruments

1. Vernier caliper

    • The Vernier caliper is used to measure length precisely to 1/10 mm (0.1 mm) or 1/100 cm (0.01 cm).
    • When reading the measurement from the Vernier caliper,

a. Read the fixed scale first from the zero mark of the Vernier scale.

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b. Look at where the divisions on the Vernier scale coincide with that on the main scale.
Reading: 4 on the main scale coincides with the 6 on the sliding scale

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c) Add the reading from the main scale to that of the Vernier scale (remember the Vernier scale is a 1/100 cm)

From Fig 1.3 and Fig 1.4:
Reading = 1.30 cm + 6/100 cm
= 1.36 cm

9. Fig. 1.1.4.jpg (107 KB)

2. Micrometer screw gauge

  • This instrument measures to the nearest 1/100 mm (0.01 mm) or 1/10 cm (0.1 cm).
  • When reading measurements from the screw gauge;
  • Hold the object being measured between the gap of the anvil and spindle by screwing using the ratchet.
10. Fig. 1.1.5.jpg (205 KB)
  • Read the value from the fixed scale (sleeve) at the edge of the barrel first.
11. Fig 1.1.6.jpg (115 KB)
  • Read the value on the barrel (revolving scale) that is directly on the centre line of the sleeve scale.
  • To find the measurement of the object, add the value on the sleeve with the value on the barrel. Measurement = 3.5 mm + 0.31 mm

    = 3.81 mm

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Derived S.I. units

  1. Units such as newtons, joules and volts are derived from base units.
  2. There are seven fundamental base units of measurements and all other S.I. units are derived by multiplying, dividing or powering the base units.

Example 1.1.3: Deriving the units of area

To find the area (A) of this rectangle you multiple its length (L) with its width (W).

A = L x W

= 5 m x 3 m

= 15 m2

The units are also multiplied together to give m2

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Deriving the unit of force:

  • Force is a push or a pull and the units for force are the newtons.
  • Force can also be describe by its effect; it causes a mass of an object to change its velocity (acceleration) hence.
  • F = m x a

    F is in N (a derived unit),m is in kg (a base unit),

    a = VelocityTime

    time is in s (a base unit),

    velocity = DistanceTime

    distance is in m (a base unit).


    N = kg x 1s x ms

    N = kgms-2

      1. When deriving units, the base units can be multiplied, divided or powered but they cannot be added or subtracted.
      2. When deriving units for volume,

volume = L x w x h
(length, width and height are all in centimetres, therefore you multiply the units together)
= cm x cm x cm
= cm3

  • When finding the perimeter of a square, you add the four sides together, but you do not add the units together.

For example, for a square measuring 2 m each side;

perimeter for the square = 2 m + 2 m + 2 m + 2 m

= 8 m [not 8 m4]

Measuring small quantities

The leaky tap

George has a continuously leaking tap in his kitchen. He is trying to figure out the time it takes for one drop of water to move from the tap and hit the base of the kitchen sink.

Here are the challenges he faces

1. The drop is moving too fast between the tap and base of the sink. (Distance too short)

2. As soon as he starts the stop watch (that is when the drop leaves the tap), he has to stop it (that is when the drop makes the sound as it hits the base of the sink). This gives him a ridiculous time interval due to how fast he reacts between starting the stop watch and stopping it (his reflex action).

How would you help George get an accurate time interval for the drops?

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  • In the laboratory we use the stopwatch to measure small fractions of time.
  • Stopwatch can measure up to 15 or 1100 of a second.
  • Accuracy of measuring time interval depends on the:
    1. Clock.
    2. Human reaction time.
  • When timing a regular (continuous) short event, accuracy can be increased by taking multiple readings

Activity: measuring small time interval.

  • In the laboratory a simple pendulum can be used to study continuous short time interval.
  • The setup is shown in Fig. 1.1.8 below:
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  1. Start with the length of 50cm.
  2. Set the pendulum swinging to an angle of no more than 100.
  3. Sit in front of the pendulum with eye at level with the bob and pointer.
  4. As the bob passes the pointer start the stopwatch, count 20 oscillations and then stop the watch.

Oscillation is a complete cycle/swing (to- and fro- motion) from the diagram below, Fig.1.1.9. One complete oscillation is done when the bob swings from A to B, then to C through A and finally back to A.

  • The time taken for one complete oscillation is obtained by dividing time taken for the twenty oscillations by 20.
  • Using 20 oscillations reduces the error of reaction time.
  • Repeat the same procedure so as to obtain the average time for one oscillation.
16. Fig. 1.1.9.png (8 KB)
  • The time taken for one complete oscillation is obtained by diving time taken for the twenty oscillations by 20.
  • Using 20 oscillations reduces the error of reaction time.
  • Repeat the same procedure so as to obtain the average time for one oscillation.
  • The time taken to make one complete oscillation is called the period and is given the symbol T.
  • Varying the length l of the pendulum will also change  of the pendulum.

Example: (Alternative to practical question, on measuring period of a pendulum)

A physics student is investigating the oscillation of a pendulum.
The apparatus is set up as shown in Fig. 1.1.10

18. Fig. 1.1.10 and Fig. 1.1.11.png (10 KB)

She proceeds as following:

  1. h is the height of the pendulum bob above the bench, it is measured and recorded.
  2. She repeats for five different values of h, by shortening the string of the pendulum but without changing the height of the clamp.
  3. For each value of h, the bob is pulled to one side by a small distance, as shown in Fig. 1.1.11. And the time t for 10 complete oscillations is measured and recorded.
  1. Describe a precaution which the physics students might have taken in order to measure h as accurately as possible. You may draw a diagram.
    Take reading with eye line perpendicular to rule / use set square to ensure rule is vertical
  2. Fig. 1.1.12 to Fig1.1.16 are scale diagrams showing the height h of the pendulum bob above the bench for each of the five experiments.
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(i) The diagrams are drawn to 1/5 scale.
Calculate, and record in Table 1.1.4, the actual heights H of the pendulum bob above the bench. [2]

Table 1.1.4

(c) (i) For each value of height h, calculate the time T for one complete oscillation, using the equation T = t10.

Record these values in Table 1.1.4.

(ii) Calculate the values of T2 and record these in the table. [1]

(d) Plot a graph of T2 / s2 (y-axis) against H / cm (x-axis).

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(e) Determine the gradient G of the graph.

Show clearly on the graph how you obtained the necessary information. [1]

G = (1.96 - 0.80)s2 ÷ (10 - 40)cm

= - 0.04 s2/cm

Volume and density

  • Volume: this is the space occupied by matter or substance
  • S.I. units for volume are cm3
  • Volume for regular shaped solid objects can be calculated as shown in the Fig 1.8 below:
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22. Fig 1.1.17_2.jpg (60 KB)
22. Fig 1.1.17_3.jpg (58 KB)
Fig 1.1.17: Formulae for common regular shapes.

  • Volume for liquids can be measured using a measuring cylinders.
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Measurements and Errors

  • In physics experiments (and other sciences), we aim to obtain results that are both accurate and precise, but in all physical measurements errors occur.
  • When a measurement is used in a calculation, the error in the measurement is carried through into the result.
  • They are two types of errors:
    1. Systematic errors
      • These errors are associated with a fault in the equipment or in the design of the experiment.
    2. Random errors
      • These errors usually result from the experimenter's inability to take the same measurement in exactly the same way to get the same number.
  • Example of a systematic error can be due to:
    1. Zero error - occurs when a measuring instrument fails to return or read the zero mark.
    2. Inaccurately calibrated instruments.
  • Example of random errors can be due to:
    1. Parallax error - not reading from the appropriate eye level.
    2. Careless in making reading or recording.
    3. Wrong techniques.

Zero error

  • Consider the following voltmeters that are not connected to a circuit. They should be reading zero but both have different types of zero error.
24. Fig. 1.8a.jpg (73 KB)
24. Fig. 1.8b.jpg (75 KB)

  • For such an error the experimenter needs to consider the value of the error in their final reading.
  • For the positive zero error, the error margin is subtracted from the final reading.
  • For the negative zero error, the error margin is added to the final reading.

Example 1.4

The following calipers have zero errors as indicated in each case. Find the actual reading for each caliper.

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Zero error = +0.02 cm Zero error = -0.04cm


  1. Actual reading = caliper reading - positive zero error
    = (1.0 cm + 0.5 cm) - 0.02 cm
    = 1.03 cm
  2. Actual reading = caliper reading + negative zero error
    = (1.3 cm + 0,03 cm) + 0.04
    = 1.37 cm

Experiment 1.1.1: Volume of irregular shaped body

The following methods can be used to find the volume of irregular shaped bodies.

1) Using the displacement can:

27. Fig 1.1.19.jpg (114 KB)
  • Placing the stone in water as shown in Fig 1.9, will cause the level of water to rise in the displacement can.
  • The water will spill over the spout into the measuring cylinder.
  • The water collected in the measuring cylinder (displaced by the stone), is equal in volume of the stone.

2) Placing the object in a measuring cylinder:

28. Fig 1.1.20.jpg (119 KB)
  • Placing the stone in water will cause the level of the water in the measuring cylinder to rise as shown in Fig 1.10.
  • The difference between the initial reading (before placing stone in water) and final reading (after placing the stone in water) will give you the volume of the stone.


  • This is the mass per given volume of substance or material.
  • The density of a material is calculated using the formula:

Density = massvolume

ρ = mv

  • From the formula, the units can be deduced; g/cm3 (or kg/m3).
  • Different materials have different densities, but density is the same (does not change) for the same material. For example, the density of pure water is 1 g/m3 and this is constant regardless of the volume and mass used.

Finding the density of a liquid:

  • Find the mass of an empty measuring cylinder.
  • In the measuring cylinder place a known volume of the liquid.
  • Place the measuring cylinder with the liquid on the scale as shown on the Fig 1.11.
  • To calculate the density divide the mass of the liquid by its volume.
  • mass of liquid = mass of measuring cylinder with liquid - mass of empty measuring cylinder
31. Fig 1.1.21.jpg (79 KB)

Float or sink

  • Density can be used to explain why some objects sink or float in water.
  • If an object has a density less than that of water, it will float.
  • If the object is denser it will sink.
  • The diagrams below show how some objects due to a density lower than water are able to float on water.
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33. Fig. 1.1.23.jpg (137 KB)
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