Chapter 1: Physical Quantities, Units and Measurement

Candidates should be able to:

show understanding that all physical quantities consist of a numerical magnitude and a unit
recall the following base quantities and their units: mass (kg), length (m), time (s), current (A), temperature (K), amount of substance (mol)
use the following prefixes and their symbols to indicate decimal sub-multiples and multiples of the SI units: nano (n), micro (µ), milli (m), centi (c), deci (d), kilo (k), mega (M), giga (G)
show an understanding of the orders of magnitude of the sizes of common objects ranging from a typical atom to the Earth
state what is meant by scalar and vector quantities and give common examples of each
add two vectors to determine a resultant by a graphical method
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
describe how to measure a short interval of time including the period of a simple pendulum with appropriate accuracy using stopwatches or appropriate instruments

Physical quantities consist of:
1. Numerical magnitude – denotes the size of the physical quantity.
2. Unit – denotes the physical quantity it is expressing.
Physical quantities can be classified into:
1. Basic quantities
  Basic Quantity Name of SI Unit SI Unit
  length metre m
  mass kilogram kg
  time second s
  thermodynamic temperature kelvin K
  amount of substance mole mol
2. Derived quantities – defined in terms of the basic quantities through equations. SI units for these quantities are obtained from the basic SI units through the equations.
  
  Example 1.1

Physical quantities can be very large, like 23 150 000 000 m, or very small, like 0.000 000 756 m. Writing down such numbers can be time consuming and error-prone. We use prefixes to indicate decimal sub-multiples and multiples of the SI units to make writing such numbers easier.

Some prefixes of the SI units are as follows:

The ones in bold are specifically required in the syllabus.
Example 1.2

When measurements are too large or too small, it is convenient to express them in standard form as follows:
M × 10^N
M lies in the range of: 1 ⩽ M < 10
N denotes the order of magnitude and is an integer.
Orders of magnitude are often being used to estimate numbers which are extremely large to the nearest power of ten.
E.g.
1. Estimate the number of strands of hair on a person’s head.
2. Estimate the number of breaths of an average person in his lifetime.
The following tables show how the orders of magnitude are used to compare some masses and lengths.
Mass/kg Factor
Electron 10^-30
Proton 10^-27
Ant 10^-3
Human 10¹
Earth 10²⁴
Sun 10³⁰

Length/m Factor
Radius of a proton 10^-15
Radius of an atom 10^-10
Height of an ant 10^-3
Height of a human 10⁰ (10⁰ = 1)
Radius of the Earth 10⁷
Radius of the Sun 10⁹
Example 1.3
Find the ratio of the height of a human to that of an ant.
Ratio of height of human to that of an ant = 10⁰/10^-3 = 10³ = 1000.

A scalar quantity – has only magnitude but does not have direction.
E.g. mass, distance, time, speed, work, power.
A vector quantity – has both magnitude and direction.
E.g. weight, displacement, velocity, acceleration, force.
Example 1.4
The velocity of a particle can be stated as: “speed of particle = 2.0 m/s and it is moving at an angle of 30° above the horizontal”.

1. Involves magnitude and direction.

Example 1.5

Find the resultant force R at point P due to F₁ and F₂.

Method 1: Trigonometric Method

Method 2: Graphical Method

(Not drawn to scale)

Step 1: Select an appropriate scale

E.g. 1 cm to 2 N.

Step 2: Draw a parallelogram of vectors to scale.

Step 3: Measure the diagonal to find R.

Step 4: Use the protractor to measure angle θ.

Choice of instrument depends on the degree of accuracy required.

How parallax errors can occur during measurement:
1. incorrect positioning of the eye

1. the object is not touching the marking of the scale (for measuring tape and metre rule, ensure that the object is in contact with the scale)

A measuring instrument can give precise but not accurate measurements, accurate but not precise measurements or neither precise nor accurate measurements.
1. Precision is how close the measured values are to each other but they may not necessarily cluster about the true value. Zero errors and parallax errors affect the precision of an instrument.
2. Accuracy is how close a reading is to the true value of the measurement. The accuracy of a reading can be improved by repeating the measurements.
Vernier calipers
A pair of vernier calipers can be used to measure the thickness of solids and the external diameter of an object by using the external jaws. The internal jaws of the caliper are used to measure the internal diameter of an object. The tail of the caliper is used to measure the depth of an object or a hole. Vernier calipers can measure up to a precision of ±0.01 cm.