Topic 1: Measurements and uncertainties

1.1 – Measurements in physics

Fundamental and derived SI units

There are seven fundamental SI units, but only six are used in the course. The seventh is a measure of luminous intensity, the Candela (cd). You need to memorise these!

Quantity	SI Unit	Symbol
Mass	Kilogram	kg
Distance	Meter	m
Time	Second	s
Electric current	Ampere	A
Amount of substance	Mole	mol
Temperature	Kelvin	K

Derived units are combinations of fundamental units. Some examples are:

• ms^-1 (Unit for velocity)
• N (kgms^-2) (Unit for force)
• J (kgm²s^-2) (Unit for energy)

Scientific notation and metric multipliers

In scientific notation, values are written in the form a×10ⁿ, where a is a number between 1 and 10 and n is an integer. Some examples are:

• The speed of light is 300,000,000 (ms^-1). In scientific notation, this is expressed as 3×10⁸
• A centimeter (cm) is 1/100 of a meter (m). In scientific notation, one cm is expressed as 1×10^-2 m.

The prefixes, abbreviations, and values are given to you in the physics data booklet nevertheless you should try to get familiar with them

Prefix	Abbreviation	Value
peta	P	1015
tera	T	1012
giga	G	109
mega	M	106
kilo	k	103
hecto	h	102
deca	da	101
deci	d	10-1
centi	c	10-2
milli	m	10-3
micro	μ	10-6
nano	n	10-9
pico	p	10-12
femto	f	10-15

Significant figures

For a certain value, all figures are significant, except leading zeros and trailing zeros if this value does not have a decimal point, for example:

• 12300 has 3 significant figures. The two trailing zeros are not significant.
• 012300 has 5 significant figures. The two leading zeros are not significant. The two trailing zeros are significant.
• 4.6340 has 5 signficant figures. The trailing zero is significant.

When multiplying or dividing numbers, the number of significant figures of the result value should not exceed the least precise value of the calculation.

The number of significant figures in any answer should be consistent with the number of significant figures of the given data in the question.

• In multiplication or division, give the answer to the lowest significant figure (S.F.).
• In addition or subtraction, give the answer to the lowest decimal place (D.P.).

Orders of magnitude

Orders of magnitude are given in powers of 10, likewise those given in the scientific notation section previously. They are used to compare the size of physical data.

Distance	Magnitude (m)	Order of magnitude
Diameter of the observable universe	1026	26
Diameter of the Milky Way galaxy	1021	21
Diameter of the Solar System	1013	13
Distance to the Sun	1011	11
Radius of the Earth	107	7
Diameter of a hydrogen atom	10-10	10
Diameter of a nucleus	10-15	15
Diameter of a proton	10-15	15

Mass	Magnitude	Order of magnitude
The universe	1053	53
The Milky Way galaxy	1041	41
The Sun	1030	30
The Earth	1024	24
A hydrogen atom	10-27	-27
An electron	10-30	-30

Time	Magnitude (s)	Order of magnitude
Age of the universe	1017	17
One year	107	7
One day	105	5
An hour	103	3
Period of heartbeat	100	0

Estimation

Estimations are usually made to the nearest power of 10. Some examples are given in the tables in the orders of magnitude section.

1.2 – Uncertainties and errors

Random and systematic errors

Random Error	Systematic Error
Caused by fluctuations in measurements centred around the true value (spread).	Caused by fixed shifts in measurements away from the true value.
Can be reduced by averaging over repeated measurements.	Cannot be reduced by averaging over repeated measurements.
Not caused by bias.	Caused by bias.
Examples: Fluctuations in room temperature, the noise in circuits, human error	Examples: Equipment calibration error such as the zero offset error, incorrect method of measurement

Absolute, fractional and percentage uncertainties

Physical measurements are sometimes expressed in the form x±Δx. For example, 10±1 would mean a range from 9 to 11 for the measurement.

Absolute uncertainty	Δx
Fractional uncertainty	Δx / x
Percentage uncertainty	Δx / x * 100%

Calculating with uncertainties

Addition/Subtraction	y = a±b	Δy=Δa+Δb (sum of absolute uncertainties)
Multiplication/Division	y = a×b or y = a÷b	Δy/y=Δa/a+Δb/b (sum of fractional uncertainties)
Power	y = an	Δy/y=|n|Δa/a (|n| times fractional uncertainty)

Error bars

Error bars are bars on graphs that indicate uncertainties. They can be horizontal or vertical with the total length of two absolute uncertainties.

Uncertainty of gradient and intercepts

Line of best fit: The straight line drawn on a graph so that the average distance between the data points and the line is minimized.

Maximum/Minimum line: The two lines with maximum possible slope and minimum possible slope given that they both pass through all the error bars.

The uncertainty in the intercepts of a straight line graph: The difference between the intercepts of the line of best fit and the maximum/minimum line.

The uncertainty in the gradient: The difference between the gradients of the line of best fit and the maximum/minimum line.

1.3 – Vectors and scalars

Vector and scalar quantities

Scalar	Vector
A quantity that is defined by its magnitude only.	A quantity that is defined by both is magnitude and direction.
Examples: distance, speed, time, energy	Examples: displacement, velocity, acceleration, force

Combination and resolution of vectors

Vector addition and subtraction can be done by the parallelogram method or the head to tail method. Vectors that form a closed polygon (cycle) add up to zero. When resolving vectors in two directions, vectors can be resolved into a pair of perpendicular components.

The relationship between two sets of data can be determined graphically.

Relationship	Type of graph	Slope	y-intercept
y=mx+c	y against x	m	c
y=kxn	log(y) against log(x)	n	log(k)
y=kxn+c with n given	y against xn	k	c