1. Physical Quantities and Units

Section 1 of 17Physical quantities and estimation

Physical quantity

• Physical quantity a measurable property that is always written as a numerical magnitude together with a unit.

Symbols need context

• Symbols need context the same letter can stand for different quantities, so the unit shows what it means.
• v with unit m s⁻¹ is velocity.
• v with unit m3 is volume.
• V with unit V is potential difference.

Estimation

• Estimation you should know typical sizes so you can make order-of-magnitude estimates.

Quantity	Typical size
Diameter of an atom	10−10 m
Wavelength of UV radiation	10 nm
Height of an adult human	2 m
Earth to Sun distance (1 AU)	1.5 × 1011 m
Mass of a hydrogen atom	10−27 kg
Mass of an adult human	70 kg
Mass of a car	1000 kg
Seconds in a day	90 000 s
Seconds in a year	3 × 107 s
Speed of sound in air	300 m s−1
Power of a light bulb	60 W
Atmospheric pressure	1 × 105 Pa

Section 2 of 17How to make an estimate

Method

• Method recall the equation that links the quantities, estimate each input from everyday experience, then substitute.

Scope

• Scope only familiar quantities are estimated, such as a walking speed of 1 to 1.5 m s⁻¹.

Range

• Range a range of sensible values is acceptable, and many constants are in the data booklet.

Section 3 of 17SI base units

• SI base units
  • every other unit is built from a small set of base units, the only system of measurement officially used almost everywhere.
  • Of the seven SI base units, five are required for this course.

Quantity	SI base unit	Symbol
Mass	kilogram	kg
Length	metre	m
Time	second	s
Current	ampere	A
Temperature	kelvin	K

Section 4 of 17Derived units in base units

Derived unit

• Derived unit a unit such as the newton, joule or pascal that is a combination of base units.

Method

• Method write the defining equation of the quantity, then replace each quantity by its base units.

Newton

• Newton force = mass × acceleration, so N = kg m s⁻².

Joule

• Joule energy = ½ × mass × velocity2, so J = kg (m s⁻¹)2 = kg m2 s⁻².

Pascal

• Pascal pressure = force ÷ area, so Pa = kg m s⁻² ÷ m2 = kg m⁻¹ s⁻².

Section 5 of 17Homogeneity of equations

• Homogeneous equation
  • an equation in which the base units on both sides are identical.
  • Every correct physical equation must be homogeneous, so a unit check can expose a wrong equation.
  • A constant with no units is called dimensionless.
  • Write the base units of every term on the left side.
  • Write the base units of every term on the right side.
  • Compare them; if they differ, the equation needs to be corrected.

Section 6 of 17Powers of ten and prefixes

• Range of sizes
  • an atom is about 10⁻¹⁰ m across and a galaxy about 1021 m, a difference of 31 powers of ten.
  • Getting a prefix wrong is a very common exam mistake, so learn them carefully.

Prefix	Symbol	Power of 10
tera	T	1012
giga	G	109
mega	M	106
kilo	k	103
centi	c	10−2
milli	m	10−3
micro	µ	10−6
nano	n	10−9
pico	p	10−12

Section 7 of 17Random and systematic errors

Random and systematic errors

True value

• True value measurements aim to find it, but it can never be known exactly, so every result has an uncertainty.

Random error

• Random error unpredictable scatter in readings from uncontrollable factors such as the environment.
• It affects precision by spreading results about the mean.
• It is reduced by repeating readings and averaging.

Systematic error

• Systematic error a repeated offset caused by a faulty instrument or a flawed method.
• It affects accuracy because every reading is shifted the same way.
• It is reduced by recalibrating the instrument or correcting the technique.

Zero error

• Zero error a systematic error where the instrument reads a non-zero value when the true value is zero.

Example: a balance showing 0.200 g with nothing on it; this offset must be corrected in every result.

Section 8 of 17Precision, accuracy and resolution

Term	Meaning	Affected by
Precision	How close repeated readings are to each other; a single reading cannot be precise	Random error
Accuracy	How close a reading is to the true value	Systematic error
Resolution	Smallest change the instrument can show; many decimal places mean high resolution only	The instrument

Precision, accuracy and resolution

Readings can be precise but inaccurate, for example when 0.5 is added to every value.

Averaging repeated readings improves accuracy when only random error is present.

Section 9 of 17Absolute, fractional and percentage uncertainty

Uncertainty

• Uncertainty a range around a measurement within which the true value is expected to lie;
  • it is an estimate and not the same as an error.

Example: readings of 12.55, 12.45 and 12.51 cm give 12.50 ± 0.05 cm.

Absolute uncertainty

• Absolute uncertainty a fixed amount with units, such as ± 0.05 cm.

Fractional uncertainty

• Fractional uncertainty the uncertainty divided by the measured value.

Percentage uncertainty

• Percentage uncertainty absolute uncertainty ÷ measured value × 10⁰%.

Section 10 of 17Estimating the uncertainty of a reading

Situation	Uncertainty to quote
Single reading on an analogue scale, such as a voltmeter	± half the smallest division
Measurement with a ruler (two ends read)	at least ± 1 smallest division
Repeated data	half the range = ½ (largest − smallest)
Digital meter	± the last significant digit unless stated

Section 11 of 17Combining uncertainties

Operation	Rule
Add or subtract values	Add the absolute uncertainties
Multiply or divide values	Add the percentage (or fractional) uncertainties
Raise to a power n	Multiply the percentage uncertainty by n

Section 12 of 17Scalars and vectors

Scalar

• Scalar a quantity with magnitude only.

Vector

• Vector a quantity with magnitude and direction, drawn as an arrow whose length shows the magnitude.
• Distance (scalar) is the total path length;
  • displacement (vector) is the straight-line separation with direction.
• Speed (scalar) is distance per unit time; velocity (vector) is displacement per unit time.
• An object moving in a circle at constant speed has a changing velocity because its direction changes.

Scalars	Vectors
distance, speed, mass, time	displacement, velocity, acceleration
energy, volume, density	force, momentum
pressure, electric charge, temperature |

Section 13 of 17Adding and subtracting vectors

Resultant

• Resultant the single vector with the same effect as the vectors being combined.

Triangle method

• Triangle method draw the vectors head to tail, then join the tail of the first to the head of the last.

Parallelogram method

• Parallelogram method draw the vectors tail to tail, complete the parallelogram, and take the diagonal.

Subtraction

• Subtraction reverse the direction of the vector to be subtracted, then add.

Equilibrium

• Equilibrium coplanar forces in equilibrium form a closed vector triangle.

Magnitude and direction

• Magnitude and direction use Pythagoras for the size and tan θ = opposite ÷ adjacent for the angle.

Section 14 of 17Resolving vectors

• Components
  • one vector can be replaced by two perpendicular vectors that together have the same effect.
  • For a vector F at angle θ to the horizontal, the horizontal component is Fx = F cos θ.
  • The vertical component is Fy = F sin θ.
  • Resolving makes problems easier because perpendicular components act independently.

Section 15 of 17More from the specification

1Calculating Uncertainties from Repeated Trials

• When repeated measurements are taken several times, the best estimate is the mean value and the absolute uncertainty is calculated as half the range (± (max - min) / 2).

2Adding Absolute Uncertainties

• When calculating the sum or difference of physical quantities, the overall absolute uncertainty is determined by adding together the absolute uncertainties of individual measurements.

3Combining Percentage Uncertainties

• When calculating quantities involving products, quotients,
• or powers, finding the overall uncertainty requires adding percentage uncertainties of each variable.

4Degree of Precision and Significant Figures

• The number of significant figures in processed experimental results should reflect the degree of precision of the measuring apparatus used.

5Calculating Uncertainties from Repeated Trials

• When repeated measurements are taken several times, the best estimate is the mean value and the absolute uncertainty is calculated as half the range (± (max - min) / 2).

6Adding Absolute Uncertainties

• When calculating the sum or difference of physical quantities, the overall absolute uncertainty is determined by adding together the absolute uncertainties of individual measurements.

7Combining Percentage Uncertainties

• When calculating quantities involving products, quotients,
• or powers, finding the overall uncertainty requires adding percentage uncertainties of each variable.

8Degree of Precision and Significant Figures

• The number of significant figures in processed experimental results should reflect the degree of precision of the measuring apparatus used.

Section 16 of 17Key terms

Term	Meaning
Physical quantity	A measurable property written as a magnitude and a unit.
SI base unit	One of the fundamental units (kg, m, s, A, K) from which all other units are derived.
Homogeneous equation	An equation whose terms have the same base units on both sides.
Random error	Unpredictable scatter in readings that reduces precision.
Systematic error	A consistent offset in all readings that reduces accuracy.
Uncertainty	The range around a measured value in which the true value is expected to lie.
Scalar	A quantity with magnitude only.
Vector	A quantity with magnitude and direction.
Resultant	The single vector equivalent to several combined vectors.

Section 17 of 17Exam technique

1

Confusing precision with accuracy — precision is the closeness of repeated readings to each other, accuracy is closeness to the true value.

2

Saying a single reading is precise — precision only describes several readings; one reading with many decimals simply has high resolution.

3

Adding percentage uncertainties when adding quantities — for addition and subtraction add absolute uncertainties; add percentages only for multiplying and dividing.

4

Forgetting to multiply the percentage uncertainty by the power — if x is cubed, the percentage uncertainty is tripled.

5

Mixing up prefixes such as milli (10-3) and micro (10-6) — check each conversion.