1. Physical Quantities and Units — CAMBRIDGE AS & A LEVEL PHYSICS 9702 — COMPLETE NOTES notes: Use SI base and derived units, check equations for homogeneity, handle uncertainties and errors, and add, resolve and combine vectors.Physical quantity — a measurable property that is a…
Symbols need context
Estimation
SI base units
Homogeneous equation
Range of sizes
True value
Section 1 of 14Physical quantities and estimation
Physical quantity
a measurable property that is always written as a numerical magnitude together with a unit.
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
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 14How to make an estimate
Method
recall the equation that links the quantities, estimate each input from everyday experience, then substitute.
Scope
only familiar quantities are estimated, such as a walking speed of 1 to 1.5 m s−1.
Range
a range of sensible values is acceptable, and many constants are in the data booklet.
Section 3 of 14SI 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 14Derived units in base units
Derived unit
a unit such as the newton, joule or pascal that is a combination of base units.
Method
write the defining equation of the quantity, then replace each quantity by its base units.
Newton
force = mass × acceleration, so N = kg m s−2.
Joule
energy = ½ × mass × velocity2, so J = kg (m s−1)2 = kg m2 s−2.
Pascal
pressure = force ÷ area, so Pa = kg m s−2 ÷ m2 = kg m−1 s−2.
Section 5 of 14Homogeneity 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 14Powers 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 14Random and systematic errors
True value
measurements aim to find it, but it can never be known exactly, so every result has an uncertainty.
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
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
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 14Precision, 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
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 14Absolute, fractional and percentage 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
a fixed amount with units, such as ± 0.05 cm.
Fractional uncertainty
the uncertainty divided by the measured value.
Percentage uncertainty
absolute uncertainty ÷ measured value × 100%.
Section 10 of 14Estimating 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 14Combining 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 14Scalars and vectors
Scalar
a quantity with magnitude only.
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 14Adding and subtracting vectors
Resultant
the single vector with the same effect as the vectors being combined.
Triangle method
draw the vectors head to tail, then join the tail of the first to the head of the last.
Parallelogram method
draw the vectors tail to tail, complete the parallelogram, and take the diagonal.
Subtraction
reverse the direction of the vector to be subtracted, then add.
Equilibrium
coplanar forces in equilibrium form a closed vector triangle.
Magnitude and direction
use Pythagoras for the size and tan θ = opposite ÷ adjacent for the angle.
Section 14 of 14Resolving 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.