Physical quantities and units

physical quantity

any property of a material or system that can be quantitatively measured. It consists of a numerical magnitude and a unit

base units

defined physical quantities in the International System of Units (SI) that are specified independently of other units.

7 base quantities

mass (kg), length (m), time (s), current (A),

temperature (K), amount of substance (mol), and luminous intensity (cd).

dimensional homogeneity

Dimensional homogeneity means that both sides of an equation must have the same units

pico

p, 10⁻¹²

nano

n, 10^-9

micro

μ, 10⁻⁶

milli

10^-3

centi

c, 10^-2

deci

d, 10^-1

kilo

k, 10³

mega

M, 10⁶

giga

G, 10⁹

tera

T, 10¹²

Systematic Errors

an error that results in ll reading being either above or below the true value, by a fixed amount and in the same direction each time the measurement is taken.

properties of systematic error

• Consistent bias in measurement.

• Due to faulty instruments, incorrect calibration, zero error.

• Affects accuracy, not precision

properties of random error

• Unpredictable fluctuations in readings.

• Due to human reaction time, air currents, thermal noise, etc.

• Affects precision.

random error

The component of measurement error that varies in an unpredictable manner from one measurement to another, producing a scatter of values that may be above or below the true value due to uncontrollable variables or inherent limitations in the measurement process

minimising random error

repeating a reading and averaging or by plotting a graph and taking a best-fit line.

examples of random error

1. reading a scale, especially judgement about interpolation between scale readings

2. timing oscillations without the use of a reference marker

3. measuring two quantities simultaneously, which vary with time

4. parallax error

examples of systematic errors

1. zero error

2. wrongly calibrated scale

3. reaction time of experimenter

reducing systematic error

improving experimental technique

instruments should be recalibrated

precision

refers to how close a set of measured values are to each other

accuracy

refers to the closeness of a measured value to the ‘true value’

difference between precision and accuracy

1. precision refers to how close a set of measured values are to each other while accuracy refers to the closeness of a measured value to the ‘true value’

2. precision is affected by random errors while accuracy is affected by systematic errors

3. accuracy is improved by calibrating instruments and correcting biases while precision is improved by refining measurement technique and reducing fluctuations

uncertainty for single measurement

Uncertainty = ± (smallest division / 2)

uncertainty For Repeated Measurements

Uncertainty = ± (range / 2)

combining uncertainty while adding quantities

If Q = A ± ΔA + B ± ΔB
Then: ΔQ = ΔA + ΔB

combining uncertainty while subtracting quantities

If Q = A ± ΔA - B ± ΔB
Then: ΔQ = ΔA + ΔB

combining uncertainty while multiplying/dividing quantities

%ΔQ = %ΔA + %ΔB

combining uncertainty with Powers and Roots

Q = Aⁿ

%ΔQ = |n| × %ΔA

true value

A perfect measurement value which reflects the quantity being measured with no errors

zero error

occurs when an instrument gives a non-zero reading when the true reading is zero

Absolute Uncertainty

where uncertainty is given as a fixed quantity

Fractional uncertainty

where uncertainty is given as a fraction of the measurement

Percentage uncertainty

where uncertainty is given as a percentage of the measurement

The uncertainty in digital readings

± the last significant digit unless otherwise quoted

Scalars

quantities that have magnitude but not direction

Vectors

quantities that have both magnitude and direction

Distance

the total length of the path taken

Displacement

the length and direction of a straight line drawn from the starting point to the finishing point

Example of scalar quantities

distance, speed, mass, time, energy, volume, density, pressure, electric charge, temperature

Examples of vector quantities

displacement, velocity, acceleration, force, momentum