Physics 3.1 Measurements and their errors

What are the fundamental SI units and what physical quantities do they measure?

Mass → kilogram (kg); length → metre (m); time → second (s); amount of substance → mole (mol); temperature → kelvin (K); electric current → ampere (A). The candela is excluded.

What are SI derived units, and how can force and energy be expressed in base units?

Derived units are combinations of fundamental SI units. Force: N = kg m s⁻². Energy: J = kg m² s⁻².

What SI prefixes must you know and what powers of ten do they represent?

T (tera) = 10¹²; G (giga) = 10⁹; M (mega) = 10⁶; k (kilo) = 10³; c (centi) = 10⁻²; m (milli) = 10⁻³; μ (micro) = 10⁻⁶; n (nano) = 10⁻⁹; p (pico) = 10⁻¹²; f (femto) = 10⁻¹⁵.

How do you convert between units using SI prefixes and standard form?

Replace the prefix with its multiplier and write the result in standard form, a × 10ⁿ where 1 ≤ a < 10. For example, 60 TΩ = 6.0 × 10¹³ Ω and 3 × 10⁻⁶ m = 3 μm.

What energy conversions must you know?

1 eV = 1.60 × 10⁻¹⁹ J and 1 kWh = 3.60 × 10⁶ J. Convert between them by converting to or from joules as required.

What is the difference between random and systematic errors, and what do they affect?

Random errors cause unpredictable variation between repeated measurements, affecting precision. Systematic errors produce a consistent bias in the same direction, affecting accuracy; they arise from faults in equipment or experimental method.

How can random and systematic errors be reduced or removed?

Random error can be reduced by repeating measurements and calculating a mean, using a data logger/computer where appropriate, or using equipment with greater resolution. Systematic error can be reduced or removed by calibrating or zeroing equipment and correcting known systematic effects. Parallax error is reduced by reading a scale at eye level.

What are precision, repeatability, reproducibility, resolution and accuracy?

Precision = how closely repeated measurements agree with each other. Repeatability = the same experimenter, using the same equipment and method, obtains consistent results. Reproducibility = different experimenters and/or different equipment or techniques give consistent results. Resolution = the smallest change in the quantity being measured that produces a recognisable change in the reading. Accuracy = closeness to the true value.

What are absolute, fractional and percentage uncertainty?

Absolute uncertainty is the uncertainty expressed in the same units as the measured quantity, e.g. 7.0 ± 0.6 V. Fractional uncertainty is the absolute uncertainty divided by the measured value. Percentage uncertainty is the fractional uncertainty × 100%.

How do you calculate the uncertainty of a single analogue reading?

Usually take the absolute uncertainty as ± half the smallest scale division.

How do you calculate the uncertainty of a quantity from repeated measurements?

Calculate the mean. The absolute uncertainty is half the range, so quote the result as mean ± half the range.

How can percentage or fractional uncertainty be reduced experimentally?

Measure a larger quantity where possible, because the same absolute uncertainty becomes a smaller fraction of the measured value.

How should values and uncertainties be written using significant figures?

Do not give more precision than the uncertainty justifies. The measured value should be rounded to the same decimal place as the uncertainty.

What are error bars and what do they show on a graph?

Error bars represent the uncertainty in a plotted measurement by showing the range of values within its uncertainty. Individual data points may or may not have associated error bars.

How should a line of best fit be drawn when data points have error bars?

Draw a line that represents the overall trend and is consistent with the uncertainties shown by the error bars. It does not have to pass through every central data point, or error bar.

worst acceptable line of best fit: the line must pass through the top of the first error bar and the bottom of the last error bar

How do you determine the uncertainty in the gradient of a straight-line graph?

Draw the best-fit line and the steepest or shallowest possible acceptable line. Calculate the gradients of the best and limiting lines. The difference between these gradients gives the absolute uncertainty in the gradient.

How do you determine the uncertainty in the y-intercept of a straight-line graph?

Draw the best-fit line and the limiting acceptable line. Find their y-intercepts. The difference between the best and limiting intercepts gives the absolute uncertainty in the y-intercept.

What is an order of magnitude?

A power of ten that describes the approximate size of a physical quantity and allows quantities to be compared by scale.

How do you find the nearest order of magnitude of a quantity?

Write the quantity in standard form and round it to the nearest power of ten. For example, 9.71 × 10⁻²¹ rounds to 1 × 10⁻²⁰, so its nearest order of magnitude is 10⁻²⁰.

What is estimation?

Finding an approximate value of a physical quantity. Estimates can be made to the nearest order of magnitude and combined with knowledge of physics to produce further derived estimates to the nearest order of magnitude.

What is the standard form of a number?

a × 10ⁿ, where 1 ≤ a < 10.

What are the numerical conversions between electronvolts, joules and kilowatt-hours?

1 eV = 1.60 × 10⁻¹⁹ J. 1 kWh = 3.60 × 10⁶ J.

How do you calculate fractional uncertainty?

Fractional uncertainty = absolute uncertainty ÷ measured value.

How do you calculate percentage uncertainty?

Percentage uncertainty = (absolute uncertainty ÷ measured value) × 100%.

How do you calculate the uncertainty of an analogue reading?

Absolute uncertainty = ±½ × smallest scale division.

How do you calculate the uncertainty from repeated measurements?

Mean = sum of measurements ÷ number of measurements. Absolute uncertainty = ½ × range. Range = maximum value − minimum value. Therefore, absolute uncertainty = (maximum − minimum) ÷ 2.

How do you combine uncertainties when quantities are added or subtracted?

Add the absolute uncertainties. If Q = A ± B, absolute uncertainty in Q = uncertainty in A + uncertainty in B.

How do you combine uncertainties when quantities are multiplied or divided?

Add the fractional or percentage uncertainties. For multiplication or division, % uncertainty in Q = % uncertainty in A + % uncertainty in B.

How do you combine uncertainty when a quantity is raised to a power?

Multiply the percentage uncertainty by the magnitude of the power. If Q = Aⁿ, % uncertainty in Q = |n| × % uncertainty in A.

How do you calculate the gradient of a straight line?

Gradient = Δy ÷ Δx. Use two well-separated points on the line when calculating the gradient.

How do you calculate the absolute uncertainty in a graph gradient?

Absolute uncertainty in gradient = |best gradient − worst gradient|. The worst gradient is obtained from the steepest or shallowest acceptable line.

How do you calculate percentage uncertainty in a graph gradient?

Percentage uncertainty = |best gradient − worst gradient| ÷ best gradient × 100%.

How do you calculate the absolute uncertainty in a y-intercept?

Absolute uncertainty in y-intercept = |best y-intercept − worst y-intercept|.

How do you calculate percentage uncertainty in a y-intercept?

Percentage uncertainty = |best y-intercept − worst y-intercept| ÷ best y-intercept × 100%.

How should a measured value be rounded when it has an uncertainty?

The measured value should be rounded to the same decimal place as the uncertainty. The uncertainty should not be quoted to unnecessary precision.

How do you determine the nearest order of magnitude?

Write the number in standard form and round the coefficient to 1 significant figure. The resulting power of ten is the nearest order of magnitude.

What is the relationship between uncertainty and the number of significant figures in a result?

The uncertainty limits the precision justified by a measurement. Usually quote the uncertainty to 1 significant figure, sometimes 2 where appropriate, and round the measured value to the same decimal place as the uncertainty.

How do you use uncertainty to assess whether two experimental results agree?

Compare the uncertainty intervals around the results. Overlapping intervals indicate that the results may be consistent within their uncertainties, but do not prove that they agree or that systematic errors are absent.

What does it mean if a percentage uncertainty is very large?

The uncertainty is large relative to the measured value, so the result is poorly constrained. A value close to zero can have a very large or undefined percentage uncertainty despite having a small absolute uncertainty.

How do you estimate a derived physical quantity to the nearest order of magnitude?

Estimate the relevant physical quantities, use an appropriate physical equation to combine them, then express the result to the nearest power of ten. Check that the estimate is physically plausible.

How can you determine the SI base units of a derived quantity?

Substitute the base units of the quantities in its defining equation and simplify. For example, power = energy ÷ time, so W = kg m² s⁻³.