Measurements: Class 11 Physics

Key points

A physical quantity is a property that can be measured, such as length, mass, time or temperature. A measurement is a number with a unit. Writing only the number, for example 5, tells us nothing. 5 m and 5 s are different quantities.

The SI system has seven base quantities. Every other quantity is derived from them:

• length, metre (m)
• mass, kilogram (kg)
• time, second (s)
• electric current, ampere (A)
• temperature, kelvin (K)
• amount of substance, mole (mol)
• luminous intensity, candela (cd)

A derived quantity is formed by multiplying or dividing base quantities. Speed is length divided by time, so its unit is m/s. Force is mass times acceleration, so its unit is kg m/s², which is called the newton (N). Energy has the unit kg m²/s², called the joule (J).

Prefixes make very large and very small values easier to write. The common ones are: 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 and pico (p) 10⁻¹². So 2.5 mm = 2.5 × 10⁻³ m.

The dimensions of a quantity show how it is built from the base quantities. We write them with square brackets: [L] for length, [M] for mass and [T] for time. Speed has dimensions [L T⁻¹], acceleration [L T⁻²], force [M L T⁻²] and energy [M L² T⁻²]. Pure numbers, angles and trigonometric ratios have no dimensions.

Dimensional analysis has three uses:

• Checking an equation. Both sides of a correct physical equation must have the same dimensions.
• Deriving the form of a relationship, for example the time period of a pendulum, up to a constant.
• Converting from one system of units to another.

Dimensional analysis has limits. It cannot give the value of a dimensionless constant such as 2π or ½. It cannot tell us whether an equation is fully right, only that it is dimensionally consistent. It also fails when a quantity depends on a sum of terms, or on a trigonometric or exponential function whose argument must itself be dimensionless.

Rules for significant figures:

• All non-zero digits are significant (452 has three).
• Zeros between non-zero digits are significant (2005 has four).
• Zeros at the start of a number are not significant (0.0042 has two).
• Trailing zeros after a decimal point are significant (3.50 has three).
• In addition and subtraction the answer has as many decimal places as the measurement with the fewest decimal places. In multiplication and division the answer has as many significant figures as the measurement with the fewest.

Accuracy tells us how close a measurement is to the true value. Precision tells us how close repeated measurements are to each other. A set of readings can be precise but not accurate, for example when an instrument has a zero error.

Errors are of two main kinds:

• A systematic error shifts all readings in the same direction. Examples are a zero error or a wrongly calibrated instrument. It can be removed or corrected once it is found.
• A random error changes from one reading to the next because of small unpredictable effects. It can be reduced by taking many readings and finding their mean.

The absolute uncertainty is the size of the possible error in a reading, such as ±0.1 cm. The fractional uncertainty is the absolute uncertainty divided by the measured value, and the percentage uncertainty is that fraction times 100. The smaller the percentage uncertainty, the better the measurement.

Combining uncertainties:

• When quantities are added or subtracted, add their absolute uncertainties.
• When quantities are multiplied or divided, add their percentage uncertainties.
• When a quantity is raised to a power n, multiply its percentage uncertainty by n.

Study tip: For an uncertainty question, first decide whether the quantities are added or multiplied. Write the rule, convert each uncertainty to the right form, combine them, and give the final answer with the same number of decimal places as its uncertainty.

Definitions and flashcards

Say the meaning first, then open the card to check.

Formula sheet

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Short questions with model answers

These are the kind of questions asked for 2, 3 and 5 marks. Try each one before opening the answer.

Numericals with worked steps

Common mistakes

• Adding percentage uncertainties for a sum, or absolute uncertainties for a product. Absolute for sums and differences, percentage for products, quotients and powers.
• Forgetting to multiply the percentage uncertainty by the power when a quantity is squared or cubed.
• Counting leading zeros as significant figures.
• Using dimensional analysis to claim an equation is certainly right. It only shows that it is consistent.
• Writing a measurement with no unit.

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