Physics · Mechanics · Grade 7

Physical quantities and measurement

How many grams does a melon at the market really weigh, and how accurate is the measurement?

The situation

At the market, the seller puts a melon in one pan of a balance and piles copper weights in the other. When will the pointer stop at zero, and how many grams does the melon really weigh? At home, if you measure the diameter of an apple with a ruler several times, you get a slightly different number each time. Which measurement is more accurate, and how do we calculate the error?

Topic

Physical quantities and measurement. Market scales and a ruler: measurement accuracy and error

Key ideas

Scale division
Absolute and relative error
Mean value

Formulas

C = \frac{a_{2} - a_{1}}{n}

Scale division value: the difference between two adjacent numbered marks ( $a_{2} - a_{1}$ ) divided by the number $n$ of divisions between them.

x = x_{meas} \pm Δ x, Δ x_{inst} = \frac{C}{2}, ε = \frac{Δ x}{x} \cdot 100 %

The measurement result, the instrument (reading) error — half a division — and the relative error: the fraction of the measured quantity that the absolute error makes up.

S = \frac{Δ n}{Δ m}, m = Σ - \frac{n}{S}

Balance sensitivity (divisions/g) and the mass corrected for the pointer deflection $n$ : $Σ$ is the sum of the weights.

\overline{x} = \frac{x_{1} + x_{2} + \dots + x_{n}}{n}

Mean of repeated measurements.

Δ x_{rand} = \frac{| x_{1} - \overline{x} | + \dots + | x_{n} - \overline{x} |}{n}

The mean deviation is the random error. Total error: $Δ x = Δ x_{inst} + Δ x_{rand}$ .

Theory

Scale division

Every instrument has a scale. The scale division value $C$ is the value between the two nearest adjacent marks. On a ruler there are 10 divisions between 0 and 1 cm: $C = 1$ mm. The extra scale of a vernier calliper — the vernier — has 10 divisions in a length of 9 mm, so it can sense a tenth of a millimetre: $C = 0.1$ mm. On the balance pointer in this experiment, one division is a difference of 5 g.

Absolute and relative error

We round the reading to the nearest mark, so the result is accurate to half a division: $x \pm C / 2$ . This is the absolute error. On its own it does not describe the accuracy: a ruler error of 0.5 mm is 0.7 % for a 75 mm apple, but 2.8 % for an 18 mm grape — the relative error is much greater. To measure a small object accurately you need an instrument with a small scale division value.

Why do we repeat measurements?

An apple is not a perfect sphere: if you turn it as you measure, the diameter changes by a few millimetres. To reduce this random error, we measure several times and take the mean value, and the mean of the deviations is the random error. In this experiment the total error is taken as $Δ x = Δ x_{inst} + Δ x_{rand}$ .

How does a pan balance work?

The balance beam is an equal-arm lever: at equilibrium the moments are equal, $m_{1} g L = m_{2} g L$ , so $m_{object} = m_{weights}$ . The centre of gravity of the beam is below the pivot, so the beam oscillates like a pendulum and returns to equilibrium. The difference between the weights in the pans tilts the beam: $\tan φ = Δ m L / (M d)$ , where $L$ is the arm, $M$ the mass of the beam and $d$ how far its centre of gravity is below the pivot. The deflection depends not on the load in the pan but only on the difference: on this balance a difference of 5 g deflects the pointer by 1 division. The model is simplified: the pans are taken as point loads, and their oscillation and the damping in the damper are not calculated exactly; if the deflection exceeds 10 divisions, the beam touches the stop. An analytical balance works in the same way, only much more sensitively: first the zero point of the empty balance is found, then a small known load is placed on it and the sensitivity $S = Δ n / Δ m$ is found (0.2 divisions/g on this balance). In weighing, weights are added until the pointer approaches zero, and the remaining deflection is taken into account by $m = Σ - n / S$ — the “corrected mass” in “Result”. When the balance is not in use, the beam is held by the supports (the beam lock), and an air damper damps the oscillation.

Physical quantities and the SI

A physical quantity = number × unit: $m = 2347$ g. The International System of Units (SI) rests on 7 base units: the metre (m), kilogram (kg), second (s), ampere (A), kelvin (K), mole (mol) and candela (cd); since 2019 all of them have been defined through exactly fixed fundamental constants ( $h$ , $c$ , $e$ , $k_{B}$ , $N_{A}$ and others). The rest are derived units: N = kg·m/s², J = N·m, Pa = N/m². Prefixes: milli ( $10^{- 3}$ ), centi ( $10^{- 2}$ ), kilo ( $10^{3}$ ), mega ( $10^{6}$ ). Quantities are linked by laws: $m = ρ V$ , weight $P = m g$ — $g$ is the same at both pans, so a balance measures mass by comparing weights. Dimensional analysis: both sides of a formula must have the same units. The oscillation period of the beam is $T = 2 π \sqrt{I / (M g d)}$ : $[I] =$ kg·m², $[M g d] =$ kg·m²/s², and under the root there is s² — the period is indeed in seconds.

Methods of research in physics

Physics begins with observing natural phenomena, proposes a hypothesis to explain them and tests it in an experiment — by measurement. A quantitative relation confirmed many times becomes a law (for example, the equilibrium of a lever); a system of laws and the ideas explaining them is a theory. Every theory works with a model: an ideal sphere, a point load, a weightless string. Measurements show how good a model is — because the apple is not an ideal sphere, the diameter comes out different each time. Modern methods: electronic sensors and automatic recording (the check balance), computer modelling (in this experiment the equation of motion of the beam is solved numerically), spectral and laser methods. The methods of physics are the foundation of chemistry, biology, medicine and engineering — the scale of sizes is covered in the “Scales of the universe” experiment.

Did you know?

From 20 May 2019 the kilogram is no longer defined by the platinum–iridium standard kept at Sèvres in France (since 1889) but by the Planck constant $h = 6.626 070 15 \cdot 10^{- 34}$ J·s.
The extra scale on a vernier calliper is called a nonius — after the Portuguese mathematician Pedro Nunes (Latin: Nonius). The sliding scale in its present form was proposed in 1631 by the Frenchman Pierre Vernier, which is why it is also called a vernier.

Inside the experiment

The experiment starts from a real-life situation: you predict the result, change the controls, run the experiment and draw a conclusion. Inside are missions, theory, a quiz and a guided tour in which the camera moves and explains the key points.

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