GRADE 7 · PHYSICS AND MEASUREMENTS
Measuring instruments and accuracy
SI units, scales, scale intervals, and direct and indirect measurements.
Read the full lesson theory here without an account. In the interactive lesson, the same theory is presented as cards, followed by questions with feedback and practice problems. Your progress is saved.
Understanding step by step
Two students both give 12 as the length of a pencil. One uses centimetres and the other uses millimetres. Why is a number alone not a complete measurement, and how accurately can we measure a pencil?
Number with unit
To compare pencils, saying “this one is longer” is not enough: we must agree on a standard for comparing lengths. A measurable property expressed by a number and a unit is a physical quantity. Length l describes how far an object extends. Its SI unit is the metre: 1 м = 100 см = 1000 мм. Thus 12 mm and 12 m differ by a factor of a thousand. The unit tells us which equal lengths we imagine laying end to end along the object.
Count the intervals
There are small scale marks between 2 and 3 cm on the ruler. The whole section is 1 cm long. If it contains five equal intervals, each interval is one fifth of a centimetre. Count them: five intervals are bounded by six marks. We therefore count intervals, not marks. More equal intervals in the same section of a scale allow us to distinguish smaller differences in readings.
Scale interval
We need the value of one step on the scale: the scale interval c. Take the difference between the labelled values x₂ − x₁ and divide it equally among n intervals: c = (x₂ − x₁)/n. Between 2 and 3 cm, five intervals give c = 0,2 см. Twice as many intervals over the same difference would halve the scale interval. This reasoning applies to a uniform scale; c has the same unit as its readings.
Take a reading
Place one end of the pencil at zero and look at the other end directly from above. Looking from the side shifts its apparent position relative to the scale marks. What if the ruler’s edge is damaged? Start at any labelled whole-number mark. The final reading then includes the extra distance from zero to the starting mark, which must be subtracted: l = x₂ − x₁. Moving the whole pencil along the ruler changes both readings but leaves their difference, the length, unchanged.
Accuracy is limited
What should we record when the end falls between two marks? The instrument does not justify unlimited digits. Record an estimated length together with its uncertainty: for example, (2,5 ± 0,5) см represents roughly the interval from 2 to 3 cm. This describes how precisely we know the length; it does not mean the pencil changes length. A finer scale can narrow the interval if other sources of uncertainty do not dominate.
Where does uncertainty come from?
Imagine two rulers with the same scale interval, but with the zero mark misplaced on one of them. A fine scale cannot correct that error. The length uncertainty Δl has the same unit as the length and depends on both the instrument and how we use it. In school measurements, half the scale interval is often used unless another rule is given. This is a simple estimate, not a guarantee: consider the width of the marks, eye position, zero mark and condition of the instrument.
Measure directly or calculate
How can we measure a sheet of paper thinner than one scale interval? Stack many identical sheets without noticeable gaps and measure the stack. Its total thickness is the sum of the sheet thicknesses, so dividing by the number of sheets gives their average thickness. Reading an instrument is a direct measurement; calculating from measured quantities is an indirect measurement. The stack makes a tiny thickness measurable with a ruler, but gaps or strong compression can distort the result.
Sheet thickness
A gap-free stack of 100 identical sheets is 10 mm thick. Find the average thickness of one sheet.
Let's look at an example · 1
One sheet is too thin to measure conveniently with a ruler. The stack makes the measured length 100 times larger.
Let's look at an example · 2
Divide 10 mm by 100 to get 0,10 mm. This is an average thickness, and its accuracy is limited by the measurement of the stack.
Let's check the result
The average sheet thickness is about 0,10 mm.
Key idea · 1
Measurement compares a property with a chosen unit, so a result without a unit is incomplete.
Key idea · 2
The scale interval is the difference in labelled readings divided by the number of intervals. Length is the difference between the readings at the two ends.
Key idea · 3
Uncertainty indicates how closely we can trust the stated value. Measuring a stack makes a tiny thickness measurable, but we must control the gaps.
Where do we go next?
Instruments can help us investigate things too small to see. Next, we will ask which observations provide evidence for particles in substances.
Basic Concepts
Physical quantity
A measurable property of a body or phenomenon that is expressed by a number and a chosen unit.
Length, time, mass and temperature allow objects to be compared quantitatively, and the SI system makes the results understandable to everyone.
Scale and scale interval
A scale is a set of marks on an instrument. The scale interval is the change in the measured quantity between adjacent marks.
To read an instrument, find two labelled marks and divide the difference between them by the number of intervals; then use that interval to take the reading.
Accuracy and uncertainty
Uncertainty estimates the possible deviation of the result from the value that best represents the measured quantity.
A direct measurement is read from an instrument; an indirect measurement is calculated. To measure small identical objects, arrange them in a row, measure the total length, and divide by their number.
Physical quantities
Length
lThe length of an object or the distance between selected points.
Unit: meter · \mathrm{м}
Scale interval
cThe change in the measured value between adjacent scale lines.
Unit: meter · \mathrm{м}
Length uncertainty
\Delta lAssessment of possible deviation of the length measurement result.
Unit: meter · \mathrm{м}
Lesson formulas
Value of a scale interval
c = \frac{x_2-x_1}{n}Divide the difference between two labelled scale values by the number of intervals between them.
Problems on this topic
Statements, answer options and the walkthrough.