Learning objectives
Measure length, volume and time with the correct instrument and use multiples to measure very small quantities
Find the period of a pendulum by timing many oscillations
Distinguish scalars from vectors and add two vectors at right angles (Extended)
Define density, use ρ = m/V and predict floating and sinking
Describe how to measure the density of liquids and of regular and irregular solids
Key definitions
Density
The mass per unit volume of a substance: ρ = m/V, in kg/m³ or g/cm³.
Scalar (Extended)
A quantity with magnitude (size) only, such as speed, distance, mass, energy, time and temperature.
Vector (Extended)
A quantity with both magnitude and direction, such as velocity, displacement, force, weight, acceleration and momentum.
Period
The time for one complete oscillation of a pendulum, found by timing many swings and dividing by the number of swings.
Explanation
Choose the instrument to suit the size of the quantity. A metre rule reads to the nearest millimetre; a measuring cylinder measures the volume of a liquid, read at the bottom of the meniscus with your eye level with the surface to avoid parallax error. The volume of a regular solid comes from its dimensions (length × width × height for a block), while an irregular solid is placed in a part-filled measuring cylinder and the rise in the water level gives its volume. When a quantity is too small to measure directly, measure a multiple of it and divide: the thickness of one sheet of paper is found by measuring the thickness of 100 sheets, and the diameter of a wire by winding 20 turns tightly around a pencil and measuring the width of the coil. This makes the instrument's ±0.5 mm uncertainty a tiny fraction of the value measured.
Time intervals are measured with stopclocks or digital stopwatches, which read to 0.01 s, but a human operator adds a reaction time of about 0.2 s at the start and the stop. This matters when the interval itself is short. For a simple pendulum, therefore, do not time one swing: time 20 complete oscillations (from the release point, through the centre and back again is one oscillation) and divide the total by 20 to give the period. The reaction-time error is spread across all twenty swings, so the error in the period is twenty times smaller. Start counting from zero as the bob passes a marker at the centre of the swing, where it moves fastest and the crossing is easiest to judge, and repeat the timing to obtain a mean. A pendulum's period depends on its length but not on the mass of the bob or, for small swings, the amplitude.
(Extended) Scalars have size only: distance, speed, time, mass, energy, temperature. Vectors have size and direction: displacement, velocity, acceleration, force, weight, momentum, gravitational field strength. The difference matters when quantities combine. Two 5 N forces acting along the same line add to 10 N or cancel to zero depending on their directions. When two vectors act at right angles, the resultant is the diagonal of the rectangle they form. By calculation, its magnitude is R = √(a² + b²) and its direction is given by tan θ = b/a measured from the first vector; by scale drawing, draw the two vectors tip-to-tail with a stated scale (for example 1 cm = 2 N), complete the triangle and measure the resultant's length and angle with a ruler and protractor. A 3 N and a 4 N force at right angles give 5 N at 53° to the 3 N force.
Density is mass per unit volume, ρ = m/V, so 1 g/cm³ = 1000 kg/m³. An object floats in a fluid if its average density is less than the fluid's density and sinks if it is greater; (Extended) the same rule tells you which of two liquids that do not mix floats on top. To find the density of a liquid, weigh an empty measuring cylinder, pour in a measured volume, weigh again and divide the mass of liquid by its volume. For a regular solid, measure the mass on a balance and calculate the volume from measured dimensions. For an irregular solid, measure the mass, then lower it into a measuring cylinder containing enough water to cover it and take the difference in readings as the volume; if it floats, push it under with a thin pin or hang a sinker beneath it and subtract the sinker's volume.
Formulae
ρ = m/V (kg/m³ or g/cm³; 1 g/cm³ = 1000 kg/m³)
Period T = total time ÷ number of oscillations
Vectors at right angles: R = √(a² + b²), tan θ = b/a (Extended)
Thickness of one sheet = thickness of stack ÷ number of sheets
Worked examples
q: A stone of mass 81 g is lowered into a measuring cylinder containing 50 cm³ of water. The water level rises to 80 cm³. Calculate the density of the stone in g/cm³ and in kg/m³, and state whether it will float in water.
steps: Volume of stone = rise in water level = 80 − 50 = 30 cm³
ρ = m/V = 81 ÷ 30 = 2.7 g/cm³
Convert: 2.7 g/cm³ × 1000 = 2700 kg/m³
2700 kg/m³ is greater than the density of water (1000 kg/m³), so the stone sinks.
In context
Ships made of steel float because the hull encloses air, making the average density of the whole ship less than that of water.
Oil spills float on the sea because oil (about 900 kg/m³) is less dense than seawater (about 1030 kg/m³), which is why floating booms can contain them.
Common misconceptions
Reading a measuring cylinder from above or below eye level — always read the bottom of the meniscus at eye level.
Counting a swing from one side to the centre as a full oscillation — one oscillation is there and back to the starting point.
Adding vectors at right angles as if they were numbers: 3 N + 4 N at right angles is 5 N, not 7 N.
Thinking heavy objects sink and light objects float — floating depends on density, not on mass.
Study tips
When asked how to improve a measurement, name the specific technique: time 20 oscillations, measure 100 sheets, repeat and average, read at eye level.
Keep units consistent in density calculations: grams with cm³, kilograms with m³.
Evaluate the idea
Timing many oscillations reduces the effect of reaction time but does not remove systematic errors such as a slow-running clock, so accuracy and precision are different ideas.
Reasoning
Measure mass on a balance → find volume by dimensions or displacement → ρ = m/V → compare with fluid density → predict float or sink
Summary
Measure a multiple of a small quantity and divide; time many oscillations and divide.
Scalars have size only; vectors have size and direction and add by scale drawing or Pythagoras when perpendicular (Extended).
Density ρ = m/V; an object floats if it is less dense than the fluid.
Check your understanding
Question 1: A student times 25 complete swings of a pendulum at 40.0 s. The period is…
1. 0.625 s
2. 40 s
3. 1.6 s
Question 2: (Extended) Which of these is a vector quantity?
1. mass
2. speed
3. weight