Measurement, Mass, Weight and Density: Revision Notes

Condensed for the final weeks. For the full explanation, use the Measurement, Mass, Weight and Density study guide.

Instruments and their resolution

Quantity	Instrument	Resolution
Length (cm–m)	Metre rule	1 mm
Length (small) (background — not named in the 5054 syllabus, which lists tapes, rules and micrometers)	Vernier callipers	0.01 cm
Length (very small)	Micrometer screw gauge	0.01 mm
Volume (liquid)	Measuring cylinder	Read the bottom of the meniscus, eye level
Time	Stopwatch	0.01 s, but human reaction ≈ 0.2 s
Mass	Balance	Depends on instrument

Parallax error — reading a scale from an angle. Avoid by looking perpendicular to the scale.

Reading a micrometer: add the main scale reading (whole and half mm) to the thimble scale reading (hundredths of a mm) at the line that aligns with the datum. Miss either scale and the answer is wrong by a large margin.

Timing tip: time 20 oscillations and divide by 20, so reaction-time error is spread across many swings.

Worked example: a pendulum takes 18.4 s for 20 oscillations.
period = total time / number of oscillations = 18.4 / 20 = 0.92 s

A single swing timed on its own would still carry almost the full ≈ 0.2 s reaction-time error; spreading that same fixed error over 20 swings and dividing shrinks its effect on the period by a factor of 20.

Mass vs weight — learn the distinction

	Mass	Weight
What it is	Amount of matter	Force due to gravity
Unit	kg	N
Changes with location?	No	Yes
Measured with	Balance	Newtonmeter / spring balance

W = m g          g = 9.8 N/kg on Earth
                 g = 1.6 N/kg on the Moon

A 60 kg astronaut has mass 60 kg everywhere, but weighs 588 N on Earth and 96 N on the Moon.

Inertia: mass is also what makes an object resist a change to its state of rest or motion. A more massive object is harder to start moving and harder to stop — this resistance is called inertia, and it is a property of mass alone, not weight.

Gravitational field strength links mass and weight:

g = W / m        (force per unit mass)

This is numerically the same as the acceleration of free fall (≈ 9.8 m/s² on Earth) — the two describe the same physical situation from different angles, one as force-per-mass and the other as an acceleration, and they turn out equal because of F = ma. A gravitational field is the region around a mass in which another mass feels a force due to gravitational attraction; g is weaker on the Moon, which is exactly why the same mass weighs less there.

Density

density = mass / volume        rho = m / V

kg/m3  or  g/cm3
1 g/cm3 = 1000 kg/m3

Measuring density:

• Regular solid — measure dimensions, calculate volume, weigh.
• Irregular solid — weigh, then use a displacement can or measuring cylinder; volume = water displaced.
• Liquid — weigh empty cylinder, add liquid, reweigh, subtract; read volume directly.

Floating rule: an object floats if its density is less than that of the fluid. Water = 1 g/cm³.

Exam traps

• Weight is measured in newtons, never kilograms.
• Mass does not change on the Moon; weight does.
• Convert cm³ to m³ by dividing by 1 000 000 — a frequent slip.
• Read the meniscus at its bottom, at eye level.
• Density of water is 1 g/cm³ or 1000 kg/m³ — check which units the question uses.
• A micrometer reads to 0.01 mm. (Vernier callipers, reading to 0.01 cm, are background — the 5054 syllabus names only tapes, rules and micrometers as measuring instruments.)
• A micrometer reading needs both the main scale and thimble scale added together — reading only one gives a large, obviously-wrong error.
• g = W/m gives gravitational field strength in N/kg, numerically equal to the acceleration of free fall in m/s² — don’t confuse the two units when asked which one a question wants.
• Inertia is a property of mass, not weight — a body still resists a push in deep space, where it is weightless but not massless.

Self-test

1. A block is 2 cm × 3 cm × 4 cm and has mass 96 g. Find its density.
2. Will it float in water?
3. State two differences between mass and weight.
4. How would you find the volume of an irregular stone?
5. Why time 20 oscillations rather than one?
6. A pendulum takes 24.0 s for 20 oscillations. Find the period.
7. Define gravitational field strength, and give its formula in terms of weight and mass.
8. Explain what inertia is, and which of mass or weight it depends on.

Answers: 1. V = 24 cm³; ρ = 96/24 = 4 g/cm³. 2. No — 4 g/cm³ is greater than water’s 1 g/cm³, so it sinks. 3. Mass is matter in kg and constant; weight is a force in newtons that varies with gravitational field strength. 4. Submerge it in a displacement can or partly filled measuring cylinder; the volume of water displaced equals the stone’s volume. 5. Human reaction time (~0.2 s) is a fixed error; spreading it over 20 swings and dividing reduces its effect on each period by a factor of 20. 6. period = 24.0/20 = 1.2 s. 7. Force per unit mass; g = W/m. 8. Inertia is a body’s resistance to a change in its state of rest or motion; it depends only on mass, since a weightless object in space still has inertia.