Mass, Weight and Density CoreExtended adds floating liquids
Mass and weight are two of the most commonly confused quantities in everyday language, but in physics they mean very different things. This note also covers density — the property that determines whether something floats or sinks — and how to measure it for liquids, regular solids and awkwardly-shaped solids alike.
1. Mass and Weight
Mass is a measure of the quantity of matter in an object at rest relative to the observer. Mass is measured in kilograms (kg) and does not change depending on where the object is — a rock has the same mass on the Earth, on the Moon, or floating in space.
Weight is the gravitational force acting on an object because it has mass. Weight is measured in newtons (N), because it is a force, and it is a vector — it always acts downward, toward the centre of the gravitational body.
| Mass | Weight | |
|---|---|---|
| What it measures | amount of matter | gravitational force on that matter |
| Unit | kilogram (kg) | newton (N) |
| Scalar or vector | scalar | vector |
| Changes with location? | no — constant everywhere | yes — depends on local gravity |
| Measured with | balance | force meter (newtonmeter/spring balance) |
2. Gravitational Field Strength
Gravitational field strength is defined as force per unit mass.
This value, g, is numerically equal to the acceleration of free fall you met in the Motion note (g ≈ 9.8 m/s² near the Earth’s surface, which is the same as saying g ≈ 9.8 N/kg — both describe the same physical quantity, just expressed in two equivalent units).
W = m × g = 5 × 9.8 = 49 N
m = W / g = 196 / 9.8 = 20 kg
W = m × g = 12 × 3.7 = 44.4 N
The rock’s mass, 12 kg, is exactly the same as it would be on Earth — only its weight is different.
Comparing weights (and masses) using a balance: a simple beam balance compares an unknown mass against known masses because it is really comparing the weights (gravitational forces) on each side — since both sides experience the same gravitational field strength, the balance effectively cancels g out and gives a direct mass comparison.
- State one difference between mass and weight in terms of the units used to measure them.
- Calculate the weight of a 15 kg suitcase on Earth (g = 9.8 N/kg).
- A block has a weight of 78.4 N on Earth. Calculate its mass.
- The same block is taken to the Moon, where g = 1.6 N/kg. Calculate its weight on the Moon, and state its mass there.
Show answers
- Mass is measured in kilograms (kg); weight is measured in newtons (N), since it is a force.
- W = m × g = 15 × 9.8 = 147 N
- m = W / g = 78.4 / 9.8 = 8 kg
- W = m × g = 8 × 1.6 = 12.8 N. The mass is unchanged at 8 kg, since mass does not depend on location.
3. Weight as the Effect of a Gravitational Field Extended only
A more precise way to describe weight is as the effect of a gravitational field on a mass. Any object with mass, placed within a gravitational field, experiences a force (its weight) that depends on both its own mass and the strength of the field it is in. This is exactly why the same object can have different weights on different planets: the mass supplying the “effect” stays constant, but the strength of the field producing that effect changes.
Weight on Earth = 4 × 9.8 = 39.2 N
Weight on Moon = 4 × 1.6 = 6.4 N
Weight in deep space = 4 × 0 = 0 N (the object would be “weightless,” even though its mass is still 4 kg)
4. Density
Density is defined as mass per unit volume.
Density tells you how tightly packed the matter in a substance is. It is measured in g/cm³ or kg/m³, and — crucially — it is a property of the material, not of a particular object: a small block of iron and a huge block of iron have the same density, even though their masses and volumes are completely different.
Measuring the density of a liquid
Weigh an empty measuring cylinder on a balance. Pour in the liquid, and read its volume directly from the scale on the cylinder. Weigh the cylinder plus liquid again. Subtract to find the mass of the liquid alone, then use ρ = m / V.
Measuring the density of a regularly shaped solid
Weigh the solid on a balance to find its mass. Measure its dimensions with a ruler (length, width, height, or diameter, depending on the shape) and use the appropriate geometric formula to calculate its volume. Then use ρ = m / V.
Measuring the density of an irregularly shaped solid (displacement method)
You cannot use a ruler to find the volume of an irregular shape like a stone, so you use displacement: partly fill a measuring cylinder with water and record the initial volume reading. Lower the solid into the water (it must sink) and record the new, higher volume reading. The volume of the solid equals the difference between the two readings. Weigh the solid separately on a balance to find its mass, then use ρ = m / V.
Mass of liquid = 77 − 45 = 32 g
Density = m / V = 32 / 40 = 0.8 g/cm³
Volume = 2 × 3 × 5 = 30 cm³
Density = m / V = 210 / 30 = 7 g/cm³
Volume of stone = 68 − 50 = 18 cm³
Density = m / V = 45 / 18 = 2.5 g/cm³
- A block of wood has a mass of 60 g and a volume of 80 cm³. Calculate its density.
- An empty measuring cylinder has a mass of 38 g. After adding some oil, the total mass is 66 g and the oil’s volume is measured as 32 cm³. Calculate the density of the oil.
- A stone is lowered into a measuring cylinder and the water level rises from 25 cm³ to 39 cm³. The stone has a mass of 42 g. Calculate its density.
Show answers
- ρ = 60/80 = 0.75 g/cm³
- Mass of oil = 66 − 38 = 28 g. ρ = 28/32 = 0.875 g/cm³
- Volume = 39 − 25 = 14 cm³. ρ = 42/14 = 3 g/cm³
5. Floating, Sinking and Floating Liquids
Whether a solid floats or sinks in a liquid depends on comparing their densities: if the solid’s density is less than the liquid’s density, it floats; if it is greater than the liquid’s density, it sinks.
Since 0.9 g/cm³ < 1.0 g/cm³, the block's density is less than the water's, so it will float.
Extended: the same idea extends to two liquids that do not mix (are immiscible). If one liquid is poured on top of another, the liquid with the lower density floats on top of the liquid with the higher density.
Since oil’s density (0.92 g/cm³) is lower than water’s density (1.0 g/cm³), the oil floats on top of the water.
- A ball has a density of 1.3 g/cm³. Will it float or sink in water (density 1.0 g/cm³)? Explain your answer.
- Three immiscible liquids have densities 0.7 g/cm³, 1.0 g/cm³, and 1.3 g/cm³. Describe, from top to bottom, the order in which they would settle if poured into the same container.
Show answers
- It will sink, because its density (1.3 g/cm³) is greater than the density of water (1.0 g/cm³).
- From top to bottom: the 0.7 g/cm³ liquid (lowest density, floats highest), then the 1.0 g/cm³ liquid, then the 1.3 g/cm³ liquid at the bottom (highest density, sinks lowest).
- Using “weight” and “mass” interchangeably, or giving a weight answer in kilograms instead of newtons (or vice versa).
- Forgetting that mass never changes with location, while weight does, because it depends on the local gravitational field strength.
- Forgetting to subtract the empty container’s mass when finding the mass of a liquid alone.
- Using the wrong volume formula for a regular solid’s shape — always check whether it’s a cube, cuboid, cylinder, or sphere before calculating.
- In the displacement method, forgetting that the “volume of the solid” is the difference between the two readings, not either reading on its own.
- Mixing up which way round density comparisons work — the object with the lower density is the one that floats (on top), not the one with the higher density.
6. Quick-Fire Challenge Round
- Mass is measured in newtons.
- An object’s weight would be different on the Moon compared to on Earth, but its mass would stay the same.
- Density is a property of a particular object rather than of the material it is made from.
- A liquid with a lower density than another, immiscible liquid will float on top of it.
Show answers
- False — mass is measured in kilograms (kg). Weight is measured in newtons (N).
- True — mass stays constant everywhere, but weight depends on the local gravitational field strength, which is weaker on the Moon.
- False — density is a property of the material itself; any amount of that material, in any size or shape, has the same density.
- True — the less dense liquid floats on top of the denser one, provided the two liquids do not mix.
- “A student says their mass is 65 N.”
- “To find the density of an irregular stone, measure its length, width and height with a ruler and use ρ = m / V.”
- “A block of density 1.2 g/cm³ placed in water (density 1.0 g/cm³) will float, because 1.2 is greater than 1.0.”
Show answers
- Mass should be measured in kilograms, not newtons — newtons are the unit of weight (a force). The student likely means their mass is 65 kg, or that their weight is around 65 kg × 9.8 N/kg ≈ 637 N.
- An irregular shape cannot be measured with a ruler in this way, since its dimensions aren’t uniform. Instead, its volume should be found using the displacement method: submerging it in a measuring cylinder and recording the rise in water level.
- The logic is backwards — an object with a higher density than the liquid (1.2 > 1.0) will sink, not float. It floats only if its density is lower than the liquid’s density.
7. Mixed Practice — Bringing It All Together
- Calculate the weight of an 8 kg object on Earth (g = 9.8 N/kg).
- A rectangular block measures 4 cm × 2 cm × 3 cm and has a mass of 96 g. Calculate its density.
- A stone is lowered into a measuring cylinder, and the water level rises from 30 cm³ to 45 cm³. Its mass is 60 g. Calculate its density.
- Will an object of density 0.85 g/cm³ float or sink in water (1.0 g/cm³)? Explain.
- Explain why an astronaut’s mass stays the same on the Moon even though their weight decreases.
Show answers
- W = m × g = 8 × 9.8 = 78.4 N
- Volume = 4 × 2 × 3 = 24 cm³. ρ = 96/24 = 4 g/cm³
- Volume = 45 − 30 = 15 cm³. ρ = 60/15 = 4 g/cm³
- It will float, since its density (0.85 g/cm³) is less than the density of water (1.0 g/cm³).
- Mass is a measure of the quantity of matter in the astronaut, which does not change just by moving to a different location. Weight, however, is the gravitational force on that mass, and since the Moon’s gravitational field strength is weaker than Earth’s, the same mass experiences a smaller gravitational force there — hence a smaller weight, while the mass itself is unchanged.
Practice at Home
Take your time with this set. Part D is Extended-only content.
- State, in one sentence each, the difference between mass and weight.
- Calculate the weight of a 25 kg object on Earth (g = 9.8 N/kg).
- A box has a weight of 58.8 N on Earth. Calculate its mass.
- Define gravitational field strength, including the equation used to calculate it.
- A 6 kg object has a weight of 22.2 N on a distant planet. Calculate the gravitational field strength on that planet.
- Explain why a balance can be used to compare the masses of two objects, even though it is technically comparing their weights.
- A liquid of mass 54 g fills a container to a volume of 60 cm³. Calculate its density.
- A metal cube has sides of length 3 cm and a mass of 108 g. Calculate its density.
- A stone is lowered into a measuring cylinder, and the water level rises from 40 cm³ to 52 cm³. Its mass is 30 g. Calculate its density.
- Two immiscible liquids have densities 0.79 g/cm³ and 1.11 g/cm³. State which one would float on top if poured into the same container, and explain why.
- Explain, using the concept of a gravitational field, why the same object would have a different weight on Earth compared to on Jupiter, while having the same mass in both places.