This note separates mass from weight, shows how weight is found from gravitational field strength, and explains density and how to measure it. These ideas explain why some objects float, why astronauts weigh less on the Moon, and how engineers choose materials.
Key ideas
Mass
Mass is the amount of matter in an object. It is measured in kilograms (kg) with a balance. Mass is a scalar, and it stays the same wherever the object is. Mass also measures inertia: the larger the mass, the harder it is to change the object's motion. A heavy trolley is harder to start and harder to stop than an empty one.
Weight
Weight is the force of gravity on an object. It is measured in newtons (N) with a newton meter (a spring balance). Weight is a vector that points towards the centre of the planet. It depends on where the object is, because the strength of gravity changes from place to place.
The gravitational field strength, g, is the force on each kilogram of mass. Near the Earth's surface g = 9.8 N/kg. On the Moon it is about 1.6 N/kg, and far from any planet it is close to zero. The same quantity can be called the acceleration of free fall, with units m/s2, because 1 N/kg = 1 m/s2.
Weight is found from W = m × g. Mass does not change on the Moon, but weight does. An astronaut has the same mass in orbit but feels "weightless" because of how they are moving, not because mass has disappeared.
Two masses can be compared using a balance. Because both feel the same g at the same place, the balance works anywhere. A newton meter would give different readings on the Moon, because it measures weight.
Density
Density tells you how much mass is packed into each unit of volume. A steel block and a wooden block of the same size have different masses because steel is denser. Density is a property of the material, not of the shape or size of the sample.
Common units are kg/m3 and g/cm3. To convert, 1 g/cm3 = 1000 kg/m3. Water has a density of 1.00 g/cm3, which is 1000 kg/m3.
Floating and sinking
An object floats in a liquid if its average density is less than the density of the liquid. It sinks if its density is greater. Ice has a density of about 920 kg/m3, which is less than water, so ice floats. A steel ship floats because its average density, including the air inside the hull, is less than that of water.
Measuring density
- Regular solid: measure the mass on a balance. Measure the dimensions with a ruler and calculate the volume (for a cuboid, length × width × height). Divide mass by volume.
- Irregular solid: find the mass on a balance. Find the volume by displacement: put the object into a measuring cylinder part-filled with water and subtract the starting reading from the final reading (or use a displacement can and collect the water displaced). Divide mass by volume.
- Liquid: place an empty measuring cylinder on a balance and zero it (or record its mass). Add some liquid. Read the volume from the cylinder and the mass of the liquid from the balance. Divide mass by volume.
Definitions and formulae
| Term or formula | Meaning / when to use |
|---|---|
| Mass | Amount of matter in an object; measured in kg; does not change with place. |
| Weight | Gravitational force on an object; measured in N; depends on g. |
| W = m × g | Weight in newtons from mass in kg and g in N/kg. |
| g | Gravitational field strength: 9.8 N/kg on Earth, about 1.6 N/kg on the Moon. |
| ρ = m ÷ V | Density (kg/m3 or g/cm3) from mass and volume. |
| m = ρ × V and V = m ÷ ρ | Rearrangements of the density equation. |
| V = l × w × h | Volume of a cuboid, in m3 or cm3. |
| 1 g/cm3 = 1000 kg/m3 | Conversion between the two common density units. |
Worked examples
Example 1 An astronaut and equipment have a total mass of 75 kg. Find the weight on Earth (g = 9.8 N/kg) and on the Moon (g = 1.6 N/kg).
Step 1: on Earth, W = m × g = 75 × 9.8 = 735 N.
Step 2: on the Moon, W = 75 × 1.6 = 120 N.
Step 3: the mass is still 75 kg in both places.
Answer: 735 N on Earth and 120 N on the Moon; mass unchanged at 75 kg.
Example 2 A stone has a mass of 156 g. It is lowered into a measuring cylinder, and the water level rises from 40.0 cm3 to 98.0 cm3. Find the density of the stone in g/cm3 and in kg/m3.
Step 1: volume of stone = 98.0 − 40.0 = 58.0 cm3.
Step 2: density = m ÷ V = 156 ÷ 58.0 = 2.69 g/cm3.
Step 3: convert: 2.69 × 1000 = 2690 kg/m3.
Answer: 2.69 g/cm3 = 2690 kg/m3.
Example 3 A steel bar measures 0.20 m by 0.050 m by 0.030 m. The density of steel is 7800 kg/m3. Find its mass and its weight on Earth (g = 9.8 N/kg).
Step 1: volume = 0.20 × 0.050 × 0.030 = 3.0 × 10⁻4 m3.
Step 2: mass = ρ × V = 7800 × 3.0 × 10⁻4 = 2.34 kg.
Step 3: weight = m × g = 2.34 × 9.8 = 22.9 N.
Answer: mass 2.34 kg, weight 22.9 N.
Common errors
- Using "weight" and "mass" as if they were the same. A person's mass is in kg and their weight is in N.
- Saying mass changes on the Moon. Only the weight changes.
- Forgetting to convert cm3 to m3 (a factor of 106) when the density is given in kg/m3.
- Reading the volume of displaced water as the final reading instead of the difference between the final and starting readings.
- Saying that large objects sink and small ones float. It depends on density, not size.
Exam tip: For a density question, write the equation first, then substitute with units. If the units of mass and volume do not match the units of the density you want, convert before dividing.
Check yourself
- Find the weight of a 2.5 kg bag of sugar on Earth (g = 9.8 N/kg).
Answer
W = 2.5 × 9.8 = 24.5 N. - A liquid has a mass of 340 g and a volume of 250 cm3. Find its density in g/cm3.
Answer
ρ = 340 ÷ 250 = 1.36 g/cm3. - An object has a density of 800 kg/m3. Will it float or sink in water of density 1000 kg/m3? Give a reason.
Answer
It floats, because its density is less than the density of water. - An aluminium block has a volume of 0.0020 m3 and a density of 2700 kg/m3. What is its mass?
Answer
m = ρ × V = 2700 × 0.0020 = 5.4 kg.