Introduction to Physics and Measurement of Physical Quantities

Lessons

Senior 1 · Physics

Introduction to Physics and Measurement of Physical Quantities

5 lessons

Lesson 4 of 5

Mass, Weight, Volume and Density

25 min

Same mass, different size

Put a 1 kg bag of maize flour beside a 1 kg bag of charcoal. The balance says they have the same mass, but the charcoal bag is several times bigger. Now drop a small stone and a large dry log into Lake Kivu: the small stone sinks and the big log floats. Size alone does not decide what sinks. To explain both observations you need three quantities, mass, volume and density, and a fourth one that people often confuse with mass: weight.

Definition

Mass ( $m$ ) is the amount of matter in an object. SI unit: the kilogram (kg). It is measured with a balance and is the same everywhere.

Weight ( $W$ ) is the force of gravity pulling an object towards the Earth (or the Moon). It is a force, so its SI unit is the newton (N). It is measured with a spring balance.

W = m g

where $g$ is the gravitational field strength: about $9.8\,\mathrm{N\,kg^{-1}}$ on Earth and $1.6\,\mathrm{N\,kg^{-1}}$ on the Moon.

Measuring mass and weight

A beam balance (such as the triple beam balance) compares the object with known masses; an electronic balance shows the mass directly. A spring balance (newton meter) measures weight: the heavier the load, the more the spring stretches.

	Mass	Weight
What it is	amount of matter	pull of gravity on the object
SI unit	kilogram (kg)	newton (N)
Measured with	beam balance, electronic balance	spring balance (newton meter)
Changes from Earth to Moon?	no	yes: about six times smaller on the Moon
Scalar or vector?	scalar	vector (it acts downwards)

Worked example 1: weight on the Earth and on the Moon

A pupil has a mass of 50 kg. Find her weight on Earth and on the Moon.

W_{\text{Earth}} = mg = 50\,\mathrm{kg} \times 9.8\,\mathrm{N\,kg^{-1}} = 490\,\mathrm{N}

W_{\text{Moon}} = 50\,\mathrm{kg} \times 1.6\,\mathrm{N\,kg^{-1}} = 80\,\mathrm{N}

Her mass is 50 kg in both places: she is made of the same amount of matter.

Warning

Mass is not weight. In everyday speech people say "I weigh 50 kg". In physics that sentence gives a MASS. Her weight is 490 N. Never give a weight in kilograms, or a mass in newtons.

Definition

Volume ( $V$ ) is the amount of space an object takes up. SI unit: the cubic metre ( $\mathrm{m^3}$ ). Smaller units: $1\,\mathrm{cm^3} = 1\,\mathrm{mL}$ and $1\,\mathrm{dm^3} = 1\,\mathrm{L} = 1000\,\mathrm{cm^3}$ .

The meniscus is the curved surface of a liquid in a narrow tube. Water curves down in the middle.

Measuring the volume of a liquid

Pour the liquid into a measuring cylinder standing on a flat bench. Bend down until your eye is level with the liquid surface, and read the scale at the bottom of the meniscus.

Figure 4.1: With the eye level with the bottom of the meniscus the reading is 52 mL. An eye above or below the surface gives a parallax error.

Experiment: the volume of an irregular solid

A stone has no length, width and height you can multiply, so measure the water it pushes aside, the displacement method.

Half fill a measuring cylinder with water and read the volume, $V_{1}$ .
Tie the stone to a thread and lower it gently until it is completely under water.
Read the new volume, $V_{2}$ . The volume of the stone is $V = V_{2} - V_{1}$ .

For a stone too big for the cylinder, use a eureka (overflow) can: fill it to the spout, lower the stone in, and collect the water that overflows in a measuring cylinder.

Figure 4.2: The water level rises from 50 to 68 cubic centimetres, so the stone's volume is 18 cubic centimetres.

Important

Safety. Lower the stone on a thread; never drop it, because it can crack the glass. Wipe up spilled water at once so nobody slips.

Definition

Density ( $\rho$ , the Greek letter "rho") is the mass per unit volume of a substance:

\rho = \frac{m}{V}

SI unit: $\mathrm{kg\,m^{-3}}$ . A common laboratory unit is $\mathrm{g\,cm^{-3}}$ , and $1\,\mathrm{g\,cm^{-3}} = 1000\,\mathrm{kg\,m^{-3}}$ . Example: water has a density of $1000\,\mathrm{kg\,m^{-3}}$ , so $1\,\mathrm{m^3}$ of water has a mass of 1000 kg.

Why things float or sink

Density depends on the SUBSTANCE, not on the size of the piece. A solid object floats in water if its density is less than that of water, and sinks if it is greater.

Substance	Density ( $\mathrm{kg\,m^{-3}}$ )	In water it...
Cork	about 240	floats
Dry pine wood	about 500	floats
Ice	920	floats
Cooking oil	about 920	floats on water
Water	1000
Quartz sand	about 2650	sinks
Aluminium	2700	sinks
Cassiterite (tin ore)	about 7000	sinks
Iron	7870	sinks
Gold	19 300	sinks

Worked example 2: the density of a stone

The stone in Figure 4.2 has a mass of 48.6 g. Find its density in $\mathrm{g\,cm^{-3}}$ and in $\mathrm{kg\,m^{-3}}$ .

\rho = \frac{m}{V} = \frac{48.6\,\mathrm{g}}{18\,\mathrm{cm^3}} = 2.7\,\mathrm{g\,cm^{-3}} = 2700\,\mathrm{kg\,m^{-3}}

That is the density of many common rocks, such as granite.

Worked example 3: the mass of water in a jerrycan

A jerrycan holds 20 L of water. Find the mass and the weight of the water ( $g = 9.8\,\mathrm{N\,kg^{-1}}$ ).

$V = 20\,\mathrm{L} = 0.020\,\mathrm{m^3}$ . Rearrange $\rho = \frac{m}{V}$ to $m = \rho V$ :

m = 1000\,\mathrm{kg\,m^{-3}} \times 0.020\,\mathrm{m^3} = 20\,\mathrm{kg}

W = mg = 20\,\mathrm{kg} \times 9.8\,\mathrm{N\,kg^{-1}} = 196\,\mathrm{N}

Tip

Cover the quantity you want in the triangle $m$ over $\rho \times V$ : covering $\rho$ leaves $\frac{m}{V}$ ; covering $m$ leaves $\rho \times V$ ; covering $V$ leaves $\frac{m}{\rho}$ . Always check that the units match: grams with $\mathrm{cm^3}$ , or kilograms with $\mathrm{m^3}$ .

Density at work in Rwanda's mines

Rwanda mines the ores of tin, tantalum and tungsten, the "3Ts": cassiterite, coltan and wolframite. It is one of the world's largest producers of tantalum. At many mines the crushed rock is washed with running water down a sloping channel. The ore grains are two to three times denser than the quartz sand around them, so they sink and collect at the bottom while the water carries the light sand away. Miners are using density to separate the valuable mineral from the waste.

Try it yourself: in the PhET: Density (version 1.2.8) simulation, measure the mass and volume of blocks of different materials, calculate each density, and predict which will float before you drop it in the water.

Measuring the density of an irregular solid (Mr Smith's Physics online). Watch how the mass is measured first and the volume is then found by displacement.

✦

Practice

Give two differences between mass and weight.
A bag of cement has a mass of 50 kg. Find its weight on Earth ( $g = 9.8\,\mathrm{N\,kg^{-1}}$ ).
A measuring cylinder holds 35 $\mathrm{cm^3}$ of water. When a key is lowered into it, the level rises to 41 $\mathrm{cm^3}$ . What is the volume of the key?
A block of wood measures 10 cm by 5 cm by 4 cm and has a mass of 120 g. Find its density in $\mathrm{g\,cm^{-3}}$ and in $\mathrm{kg\,m^{-3}}$ . Will it float in water?
A piece of metal has a mass of 540 g and a volume of 200 $\mathrm{cm^3}$ . Find its density and use the table to suggest which metal it is.
Cooking oil has a density of $920\,\mathrm{kg\,m^{-3}}$ . What mass of oil fills a 5.0 L container?
An astronaut has a mass of 70 kg on Earth. What are her mass and her weight on the Moon ( $g = 1.6\,\mathrm{N\,kg^{-1}}$ )?
Why must you read a measuring cylinder with your eye level with the bottom of the meniscus?

Show answersHide answers

Any two: mass is the amount of matter, weight is a force; mass is in kg, weight in N; mass is measured with a balance, weight with a spring balance; mass is the same everywhere, weight changes with $g$ ; mass is a scalar, weight a vector.
$W = 50 \times 9.8 = 490\,\mathrm{N}$
$41 - 35 = 6\,\mathrm{cm^3}$
$V = 10 \times 5 \times 4 = 200\,\mathrm{cm^3}$ ; $\rho = \frac{120}{200} = 0.60\,\mathrm{g\,cm^{-3}} = 600\,\mathrm{kg\,m^{-3}}$ . It floats, because it is less dense than water.
$\rho = \frac{540}{200} = 2.7\,\mathrm{g\,cm^{-3}} = 2700\,\mathrm{kg\,m^{-3}}$ : aluminium.
$V = 5.0\,\mathrm{L} = 0.0050\,\mathrm{m^3}$ ; $m = \rho V = 920 \times 0.0050 = 4.6\,\mathrm{kg}$
Mass: still 70 kg. Weight: $70 \times 1.6 = 112\,\mathrm{N}$ .
Looking from above or below gives a parallax error: the reading would be too high or too low. The bottom of the meniscus is the agreed level for water.

Key points

Mass (kg) is the amount of matter; weight (N) is the pull of gravity: $W = m g$ .
Read a measuring cylinder at the bottom of the meniscus with your eye level with it. Find the volume of an irregular solid by displacement.
Density $\rho = \frac{m}{V}$ , in $\mathrm{kg\,m^{-3}}$ or $\mathrm{g\,cm^{-3}}$ ; $1\,\mathrm{g\,cm^{-3}} = 1000\,\mathrm{kg\,m^{-3}}$ .
Objects less dense than water float; denser objects sink.

Sources and credits

11.2: Density, from College Physics (1st edition) by Paul Peter Urone and Roger Hinrichs, OpenStax, hosted by LibreTexts, licensed under CC BY 4.0. Adapted by Ganzaa.
Photo: Triple beam balance 1.jpg by GOKLuLe 盧樂, Wikimedia Commons, licensed under CC BY-SA 3.0.
Photo: A measuring cylinder.jpg by Jeremyida002, Wikimedia Commons, licensed under CC BY-SA 4.0.
Density (version 1.2.8). Simulation by PhET Interactive Simulations, University of Colorado Boulder, licensed under CC BY 4.0 (https://phet.colorado.edu)
Video: MEASURING THE DENSITY OF AN IRREGULAR SOLID - science experiments to try yourself, Mr Smith's Physics online, embedded from YouTube.

The worked examples, diagrams, Rwandan applications and practice exercises are written by Ganzaa.

1Same mass, different size

2Measuring mass and weight

3Measuring the volume of a liquid

4Experiment: the volume of an irregular solid

5Why things float or sink

6Density at work in Rwanda's mines

Same mass, different size

Measuring mass and weight

Measuring the volume of a liquid

Experiment: the volume of an irregular solid

Why things float or sink

Density at work in Rwanda's mines