D2.8 Investigate conservation of mass and energy using mass-energy equivalence

Ontario Grade 11 Physics

Energy and Society

Investigating conservation using mass–energy equivalence

In many familiar changes, such as warming an object, the object's mass change is far too small to notice. Mass–energy equivalence explains why: a change in a system's energy is associated with a change in its mass. This lesson uses one relationship to keep track of both. The focus is on how to investigate the idea and how to judge what calculations do—and do not—show.

What you will learn

Explain why mass and energy must be considered together when studying conservation.
Use mass–energy equivalence to calculate the energy equivalent of a mass change, or the mass equivalent of an energy change.
Plan an investigation that distinguishes measured evidence from a model or calculation.

1. Prerequisite bridge: define the system and the quantities

A system is the object or group of objects being studied. Before discussing conservation, draw an imaginary boundary around it. Energy can cross that boundary; matter may also cross unless the system is sealed. A sealed system has no matter entering or leaving. For a clear energy account, state whether energy can enter or leave.

Mass is a scalar quantity: it has a size but no direction. Its SI unit is the kilogram (kg). Energy is also a scalar. Its SI unit is the joule (J). A vector, such as displacement, has both a size and a direction, but vectors are not needed to use the mass–energy relationship in this lesson.

Conservation means that the total amount in a stated account stays constant. In this topic, the account includes mass and energy together. A system may gain or lose energy, so its energy and associated mass can change. The combined account is conserved when transfers across the system boundary are included.

Define the system boundary before deciding what is conserved.
Mass and energy are scalars; use kg for mass and J for energy.
Track transfers into and out of the system.

2. The model: mass–energy equivalence

Mass–energy equivalence is the relationship between a system's mass and its energy. The symbol

E

means energy in joules,

m

means mass in kilograms, and

c

is the speed of light in a vacuum. Use

c = 3.00 \times 10^8\ \mathrm{m/s}

. The square on

c

means multiply the speed by itself.

The relationship lets us express an energy change as an equivalent mass change. If energy enters a system, the system's mass increases by the corresponding amount. If energy leaves, its mass decreases by that amount. This change is usually extremely small in everyday situations.

For an investigation, choose a system and record the energy transfer you are considering. Use the relationship to predict the associated mass change. A calculation or computer model is a prediction; it is not a physical measurement. A proposed experiment would need suitable equipment and a clear account of energy and matter crossing the boundary. Do not report predicted values as measured evidence.

A useful investigation can compare a calculated mass change with the precision of an available balance. If the predicted change is much smaller than the balance can resolve, the balance cannot provide direct evidence of that change. This does not disprove the relationship; it shows a limit of the measurement.

E = mc^2

Use $E = mc^2$ and rearrange it to find either quantity.
Energy entering corresponds to an increase in system mass; energy leaving corresponds to a decrease.
A prediction from a model is not the same as a measurement.

3. Investigating conservation with a careful account

First state the system. Next identify any energy transfer and its direction: into the system or out of it. Then calculate the mass equivalent of that transfer. Keep the sign description in words, since mass and energy here are scalar quantities rather than direction vectors.

For a system that receives energy, compare its final mass with its initial mass. For a system that gives energy to its surroundings, compare its initial and final mass in the same way. The total account remains consistent when the transferred energy is included. Do not claim that mass alone is unchanged in every energy transfer.

A practical investigation might use a proposed procedure, a computer model, or measurements from suitable equipment. For a model, enter a stated energy transfer and calculate its mass equivalent. For a physical investigation, identify what instruments would measure, their resolution, and possible transfers that are not included. Report only actual measurements as measured evidence. If the expected mass change is below the instrument's resolution, say that the measurement cannot resolve it.

The check is not simply whether a balance shows no change. Ask whether the system boundary was defined, whether energy transfers were counted, and whether the instrument could detect the predicted change. These steps make the investigation honest and the conclusion proportional to the evidence.

\Delta m = \frac{\Delta E}{c^2}

Describe transfer direction as into or out of the system.
Include energy transfers when checking the total account.
Separate model predictions, proposed methods, and actual measurements.

Worked example

Energy added to a system

A defined system receives

450\ \mathrm{J}

of energy. Find the associated mass increase. Use

c = 3.00 \times 10^8\ \mathrm{m/s}

.

Set the account
The system is the object or group receiving energy. The transfer is $450\ \mathrm{J}$ into the system, so the associated mass change is positive. The unknown is the mass increase.
Choose the relationship
Rearrange mass–energy equivalence to find mass from energy. The units of $c^2$ are square metres per square second.
$\Delta m = \frac{\Delta E}{c^2}$
Substitute and calculate
Substitute the energy in joules and the speed in metres per second. Since $1\ \mathrm{J} = 1\ \mathrm{kg\,m^2/s^2}$ , the units reduce to kilograms.
$\Delta m = \frac{450\ \mathrm{J}}{(3.00 \times 10^8\ \mathrm{m/s})^2} = 5.00 \times 10^{-15}\ \mathrm{kg}$

Answer: The system's mass increases by

5.00 \times 10^{-15}\ \mathrm{kg}

.

Check: The result is positive because energy enters. Its unit is kg. It is extremely small compared with everyday masses, which is reasonable for a transfer of only a few hundred joules.

Worked example

Energy leaving a system

A system transfers

1.20 \times 10^6\ \mathrm{J}

of energy to its surroundings. Find the associated change in the system's mass.

Set the system and sign
The system is the object or group that gives energy to its surroundings. Energy leaves, so the system's energy change and associated mass change are negative. The amount transferred is $1.20 \times 10^6\ \mathrm{J}$ .
Apply mass–energy equivalence
Use the signed energy change so that the result shows the direction of the change. The speed of light is $3.00 \times 10^8\ \mathrm{m/s}$ .
$\Delta m = \frac{\Delta E}{c^2}$
Substitute with units
Energy leaves, so substitute a negative energy change. The joule and speed units reduce to kilograms.
$\Delta m = \frac{-1.20 \times 10^6\ \mathrm{J}}{(3.00 \times 10^8\ \mathrm{m/s})^2} = -1.33 \times 10^{-11}\ \mathrm{kg}$

Answer: The system's mass decreases by

1.33 \times 10^{-11}\ \mathrm{kg}

.

Check: The negative sign matches energy leaving the defined system. The unit is kg. The small change is reasonable because even a large everyday energy transfer corresponds to a very small mass change.

Worked example

Energy equivalent of a small mass change

A calculation for a defined system gives a mass decrease of

2.00 \times 10^{-9}\ \mathrm{kg}

. Find the energy equivalent that left the system.

Identify the change
The system is the one whose mass change was calculated. A decrease means the signed mass change is negative. The unknown is the associated energy change.
Use the direct relationship
Multiply the signed mass change by the square of the speed of light. This keeps the sign consistent with energy leaving.
$\Delta E = \Delta m c^2$
Calculate with SI units
Substitute mass in kilograms and speed in metres per second. The units become kilogram metres squared per second squared, which is a joule.
$\Delta E = (-2.00 \times 10^{-9}\ \mathrm{kg})(3.00 \times 10^8\ \mathrm{m/s})^2 = -1.80 \times 10^8\ \mathrm{J}$

Answer: The system loses

1.80 \times 10^8\ \mathrm{J}

of energy.

Check: The negative sign indicates energy leaves the system. The unit is J. The large energy is consistent with the very large value of

c^2

, even though the mass change is tiny.

Common mistakes and how to avoid them

Treating mass as conserved on its own during every energy transfer.

Correction: For this expectation, account for mass and energy together. A system's mass can change when energy enters or leaves.

Using

c

instead of

c^2

.

Correction: The relationship uses the speed of light squared. Check that the calculation uses

(3.00 \times 10^8\ \mathrm{m/s})^2

.

Calling a calculated or simulated result a measurement.

Correction: Label calculations and simulations as predictions or model results. Call a value measured evidence only when it comes from an actual measurement.

Ignoring the system boundary or the direction of an energy transfer.

Correction: State what is inside the system and whether energy enters or leaves. This determines the sign of the associated change.

Lesson summary

Mass and energy are scalar quantities measured in kilograms and joules.
Mass–energy equivalence is expressed as $E = mc^2$ .
Energy entering a system corresponds to increased mass; energy leaving corresponds to decreased mass.
A sound investigation defines its system, tracks transfers, and distinguishes model predictions from measured evidence.

Check your understanding

Question 1

A system receives energy. What is the direction of its associated mass change?

The mass increases.
The mass decreases.
The mass must stay exactly the same.
correctIndexов 0,

Show answer and explanation

The mass increases.

Energy entering the defined system corresponds to a positive mass change through mass–energy equivalence.

Question 2

A model predicts a mass change smaller than the resolution of the balance being considered. What is the best conclusion?

The balance can confirm the predicted change exactly.
The prediction is a measured result.
That balance cannot resolve the predicted change.
correctIndex

Show answer and explanation

That balance cannot resolve the predicted change.

An instrument cannot distinguish a change smaller than its resolution. A model prediction is not a measured result.

Question 3

A system loses

3.00\ \mathrm{J}

of energy. What is the sign of its associated mass change?

Positive, because joules are positive units.
Negative, because energy leaves the system.
Zero, because mass and energy are different quantities.
correctIndex

Show answer and explanation

Negative, because energy leaves the system.

The energy change is negative for the defined system, so the associated mass change is also negative.

Key terms

System: The object or group of objects chosen for study.
System boundary: The stated edge that separates the system from its surroundings.
Conservation: Keeping a stated total account consistent, including transfers across the system boundary.
Mass–energy equivalence: The relationship between a mass and its energy, expressed by $E = mc^2$ .
Resolution: The smallest change an instrument can distinguish.

Continue through SPH3U

View the complete SPH3U Ontario Grade 11 Physics curriculum and lessons

About this lesson and its review

Published by DoAssignment. This AI-assisted lesson follows Ontario Grade 11 Physics (SPH3U), expectation D2.8. It is a study resource, not an official curriculum publication.

Before publication, the draft is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability, and then requires administrator approval. Errors can still occur, so corrections are welcomed.

Official curriculum reference

Report a correction or ask a question

D2.8 · Investigate conservation of mass and energy using mass-energy equivalence

Ontario Grade 11 Physics

Investigating conservation using mass–energy equivalence

What you will learn

1. Prerequisite bridge: define the system and the quantities

2. The model: mass–energy equivalence

3. Investigating conservation with a careful account

Energy added to a system

Energy leaving a system

Energy equivalent of a small mass change

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Key terms

Continue through SPH3U

About this lesson and its review