C2.4 Test conservation of energy during transformations

Ontario Grade 12 Physics

Energy and Momentum

SPH4U C2.4 | Tracking energy into, out of, and within a system

Energy can change form. A moving object can slow as its energy is transferred to its surroundings. A lamp can transfer electrical energy into light and thermal energy. Conservation of energy is the idea that energy is not created or destroyed; it is transferred or transformed. To test this idea, define the system, identify energy entering and leaving it, and compare the total energy before and after. A mismatch in measurements is evidence to investigate, not proof that energy vanished.

What you will learn

Describe how energy changes form during a transformation.
Choose a physical system and identify energy transfers across its boundary.
Use energy measurements or calculations to test whether total energy is conserved.
Explain why measured values may not match exactly without claiming that energy has disappeared.

1. Prerequisite bridge: energy, systems, and direction

In earlier physics, you used energy to describe motion and position. Kinetic energy is associated with motion. Gravitational potential energy is associated with an object's position in a gravitational field. Thermal energy is associated with the motion of particles within materials. Electrical energy can be transferred in a circuit. These are different ways to account for energy; they are not different substances.

A scalar has magnitude but no direction. Energy is a scalar, so it does not point north, up, or down. A vector has both magnitude and direction. Velocity and force are vectors. Keep this distinction clear: an object’s velocity can have a direction, while its kinetic energy does not.

The physical system is the object or group of objects being studied. The surroundings are everything outside that chosen system. The system boundary is the imagined line separating the two. Energy can cross that boundary through a transfer. A system is isolated for an analysis when no energy crosses its boundary. In real investigations, perfect isolation is difficult, so the boundary and possible transfers must be considered.

A reference frame is the viewpoint used to describe position and motion. For the examples here, use a frame fixed to the ground. Choose upward as the positive vertical direction when height or vertical motion is relevant. The sign of a velocity depends on this choice, but energy values are non-negative.

Define the system before deciding which energy forms belong in the comparison.
Energy is a scalar; velocity and force are vectors.
Energy crossing the system boundary must be included in the account.

2. The model: compare energy before and after

For a defined system, conservation of energy means the total energy remains constant when the system is isolated. Energy may change form inside the system. If energy enters or leaves, compare the change in the system's energy with that transfer rather than expecting the system's energy alone to stay constant.

For a simple transformation with no energy transfer across the boundary, add the relevant energy forms before and after. For example, as an object falls, gravitational potential energy can decrease while kinetic energy increases. If thermal energy and transfers are negligible in the model, the sum of these two forms is the same at the beginning and end.

A useful test is to calculate or measure the total energy at two stages. Use the same system boundary at both stages. Record the energy forms that matter, and state any assumptions, such as negligible energy transfer to the surroundings. If evidence does not match the prediction, check the boundary, the instruments, and the energy forms included.

In SI, energy is measured in joules, symbol

\mathrm{J}

. For motion, kinetic energy is calculated from mass in kilograms and speed in metres per second. Near Earth's surface, a change in gravitational potential energy can be calculated using mass, gravitational field strength, and vertical height change. The speed is a magnitude in the kinetic-energy equation; direction is not inserted as a negative sign.

A model is a simplified description used to make a prediction. A measured result is a value obtained from an instrument or recorded observation. A calculated example using supplied values is not a completed experiment. In a real test, report what was measured and distinguish it from the model's assumptions.

E_{\mathrm{total, before}}=E_{\mathrm{total, after}}

For an isolated system, total energy before equals total energy after.
For a system that exchanges energy, include the transfers across its boundary.
Compare like quantities in joules and state the limits of the model.

3. Turning conservation into a test

A fair test begins by stating the system and the two stages being compared. A stage is a chosen instant or condition, such as just before release and just after a motion ends. List the energy forms expected at each stage. Then calculate or measure each relevant amount and add them.

For a proposed investigation, one might use a falling object and measure its mass, starting height, and speed at selected points. Those measurements could be used to compare gravitational potential and kinetic energy. This is a proposed procedure, not evidence that an experiment has been performed. A real investigation would need suitable instruments and recorded data.

Measurements have uncertainty: an instrument's resolution and the method of reading it limit how precisely a value is known. Therefore, small differences between totals may be consistent with measurement limits. A larger difference may indicate an omitted energy transfer, an unsuitable system boundary, or an error in measurement or calculation. Do not label a difference as energy lost until you have checked where energy may have gone.

A practical test can also use electrical energy supplied to a device and energy transferred into other forms. If the device is the system, energy can cross into it electrically and leave as light, sound, or thermal transfer. A complete comparison must account for those outputs. Counting only the most visible output would give an incomplete energy account.

\Delta E_{

\mathrm{system}

}=E_{transferred\ in}-E_{

\mathrm{transferred\ out}

}

Plan which values are measured and which are calculated.
Include likely transfers and energy forms, not only the easiest ones to observe.
Use measurement limits when interpreting a mismatch.

4. Interpreting results responsibly

A result supports conservation when the accounted total before and after agrees within the precision of the evidence, or when a change in the system is accounted for by energy transfers. This is a test of a model against evidence, not a claim that every measurement must match perfectly.

Keep assumptions visible. If a calculation ignores thermal transfer to the surroundings, say so. If observed motion suggests that energy was transferred to the surroundings, the model may need a wider system or an additional energy term. Do not silently change the system between the start and end of the comparison.

Report a final value with sensible significant figures, the unit, and a brief reasonableness check. Joules must result from the energy calculations. Check that a computed energy is not negative when using a non-negative energy form such as kinetic energy. For changes in height, be clear whether the object moved up or down; for a positive energy amount, use the magnitude of the height change where appropriate.

A mismatch is a prompt to examine the model and evidence.
Use one consistent system boundary throughout the test.
State units, precision, assumptions, and relevant transfers.

Worked example

A falling object: predicted transformation

A

0.80\,\mathrm{kg}

object is released from rest and falls

2.0\,\mathrm{m}

. Use a ground-fixed frame, with upward positive. Neglect energy transfer to the surroundings. Compare the initial gravitational potential energy change in magnitude with the predicted kinetic energy gain. Use

g=9.8\,\mathrm{m/s^2}

.

Set the system and known values
Choose the object and Earth as the system so gravitational potential energy is included. The initial speed is zero. The object moves downward, opposite the positive vertical direction; however, the energy calculation uses the magnitude of the height change.
$m=0.80\,\mathrm{kg},\quad |\Delta h|=2.0\,\mathrm{m},\quad g=9.8\,\mathrm{m/s^2}$
Calculate the potential-energy decrease
The magnitude of the gravitational potential-energy change is mass times gravitational field strength times the magnitude of the height change. This gives the energy available for transformation in the stated model.
$|\Delta E_g|=(0.80\,\mathrm{kg})(9.8\,\mathrm{m/s^2})(2.0\,\mathrm{m})=16\,\mathrm{J}$
Compare with the predicted final motion
With no energy transfer to the surroundings in this model, the decrease in gravitational potential energy equals the increase in kinetic energy. The result is positive and has energy units.
$\Delta E_k=16\,\mathrm{J}$

Answer: The model predicts a kinetic-energy gain of

16\,\mathrm{J}

.

Check: The units reduce to joules because

\mathrm{kg\,m^2/s^2}=\mathrm{J}

. The energy is positive and reasonable for a small object falling a short distance. This is a prediction, not a measured result.

Worked example

Compare supplied before-and-after energy values

A data set for a defined system gives a total energy of

48\,\mathrm{J}

before a transformation and

45\,\mathrm{J}

after it. A recorded transfer of

3\,\mathrm{J}

leaves the system during the interval. Test whether the energy account is consistent. Treat these as supplied values, not as measurements made here.

Identify the system and transfer
The system is the object or group represented by the data set. Its stored total decreases, and the given transfer is directed out of the system. Use positive magnitudes for the energy amounts and state the transfer direction in words.
$E_{\mathrm{before}}=48\,\mathrm{J},\quad E_{\mathrm{after}}=45\,\mathrm{J},\quad E_{\mathrm{out}}=3\,\mathrm{J}$
Account for the energy leaving
For this system, the decrease in its energy should equal the energy transferred out if no other transfer is present in the account. Subtract the after value from the before value.
$E_{\mathrm{before}}-E_{\mathrm{after}}=48\,\mathrm{J}-45\,\mathrm{J}=3\,\mathrm{J}$
Judge the agreement
The calculated decrease matches the stated outgoing transfer. On these supplied values, the account is consistent. In a real test, the judgment would also consider measurement uncertainty and whether other transfers were omitted.
$3\,\mathrm{J}=E_{\mathrm{out}}$

Answer: The energy account is consistent: the system's energy decreases by

3\,\mathrm{J}

, equal to the stated energy transfer out.

Check: Both sides are in joules. The decrease has the correct direction for energy leaving the system. The conclusion is limited to the supplied account.

Worked example

Check a transformation with two energy forms

A cart is part of a defined system. At one stage it has

12\,\mathrm{J}

of kinetic energy and

5.0\,\mathrm{J}

of gravitational potential energy. At a later stage it has

9.0\,\mathrm{J}

of kinetic energy. Assuming no energy crosses the system boundary, find the later potential energy and test conservation.

Add the initial energy forms
The total at the first stage is the sum of the two energy forms specified for the system. Energy is scalar, so the amounts are added rather than combined as directions.
$E_{\mathrm{initial}}=12\,\mathrm{J}+5.0\,\mathrm{J}=17\,\mathrm{J}$
Use the unchanged total
The system is assumed isolated, so its total remains $17\,\mathrm{J}$ . Subtract the later kinetic energy to find the remaining gravitational potential energy.
$E_{g,\mathrm{later}}=17\,\mathrm{J}-9.0\,\mathrm{J}=8.0\,\mathrm{J}$
Verify the final total
Add the later energy forms. Matching the initial total tests the stated model. The potential energy is positive and the sum has units of joules.
$9.0\,\mathrm{J}+8.0\,\mathrm{J}=17\,\mathrm{J}$

Answer: The later gravitational potential energy is

8.0\,\mathrm{J}

, and the total remains

17\,\mathrm{J}

.

Check: The final sum agrees with the initial total. The values are physically consistent with the no-transfer assumption in the problem.

Common mistakes and how to avoid them

Saying that energy disappeared because one measured energy form became smaller.

Correction: Check for another energy form or an energy transfer across the system boundary.

Changing the system boundary between the beginning and end of a comparison.

Correction: Keep the same defined system, or clearly account for what crossed its boundary.

Treating a predicted value or a proposed procedure as experimental evidence.

Correction: Label calculations as predictions and report measurements only when they have actually been recorded.

Using a negative sign for kinetic energy because velocity points downward.

Correction: Kinetic energy is a scalar and uses speed, which is non-negative. Direction belongs to velocity.

Lesson summary

Energy can change form or cross a system boundary.
For an isolated system, the total energy before and after a transformation is equal.
A useful test identifies the system, accounts for relevant energy forms and transfers, and compares values in joules.
Interpret differences using the system boundary, assumptions, and measurement limits.

Check your understanding

Question 1

A system's initial total is

30\,\mathrm{J}

. It transfers

4\,\mathrm{J}

out and receives no energy. What should its final total be?

$34\,\mathrm{J}$
$26\,\mathrm{J}$
$30\,\mathrm{J}$
correctIndex

Show answer and explanation

26\,\mathrm{J}

The system's stored total falls by the energy transferred out:

30\,\mathrm{J}-4\,\mathrm{J}=26\,\mathrm{J}

.

Question 2

Which statement best describes kinetic energy in a falling-object calculation?

It is a vector that points downward.
It is a scalar calculated using the object's speed.
It is negative whenever velocity is downward.
correctIndex

Show answer and explanation

It is a scalar calculated using the object's speed.

Kinetic energy is a scalar. The object's velocity has a direction, but kinetic energy does not.

Question 3

A before-and-after comparison does not match exactly. What is the best first response?

Conclude that conservation of energy failed.
Check the system boundary, omitted transfers, measurements, and calculation.
Change the system boundary without explaining the change.
correctIndex

Show answer and explanation

Check the system boundary, omitted transfers, measurements, and calculation.

A mismatch calls for checking the evidence and energy account before drawing a conclusion.

Key terms

System: The object or group of objects chosen for analysis.
System boundary: The imagined separation between the system and its surroundings.
Energy transfer: Energy moving into or out of a system.
Isolated system: A system treated as having no energy transferred across its boundary.
Reference frame: The viewpoint used to describe position and motion.
Scalar: A quantity with magnitude but no direction.
Vector: A quantity with both magnitude and direction.
Uncertainty: A limit on how precisely a measured value is known.

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About this lesson and its review

Published by DoAssignment. This AI-assisted lesson follows Ontario Grade 12 Physics (SPH4U), expectation C2.4. 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.

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C2.4 · Test conservation of energy during transformations

Ontario Grade 12 Physics

SPH4U C2.4 | Tracking energy into, out of, and within a system

What you will learn

1. Prerequisite bridge: energy, systems, and direction

2. The model: compare energy before and after

3. Turning conservation into a test

4. Interpreting results responsibly

A falling object: predicted transformation

Compare supplied before-and-after energy values

Check a transformation with two energy forms

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Key terms

Continue through SPH4U

About this lesson and its review