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C2.3 · Analyse mechanical and thermal energy systems through inquiry

Learn to analyse mechanical and thermal energy systems through inquiry through clear examples and targeted practice.

Ontario Grade 12 Physics

Energy and Momentum

A Grade 12 guide to defining a system, gathering evidence, and interpreting energy changes

In SPH3U, you studied motion, forces, and forms of energy. A scalar, such as energy, has magnitude but no direction. A vector, such as displacement or force, has both magnitude and direction. This lesson uses those ideas to analyse mechanical and thermal energy systems through inquiry. An inquiry is a planned way to ask a question, gather evidence, and use that evidence to support a conclusion. Values in worked examples are supplied for practice; they are not presented as measurements from a completed student investigation.

What you will learn

Define a physical system and choose a reference frame and positive direction before analysing energy.
Use energy relationships to describe changes in mechanical and thermal energy.
Plan or interpret an inquiry using observations, measurements, and a clear conclusion.
Check calculations for units, direction, and physical reasonableness.

1. Set up the system and the inquiry

A physical system is the object or group of objects you choose to study. State what is inside the system and what is outside it, in the surroundings. This choice matters because energy can move between the system and its surroundings. For example, a cart and track may form one system. If you include only the cart, the track is outside the system.

A reference frame is the viewpoint used to describe position and motion. For a cart moving along a level track, use the track as the frame. Choose one direction, such as right, as positive. Position and velocity along the track can then be positive or negative. Energy is a scalar, so it has no direction. A change in energy can have a positive or negative sign.

An inquiry begins with a focused question. Identify what you will change, what you will observe or measure, and what you will keep the same. A measured value is evidence collected with an instrument or observation method. A proposed procedure describes what someone could do; it is not evidence that the procedure has been performed. A simulation produces model output, not direct measurements of a physical system.

Name the system and its surroundings.
State the reference frame and positive direction when describing motion.
Separate measured evidence from proposed procedures and simulated results.

2. Model mechanical energy changes

Mechanical energy includes kinetic energy, the energy of motion, and gravitational potential energy, which depends on an object's height relative to a chosen level. Mass is measured in kilograms, speed in metres per second, and height in metres. Both forms of energy are measured in joules.

For a system in which these are the relevant energy forms, total mechanical energy is their sum. If no energy is transferred into or out of the system and no other energy forms need to be included, mechanical energy remains constant. In many real setups, friction or another interaction transfers some mechanical energy into thermal energy. Thermal energy is associated with the internal motion and interactions of particles in matter. A temperature change can be evidence of a thermal energy change, but temperature and thermal energy are not the same quantity.

A useful energy-accounting model says that the change in energy stored in a system equals energy transferred into it minus energy transferred out. Here, the symbols for transferred energy refer to amounts, so they are not directions or vectors. The system boundary determines which transfers count as entering or leaving.

If the system includes both the moving object and the surface that warms, energy can change form within that system. If only the object is included, energy may instead appear to leave the system. State the boundary before interpreting an energy change.

E_k=

\frac{1}{2}

mv^2, E_g=mgh, E_{mech}=E_k+E_g, \Delta E_{system}=E_{in}-E_{out}

Kinetic and gravitational potential energy are measured in joules.
Energy can be transferred between a system and its surroundings or change form within the system.
Friction may reduce mechanical energy while increasing thermal energy in a suitably defined system.

3. Make inquiry evidence useful

A fair investigation changes one chosen factor while keeping other relevant conditions as consistent as possible. Use a clear procedure, suitable measuring tools, and repeated trials when appropriate. Record units and describe measurement limits honestly. Do not alter or invent data to make a pattern clearer.

A prediction is what you expect before collecting evidence. A conclusion is an evidence-based response after examining results. Identify a pattern in the observations, compare it with the prediction or energy model, and state limits. A trend in a small data set can support a claim, but it does not show that every possible setup will behave identically.

For a thermal energy inquiry, a possible question is how a cart's starting height affects the warming of a surface during a run. A proposed procedure could use the same cart, track, release method, and surface while changing the starting height. A student could record relevant observations, including temperature readings before and after each run. Until the procedure is carried out, it is only a proposal.

When interpreting results, look for missing energy pathways. Sound, deformation, and energy transferred to surroundings may matter in some setups. Do not claim that all lost mechanical energy became thermal energy unless the system boundary and evidence support that claim.

Control relevant conditions and record the method and units.
A conclusion must refer to evidence, not only to a prediction.
Consider measurement limits and possible energy transfers before making a broad claim.

4. Calculate, interpret, and check

For a calculation, list known values and the unknown before substituting. Keep units through the calculation. Round only at the end, using precision appropriate to the supplied values. A calculated energy should have units of joules. A negative change in a particular energy store means that store decreased; it does not mean the energy itself has a negative direction.

A complete analysis connects the number to the system. Explain which energy store changed and where energy may have gone. Check that the units match the quantity, the sign agrees with the comparison, and the size is plausible for the situation. In an inquiry, also ask whether the evidence is sufficient to support the conclusion.

Substitute values with SI units and report a sensible number of significant figures.
Explain what a calculated change means for the chosen system.
Check units, sign, reasonableness, and evidence quality.

Worked example

Compare gravitational and kinetic energy

A 0.60 kg cart moves at 2.0 m/s at a point 0.50 m above the chosen zero-height level. Find its kinetic energy, gravitational potential energy, and total mechanical energy. Use the floor as the reference level and upward as positive for height. Ignore energy transfers for this calculation.

Define the system
The system is the cart and Earth, so gravitational potential energy is included. The track is treated as outside the system. The cart's direction of travel does not affect these scalar energy values.
Calculate kinetic energy
Use the cart's mass and speed. The speed is squared, so the result is positive regardless of direction. The governing relationship is kinetic energy equals one-half mass times speed squared.
$E_k=\frac{1}{2}(0.60\,\mathrm{kg})(2.0\,\mathrm{m/s})^2=1.2\,\mathrm{J}$
Calculate gravitational potential energy
Use $g=9.8\,\mathrm{m/s^2}$ and the positive height above the chosen level. The governing relationship is gravitational potential energy equals mass times gravitational field strength times height.
$E_g=(0.60\,\mathrm{kg})(9.8\,\mathrm{m/s^2})(0.50\,\mathrm{m})=2.94\,\mathrm{J}$
Add the energy stores
Total mechanical energy is the sum of kinetic and gravitational potential energy. Round to two significant figures because the given mass, speed, and height have two significant figures.
$E_{\mathrm{mech}}=1.2\,\mathrm{J}+2.94\,\mathrm{J}\approx4.1\,\mathrm{J}$

Answer: The cart has kinetic energy of 1.2 J, gravitational potential energy of 2.94 J, and total mechanical energy of approximately 4.1 J.

Check: Each energy is in joules. The total is greater than either component and is reasonable for a small cart at modest speed and height. Energy has no direction.

Worked example

Interpret a supplied energy-change data set

For a practice analysis, a supplied data set gives a cart's initial mechanical energy as 3.6 J and its final mechanical energy as 2.8 J. The surrounding track's measured thermal energy increase is reported as 0.5 J. Find the mechanical energy change and compare the two changes. These are values supplied for the example, not results claimed from a student experiment.

Define the system and comparison
First consider the cart alone for the mechanical energy change. Then compare that decrease with the supplied thermal energy increase of the track. The system boundary differs between those statements, so the comparison is informative but does not by itself account for every energy transfer.
Find the mechanical energy change
Change means final value minus initial value. A negative result indicates a decrease in the cart's mechanical energy.
$\Delta E_{\mathrm{mech}}=2.8\,\mathrm{J}-3.6\,\mathrm{J}=-0.8\,\mathrm{J}$
Compare the reported changes
The track's reported thermal increase is 0.5 J, which is less than the 0.8 J mechanical decrease. The difference may reflect energy transferred to other surroundings or limits in the supplied measurements. The values alone do not identify which explanation is correct.
$0.8\,\mathrm{J}-0.5\,\mathrm{J}=0.3\,\mathrm{J}$

Answer: The cart's mechanical energy decreases by 0.8 J. The track's reported thermal energy increases by 0.5 J, leaving a 0.3 J difference that requires further evidence to interpret.

Check: All energy changes are in joules. The negative sign applies to the cart's mechanical energy change, not to energy as a direction. The comparison does not justify claiming that the full decrease became thermal energy in the track.

Worked example

Plan a thermal energy inquiry

You want to investigate whether release height is associated with a change in a track's temperature after a cart passes. Give a controlled inquiry plan and explain what evidence would be needed before making a conclusion.

State the question and system
Use the cart and track as the physical system. Use the track as the reference for the cart's motion, with downhill chosen as positive. The inquiry question is whether changing release height is associated with a different measured track temperature change.
Identify the changed and controlled conditions
Change release height. Keep the cart, track surface, release method, temperature measurement locations, and time between the run and readings as consistent as possible. These controls make comparisons more meaningful.
Describe evidence collection
A proposed procedure would record the track temperature before and after each run, the release height, and repeated trials at each height. Record the thermometer's resolution and units. This describes a plan; it does not claim that measurements have been collected.
Set a conclusion rule
Compare the temperature changes across heights and consider variation between repeated trials and instrument resolution. Conclude only that the results support or do not support an association in this setup. Temperature evidence alone does not measure all thermal energy transferred.

Answer: A valid plan changes release height, controls other relevant conditions, and records repeated before-and-after temperature readings with units and instrument limits. A conclusion must be based on those actual observations.

Check: The direction convention is stated, but energy and temperature remain scalar quantities. No outcome can be claimed until data are collected and assessed.

Common mistakes and how to avoid them

Treating a decrease in mechanical energy as energy disappearing.

Correction: State the system boundary and consider energy transferred to surroundings or changed into thermal energy or another form.

Saying that a temperature increase directly gives the amount of thermal energy gained.

Correction: Temperature is an observation about a system, not the same quantity as thermal energy. Use suitable evidence and avoid claiming an energy amount without a supported method.

Using a proposed procedure or simulated output as if it were measured evidence.

Correction: Label plans, simulations, and measurements accurately. Draw conclusions only from evidence that was actually obtained or explicitly supplied.

Reporting a negative energy value as a direction.

Correction: Energy is scalar. A negative change means the selected energy store decreased from its initial value.

Lesson summary

Define the system, surroundings, reference frame, and positive direction before analysing motion.
Mechanical energy can include kinetic and gravitational potential energy; energy may transfer or change form.
Inquiry connects a clear question and controlled method to measured evidence and a limited conclusion.
Check units, signs, physical reasonableness, and whether the evidence supports the claim.

Check your understanding

Question 1

A cart's speed is doubled while its mass stays the same. What happens to its kinetic energy?

It doubles.
It becomes four times as large.
It is halved.
It stays the same.

Show answer and explanation

It becomes four times as large.

Kinetic energy depends on the square of speed. Doubling speed multiplies kinetic energy by four.

Question 2

A practice data set shows a system's mechanical energy decreases by 0.6 J and a nearby surface's thermal energy increases by 0.4 J. What is the best conclusion?

All 0.6 J became thermal energy in the surface.
Energy has disappeared.
The reported increase is smaller than the mechanical decrease, so other transfers or measurement limits may need investigation.
The surface gained 1.0 J of thermal energy.

Show answer and explanation

The reported increase is smaller than the mechanical decrease, so other transfers or measurement limits may need investigation.

The two reported changes do not fully account for each other. More information is needed to identify other transfers or measurement limits.

Question 3

Which statement is accurate before an investigation has been carried out?

The proposed procedure proves the predicted result.
A simulation is a direct measurement of the physical setup.
The procedure is a plan, and no measured conclusion is available yet.
A prediction can replace the need to record observations.

Show answer and explanation

The procedure is a plan, and no measured conclusion is available yet.

A proposed method is not collected evidence. A conclusion needs observations or measurements from the inquiry.

Key terms

Physical system: The object or group of objects selected for analysis.
Reference frame: The viewpoint used to describe position and motion.
Scalar: A quantity with magnitude but no direction, such as energy.
Vector: A quantity with both magnitude and direction, such as displacement.
Inquiry: A planned process of asking a question, gathering evidence, and using it to support a conclusion.
Thermal energy: Energy associated with the internal motion and interactions of particles in matter.

Continue through SPH4U

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Published by DoAssignment. This AI-assisted lesson follows Ontario Grade 12 Physics (SPH4U), expectation C2.3. 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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