Conservation laws

Name what is being conserved

A conservation law concerns a specified physical quantity. It can constrain a total even while the system's parts change. To use one, name the quantity, select the included objects or region, and declare the boundary and time interval. System choice determines which interactions are internal and which involve the surroundings.

This lesson uses energy and horizontal linear momentum in ordinary classical models. Energy balances track storage and transfers. Momentum balances track a quantity associated with motion and direction. They are different balances, with different units and conditions; neither says that every property of every object stays unchanged.

Energy is expressed here in joules, symbol J. Values being added must concern the same quantity and use compatible units. All numerical cases below are invented, with exact stipulated model inputs and consistent fixed energy references. Their arithmetic does not establish the precision of real measurements.

Balance energy storage and transfers

In a closed thermodynamic system, matter does not cross the boundary, but energy can. For the classical accounts here, include every relevant energy contribution in E. With nonnegative amounts received and delivered over the stated interval, write E_final = E_initial + energy received - energy delivered. The change is E_final - E_initial.

Heating and work can transfer energy across the boundary. A motor turning a shaft into a sealed vessel can transfer energy by work even when no matter crosses. Heat and work describe transfers, rather than extra stores to add again after the transferred energy has entered the system.

For a stationary closed-system process with no change of its overall kinetic or potential energy, the net transferred energy changes internal energy. Internal energy concerns the microscopic energy within the materials. If the whole system's motion or position changes, include the associated energy changes too; do not automatically assign every transfer to internal energy alone.

No net energy transfer gives equal initial and final energy in this balance. An isolated model excludes energy and matter exchange; a merely closed model need not. Equal incoming and outgoing amounts can also give zero net change over an interval even while transfers occur. The system's energy need not have stayed constant at every intermediate time.

A conserved total can have changing contributions

In these mechanical models, kinetic energy is associated with motion. Potential energy is associated with an interaction and its arrangement, such as the deformation of a spring. Mechanical energy is their sum. A conversion between those contributions need not change their sum under the appropriate mechanical conditions.

Mechanical energy is not every energy contribution. Friction or deformation can change mechanical energy into internal energy within an included system. A decline in mechanical energy therefore need not be a decline in total energy. Include the bodies receiving the energy, or account for transfer across the selected boundary.

For a cart, spring and track taken together, rubbing between cart and track is internal. For the cart alone, the track is outside. If spring potential energy is counted separately, the accompanying internal-energy entry must exclude it. We call that entry remaining internal energy. Use one accounting scheme consistently so its categories do not overlap.

Add momentum with direction included

For our slow-moving carts with constant masses in a Newtonian model, a horizontal momentum component is p = m v: mass multiplied by the signed horizontal velocity. Its unit is kg m/s. Choose positive to the right in one fixed reference frame treated as non-accelerating. A leftward velocity then has a negative sign. Add signed components, not speeds, to obtain total horizontal momentum.

When the included carts interact, each can change the other's momentum. In the stated Newtonian model, the changes due to their mutual interaction are equal and opposite. If the net external horizontal force is zero throughout the interval, their combined horizontal momentum stays constant. External forces need not all vanish; their net component in the direction being checked is what matters.

If outside interactions change the total horizontal momentum, include that change in the balance instead of setting it to zero. In a real short interaction, neglecting a small external effect may support an approximation. Check its size for the chosen interval and required accuracy; a track described as frictionless does not by itself exclude every other external force.

State the conditions behind a constant total

Closed has different uses in different discussions. Here the thermodynamic meaning excludes matter exchange and permits energy transfer. Some mechanics texts use closed for a system meeting momentum-conservation conditions. State the actual exchanges and external-force conditions rather than relying on the label alone.

A sealed vessel can receive energy. A cart can be pushed from outside without exchanging matter. Neither case justifies a constant total merely because the system is closed in the thermodynamic sense. Likewise, zero energy change over an interval does not by itself establish zero momentum change.

The fixed masses in our cart model are an ordinary classical approximation, not a universal separate conservation law for the sum of individual rest masses. Special relativity relates rest mass to rest energy; in particle or nuclear processes, individual rest masses and their sum can change while the full energy account includes rest energy. No relativistic calculation is needed for the supplied slow-cart model.

Use the balance to constrain and check a model

Write the initial and final totals, list boundary contributions during the same interval, and check units and signs. An unexplained difference may reflect an omitted transfer, a changing contribution, an unsuitable approximation or a measurement problem. A failed partial account does not by itself establish a failure of the conservation law.

A conserved sum also need not determine every part separately. Knowing a combined momentum gives one constraint on the individual momenta; further information about the interaction or one final motion may still be needed. Conservation narrows the possible accounts and helps expose inconsistencies, while a complete prediction can require additional physics.

Worked examples

Try it yourself

1. Two included bodies exchange energy internally. For the combined system, no energy crosses the outer boundary and all relevant contributions are included. Must each body's energy stay constant? What does the combined balance say?

   Show the answer and reasoning

   Each body's energy can change. The combined total stays constant under the stated conditions.

   An internal transfer leaves one part and enters another, so it cancels in the combined account. For either body alone, the other is outside and that transfer crosses the smaller system's boundary.

2. A stipulated classical energy account starts at 40 J, receives 5 J and delivers 11 J during an interval. Every relevant contribution and transfer is included, with the same reference at both endpoints. Find the final energy and its change. Has 6 J been destroyed?

   Show the answer and reasoning

   The final energy is 34 J and the change is -6 J. The account explains the change through net delivery to the surroundings.

   Calculate 40 J + 5 J - 11 J = 34 J, then 34 J - 40 J = -6 J. More energy leaves than enters. The selected system loses energy through its boundary; the balance supplies no destruction term.

3. In an invented isolated classical account, the complete changing energy contributions are a mechanical entry and a non-overlapping remaining internal entry. Mechanical energy changes from 25 J to 19 J. What change of the remaining internal entry is required? Does this alone identify a particular friction process?

   Show the answer and reasoning

   The remaining internal entry must increase by 6 J. The balance alone does not identify the mechanism.

   Mechanical energy changes by 19 J - 25 J = -6 J. With no boundary transfer and no other changing contribution, the remaining internal change must be +6 J. Additional physical information is needed to identify how the conversion occurred.

4. A stipulated Newtonian system has initial total horizontal momentum -3 kg m/s, with positive to the right. During the chosen interval, external interactions produce a signed total momentum change of +5 kg m/s. There is no matter exchange. What is the final total, and was it constant over the interval?

   Show the answer and reasoning

   The final total is +2 kg m/s. It was not constant over the interval.

   Add the specified external contribution: -3 kg m/s + 5 kg m/s = +2 kg m/s. No matter exchange alone does not remove the external force effect or justify setting the momentum change to zero.

5. A closed thermodynamic system first receives 8 J and later delivers 8 J. These are its only energy transfers and all relevant storage is included. Are the initial and final energies equal? Was it isolated, and must its energy have been constant at every intermediate time?

   Show the answer and reasoning

   Its endpoint energies are equal. It was not isolated, and its energy was not constant throughout this sequential transfer process.

   The net transfer over the full interval is 8 J - 8 J = 0. After receiving the first 8 J and before delivering it, the system's energy is above its initial value. Equal endpoint totals do not erase the boundary crossings.

6. A successful slow-cart calculation treats each cart's mass as constant and uses p = m v. Does that establish an exact law conserving the sum of individual rest masses in every particle or nuclear process?

   Show the answer and reasoning

   No. The calculation uses a stated classical approximation; it does not establish that universal mass claim.

   Relativistic energy accounts include rest energy, and the sum of individual rest masses can change in such processes. The conditions of the cart model must remain attached to its formula and result.

What to carry forward

• Name the quantity, system, boundary, interval and reference before using a balance.
• A selected system's energy change is accounted for by net boundary transfer when all relevant contributions are included.
• Constant total energy does not require constant mechanical energy or constant energy of every part.
• In the stated Newtonian model, add signed momenta and check the net external force in the chosen direction.
• Thermodynamic closure excludes matter exchange; it does not by itself guarantee constant energy or momentum.
• Keep domain, approximation and completeness conditions attached to a conservation calculation.