Unit 4 · Topic 06

Conservative forces and energy conservation

Conservative forces allow their work to be represented through potential energy. Mechanical energy is conserved under specific conditions, while total-energy conservation is more general.

Unit 4ConservationOpen navigation

Concept 01

Path-independent work

Essential The minimum you should retain

A conservative force does work between two positions that depends only on the endpoints.

UnderstandInterpret and connect

For that force, Wc=−ΔU W_c=-\Delta U .

DeepenFormulation and conditions

Its total work over a closed path is zero.

ExploreConnections for further study

Two different paths from A to B give the same W_c even when their geometries differ.

Mathematical relation

Work by a conservative force

Wc=−ΔU W_c=-\Delta U
Represents

The relation between conservative work and potential-energy change.

Physical interpretation

Work depends only on endpoints and is zero over a closed path.

DeepenVariables, conditions, and checks

Variables

W_c
work by the conservative force; usual unit: J
ΔU
potential-energy change; usual unit: J

Conditions of application

  • The interaction is represented by U.

Dimensional check

J on both sides.

Errors it helps prevent

  • Giving W_c and ΔU the same sign.
Two different curves connect the same points A and B.ABsame W_c(A→B)

For a conservative force, both paths from A to B give the same work despite their different shapes.

Concept 02

When K+U is conserved

Essential The minimum you should retain

Mechanical energy is E_mech=K+U.

UnderstandInterpret and connect

If only conservative forces do relevant work, Ki+Ui=Kf+Uf K_i+U_i=K_f+U_f .

DeepenFormulation and conditions

Mechanical-energy conservation is a model condition, not an automatic rule for every system.

ExploreConnections for further study

Comparing states through energy often avoids calculating intermediate forces and times.

Worked example

Descent with friction

A 5.0 kg block starts from rest, descends 2.0 m, and friction does -30 J of work. Use g=9.8 m/s².

Given
  • m=5.0 kg
  • h=2.0 m
  • W_f=-30 J
Target

Calculate final speed.

  1. Reference

    Choose U_g=0 at the final point.

  2. Initial state

    K_i=0 and U_i=mgh=98 J.

  3. Balance

    K_f+U_f=K_i+U_i+W_f=68 J.

  4. Calculation

    (1/2)(5)v²=68, so v≈5.22 m/s.

Conclusion

Final speed is about 5.22 m/s; mechanical energy falls by 30 J and may appear as internal energy.

Mathematical relation

Mechanical energy and conservation

Emech=K+U,Ki+Ui=Kf+Uf E_{mech}=K+U,\quad K_i+U_i=K_f+U_f
Represents

Mechanical energy and its conservation when only conservative forces do relevant work.

Physical interpretation

Mechanical energy is not universally identical to total energy.

DeepenVariables, conditions, and checks

Variables

E_mech
mechanical energy; usual unit: J
K
kinetic energy; usual unit: J
U
potential energy; usual unit: J

Conditions of application

  • Only conservative forces do relevant work in the mechanical balance.

Dimensional check

Every term is in J.

Errors it helps prevent

  • Conserving K+U with unaccounted non-conservative work.

Concept 03

Balance with other forces

Essential The minimum you should retain

Forces not included in U can change mechanical energy.

UnderstandInterpret and connect

A useful balance is Δ(K+U)=Wother \Delta(K+U)=W_{other} .

DeepenFormulation and conditions

The system boundary determines which interaction appears as external work and which as internal energy.

ExploreConnections for further study

Different system boundaries can describe the same process correctly if their terms are not mixed.

Mathematical relation

Energy balance with other transfers

Δ(K+U)=Wother,ΔK+ΔU+ΔEint=0 \Delta(K+U)=W_{other},\quad\Delta K+\Delta U+\Delta E_{int}=0
Represents

Two compatible balances for declared system boundaries.

Physical interpretation

Friction can reduce mechanical energy without destroying total energy.

DeepenVariables, conditions, and checks

Variables

W_other
work by forces not included in U; usual unit: J
E_int
internal energy; usual unit: J

Conditions of application

  • The system boundary and transfers are declared.
  • The two forms are not mixed without redefining the boundary.

Dimensional check

Every term is in J.

Errors it helps prevent

  • Saying friction destroys energy.
  • Confusing mechanical with total energy.

Concept 04

Mechanical energy is not total energy

Essential The minimum you should retain

Friction can reduce K+U while internal energy increases.

UnderstandInterpret and connect

For a suitable isolated system, ΔK+ΔU+ΔE_int=0.

DeepenFormulation and conditions

Non-conservative does not mean that a force violates energy conservation.

ExploreConnections for further study

Mechanical energy and total energy must be distinguished explicitly when dissipation is present.

Two states compare mechanical- and internal-energy bars with the same total.K+UE_intK+UE_intinitial statefinal stateconstant total

Mechanical energy K+U decreases while E_int increases; each state's total height remains constant.

Concept review

Common errors

Each warning includes a concrete way to review the reasoning, not only an incorrect-answer marker.

Treating mechanical and total energy as synonyms with dissipation.

K+U can change while the system's total energy is conserved.

Saying friction destroys energy.

Mechanical energy can become internal energy or another form.