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Condition for conservation

Mechanical energy is conserved when only conservative forces transfer energy within the system and friction or other resistive transfers are absent or negligible.

Write the balance

$E_{k,i}+E_{p,i}=E_{k,f}+E_{p,f}$

Choose a convenient zero for potential energy and keep the same reference throughout.

When it is not conserved

Friction, drag or deformation transfer mechanical energy to internal energy. Total energy is still conserved, but the mechanical-energy equation needs an additional transfer term.

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An object is released from rest and slides down a frictionless ramp. The object then leaves the ramp and slides along a rough horizontal surface. The object stops in a distance s along the ramp.

The coefficient of dynamic friction between the object and the rough horizontal surface is $\mu$ .
What is the height of the ramp?

Account for energy

Define the system, identify energy stores and describe transfers. Work done by a force transfers energy; total energy is conserved even when mechanical energy is not.

Use the mechanical model

$E_k= rac12mv^2,\quad \Delta E_{p,g}=mg\Delta h,\quad E_{p,elastic}= rac12k(\Delta x)^2$

Conserve their sum only when resistive transfers are absent or included explicitly.

Use rates and ratios

$P= rac{\Delta E}{\Delta t}=Fv,\qquad \eta= rac{E_{useful}}{E_{input}}= rac{P_{useful}}{P_{input}}$

Fuel energy density connects available input energy to a chosen volume.

Final checks

Check the system boundary, signs of work and potential-energy changes, the reference height, extension from natural length, and whether the quantity is energy, power, efficiency or energy density.

A.3.7—Mechanical energy conservation

Conserve Mechanical Energy Without Resistive Forces

Retrieve the A.3 Work, Energy and Power Model

A.3.7 Exam Analysis