Energy Conservation with Non-Conservative Forces

When only conservative forces act, mechanical energy $E = K + U$ is conserved. When non-conservative forces like friction or air resistance are present, mechanical energy decreases. These forces convert mechanical energy primarily into thermal energy.

Energy Change from Non-Conservative Forces

When non-conservative forces act, the total energy of the universe is still conserved, but mechanical energy within our system decreases. The change in mechanical energy equals the work done by non-conservative forces:

Here $\Delta E = E_{\text{final}} - E_{\text{initial}} = (K_f + U_f) - (K_i + U_i)$ is the change in total mechanical energy. Since non-conservative forces like friction typically dissipate energy, $W_{\text{nc}}$ is usually negative, so $\Delta E < 0$ and mechanical energy decreases.

Energy Balance Perspective

Another useful way to think about energy conservation with non-conservative forces is to recognize that the initial mechanical energy equals the final mechanical energy plus the energy dissipated:

Expanding this in terms of kinetic and potential energies:

Since $E_{\text{dissipated}} = -W_{\text{nc}}$, this is equivalent to our original statement $\Delta E = W_{\text{nc}}$. The dissipated energy is primarily thermal energy at the contact surface.

Visualizing Energy Transformation

Consider a box sliding across a rough horizontal surface. The initial kinetic energy splits into final kinetic energy and thermal energy:

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