In ideal conservative systems, mechanical energy stays constant. Real systems often include friction, drag,
motors, braking, deformation, or other non-conservative effects. These effects change the total mechanical energy.
1. Mechanical energy
Mechanical energy is the sum of kinetic energy and potential energy:
\[
E_{\mathrm{mech}}=K+U.
\]
For example, near Earth’s surface,
\[
K=\frac12mv^2,\qquad U_g=mgh.
\]
If springs are included, the elastic potential energy may also appear:
\[
U_s=\frac12kx^2.
\]
2. Conservative-force case
If only conservative forces act, total mechanical energy is conserved:
\[
E_i=E_f.
\]
This means kinetic and potential energy can transform into each other, but their sum remains constant.
3. Non-conservative work-energy balance
When non-conservative forces act, the general balance is
\[
W_{\mathrm{nc}}=\Delta E_{\mathrm{mech}}.
\]
Since
\[
\Delta E_{\mathrm{mech}}=E_f-E_i,
\]
the most useful form is
\[
E_f=E_i+W_{\mathrm{nc}}.
\]
Negative non-conservative work removes mechanical energy. Positive non-conservative work adds mechanical energy.
4. Common non-conservative forces
| Effect |
Typical sign |
Meaning |
| Kinetic friction |
\(W_f<0\) |
Converts mechanical energy into thermal energy. |
| Air drag |
\(W_d<0\) |
Removes mechanical energy from a moving object. |
| Braking |
\(W_{\mathrm{brake}}<0\) |
Reduces mechanical energy, often converting it to heat. |
| Motor or applied pull |
Can be positive |
Adds mechanical energy when it acts along the motion. |
| External push opposite motion |
Can be negative |
Removes mechanical energy from the system. |
5. Component accounting
If several non-conservative effects are present, add their work contributions:
\[
W_{\mathrm{nc}}
=
W_f+W_d+W_{\mathrm{ext}}+W_{\mathrm{other}}.
\]
Then insert the net work into the energy balance:
\[
E_f=E_i+W_f+W_d+W_{\mathrm{ext}}+W_{\mathrm{other}}.
\]
In many practical problems, friction and drag are entered as positive loss magnitudes, but their work values are negative:
\[
W_f=-E_{\mathrm{friction}},\qquad W_d=-E_{\mathrm{drag}}.
\]
6. Worked example
A system initially has
\[
E_i=1200\ \mathrm{J}.
\]
Friction removes \(450\ \mathrm{J}\) of mechanical energy, so
\[
W_{\mathrm{nc}}=-450\ \mathrm{J}.
\]
Use
\[
E_f=E_i+W_{\mathrm{nc}}.
\]
Substitute:
\[
E_f=1200+(-450)=750\ \mathrm{J}.
\]
Therefore,
\[
\boxed{E_f=750\ \mathrm{J}}.
\]
The mechanical energy decreased because friction did negative work.
7. Solving for missing quantities
The same equation can be rearranged to find any missing quantity:
\[
W_{\mathrm{nc}}=E_f-E_i,
\]
\[
E_i=E_f-W_{\mathrm{nc}},
\]
\[
E_f=E_i+W_{\mathrm{nc}}.
\]
If the goal is to find how much external work is needed to reach a target final energy, use
\[
W_{\mathrm{ext}}
=
E_f-E_i-W_f-W_d-W_{\mathrm{other}}.
\]
Formula summary
| Concept |
Formula |
Use |
| Mechanical energy |
\(E_{\mathrm{mech}}=K+U\) |
Total energy stored in motion and position/configuration. |
| Energy change |
\(\Delta E_{\mathrm{mech}}=E_f-E_i\) |
Difference between final and initial mechanical energy. |
| Non-conservative work balance |
\(W_{\mathrm{nc}}=\Delta E_{\mathrm{mech}}\) |
Main equation for friction, drag, motors, and external work. |
| Final energy |
\(E_f=E_i+W_{\mathrm{nc}}\) |
Find final mechanical energy after non-conservative work acts. |
| Net non-conservative work |
\(W_{\mathrm{nc}}=W_f+W_d+W_{\mathrm{ext}}+W_{\mathrm{other}}\) |
Add all signed non-conservative work terms. |
| Required external work |
\(W_{\mathrm{ext}}=E_f-E_i-W_f-W_d-W_{\mathrm{other}}\) |
Find how much extra work is needed to reach a target final energy. |
Key idea: non-conservative work is exactly the change in mechanical energy, not the total energy of the universe.
Lost mechanical energy usually becomes thermal energy, sound, deformation, or internal energy.
