Conservation of Energy

Conservation laws are physics’ accounting tricks: instead of following every push and pull through time, compare two snapshots — energy before, energy after. If only conservative forces acted in between, the books must balance, whatever happened in the middle.

In an isolated system where only conservative forces act, the total mechanical energy is conserved:

$$E = KE + PE = \text{constant}$$

Potential energy for common forces:

Energy conservation equation

$$\frac{1}{2}mv_i^2 + PE_i = \frac{1}{2}mv_f^2 + PE_f$$

When non-conservative forces (e.g. friction) are present:

$$E_f = E_i + W_{\text{nc}}$$

where $W_{\text{nc}} < 0$ for friction (energy lost as heat).

Conservative vs. non-conservative

• A force is conservative if the work it does is path-independent (depends only on start and end points).
• Equivalently: the work around any closed loop is zero.
• Gravity and spring forces are conservative; friction and air resistance are not.

Problem-solving tip: Energy methods bypass the need for acceleration and time. If the problem asks for speed at a given position, try energy conservation before $F=ma$.

Common pitfall: "Energy is conserved" does not mean "mechanical energy is conserved". Friction quietly converts $KE$ into thermal energy: total energy survives, but the mechanical books alone stop balancing the moment sliding friction appears.

Conservation of Mechanical Energy

For a system with only conservative forces, total mechanical energy is constant:

$$E = K + U = \text{const}$$

where $K = \tfrac{1}{2}mv^{2}$ is kinetic energy and $U = mgh$ is gravitational potential energy.

On an inclined plane, the forces on a body are:

• Weight $P = mg$ (vertically downward)
• Normal $N$ (perpendicular to the surface)
• Friction $F_f$ (parallel, opposing motion)

When friction is present: $\Delta E = -F_f \cdot d$.

Energy is never created or destroyed — only converted between kinetic, potential, and thermal forms.