Gravitational Potential Energy

$PE_g = mgh \quad [\text{M}^1\text{L}^2\text{T}^{-2}]\ (\text{J})$

• h = height above chosen reference level
• Reference level is arbitrary; only $\Delta P$E matters
• For conservative forces: $W = -\Delta PE$

Spring Potential Energy

$PE_s = \frac{1}{2}kx^2 \quad [\text{M}^1\text{L}^2\text{T}^{-2}]\ (\text{J})$

• Always ≥ 0 (whether stretched or compressed)
• k = spring constant [M^{1}$$L^{0}$$T^{-2}] (N/m)
• x = extension or compression (m)

Conservative vs. Non-Conservative Forces

Conservation of Mechanical Energy

When only conservative forces act: $KE + PE = \text{constant (total mechanical energy E)}$

When friction acts: $\Delta(KE + PE) = W_{\text{friction}} = -f_k \cdot d < 0$

The mechanical energy decreases by the heat generated by friction.