Spring Force and PE Derivation

From Hooke's Law: $F_{\text{spring}} = kx \quad (F_{\text{external}} = kx; F_{\text{restoring}} = -kx)$

Work done by external agent to stretch from 0 to x: $W_{\text{ext}} = \int_0^x kx'\, dx' = \frac{1}{2}kx^2 = PE_s$

This is always positive — same for compression ($x^{2}$ is always positive).

Spring Constant

$k = \frac{F}{x} \quad [\text{M}^1\text{L}^0\text{T}^{-2}]\ (\text{N/m})$

Stiffer spring → larger k. Units: N/m = kg/$s^{2}$.

Spring Combinations

Parallel springs are stiffer (k_eff > either). Series springs are softer (k_eff < either).

Energy Conservation in Spring-Block System

At natural length (x = 0): PE_s = 0, KE = maximum = ½mv_m$ax^{2}$ At extremes (x = ±A): KE = 0, PE = maximum = ½$kA^{2}$

$\frac{1}{2}mv_{\max}^2 = \frac{1}{2}kA^2 \implies v_{\max} = A\sqrt{\frac{k}{m}} = A\omega$

where ω = √(k/m) is the angular frequency of SHM.