• $word_{count}$: 220

Step 1 — Identify the type: Is it about elongation (use Y), shape change (use G), or volume change (use B)? Most wire problems use Y.

Step 2 — Write the appropriate formula: Y = $\frac{FL}{A*Delta L}$ is the workhorse. Remember A = pi*$d^2$/4 = pi*$r^2$.

Step 3 — Series vs parallel: Wires in series (same force, different extensions) — add elongations. Wires in parallel (same extension, different forces) — add spring constants k = $\frac{YA}{L}$.

Step 4 — Energy problems: Choose the form U = F^2$$\frac{L}{2AY} when force is given, or U = YA(Delta L)^$\frac{2}{2L}$ when extension is given.

Step 5 — Unit consistency: Always convert mm to m, $cm^2$ to $m^2$ before calculating. Common conversions: 1 mm = 10^{-3} m, 1 $cm^2$ = 10^{-4} $m^2$, 1 GPa = 10^9 Pa.

Common traps: (1) Elongation is inversely proportional to Y, not directly. (2) Breaking force depends on area, not length. (3) When a wire is drawn to n times its length (volume constant), new elongation under same force = $n^2$ times original. (4) Wire under its own weight: use half the weight and full length, giving Delta L = $\frac{MgL}{2AY}$. (5) Thermal stress is independent of length. (6) Young's modulus is a material property — changing dimensions doesn't change Y.