The energy stored per unit volume in a wire under stress sigma is:

Two wires of the same material, same length, but cross-sections A and 2A, are stretched by the same force. The ratio of energies stored (wire 1 : wire 2) is:

Two wires of the same material and cross-section, but lengths L and 2L, are given the same elongation. The ratio of energies stored (wire 1 : wire 2) is:

A wire stores energy U when stretched by Delta L. If it is further stretched to 2*Delta L, the total energy stored is:

A rod of length 1 m and cross-section 1 cm^2 is made half of steel (Y = 200 GPa) and half of copper (Y = 120 GPa) placed end to end. A force of 12000 N is applied. The total elongation is: <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 400 130" style="max-width:400px;width:100%;height:auto;background:#faf9f5;border-radius:12px;border:1px solid rgba(20,20,19,0.10)"> <title>Composite rod — steel and copper in series under tension</title> <defs><marker id="ah-q092-08" markerWidth="8" markerHeight="6" refX="8" refY="3" orient="auto"><polygon points="0,0 8,3 0,6" fill="#c6613f"/></marker></defs> <!-- Wall --> <rect x="30" y="30" width="8" height="30" fill="#14140f"/> <line x1="30" y1="32" x2="24" y2="26" stroke="#14140f" stroke-width="1"/> <line x1="30" y1="42" x2="24" y2="36" stroke="#14140f" stroke-width="1"/> <line x1="30" y1="52" x2="24" y2="46" stroke="#14140f" stroke-width="1"/> <line x1="30" y1="58" x2="24" y2="52" stroke="#14140f" stroke-width="1"/> <!-- Steel section --> <rect x="38" y="37" width="120" height="16" rx="2" fill="#14140f" fill-opacity="0.08" stroke="#14140f" stroke-width="1.5"/> <text x="98" y="33" text-anchor="middle" font-size="9" fill="#14140f" font-family="sans-serif">Steel (200 GPa)</text> <!-- Copper section --> <rect x="158" y="37" width="120" height="16" rx="2" fill="#788c5d" fill-opacity="0.12" stroke="#788c5d" stroke-width="1.5"/> <text x="218" y="33" text-anchor="middle" font-size="9" fill="#788c5d" font-family="sans-serif">Copper (120 GPa)</text> <!-- Joint line --> <line x1="158" y1="30" x2="158" y2="60" stroke="#c6613f" stroke-width="1" stroke-dasharray="3,2"/> <!-- Force arrow --> <line x1="278" y1="45" x2="320" y2="45" stroke="#c6613f" stroke-width="2" marker-end="url(#ah-q092-08)"/> <text x="328" y="49" font-size="10" fill="#c6613f" font-family="sans-serif">F</text> <!-- Length labels --> <text x="98" y="68" text-anchor="middle" font-size="9" fill="#14140f" font-family="sans-serif">0.5 m</text> <text x="218" y="68" text-anchor="middle" font-size="9" fill="#788c5d" font-family="sans-serif">0.5 m</text> <!-- Formula --> <text x="200" y="90" text-anchor="middle" font-size="11" fill="#14140f" font-family="sans-serif">Same F through both → series: $\Delta L$ = $\Delta$$L_{1}$ + $\Delta$$L_{2}$</text> <text x="200" y="110" text-anchor="middle" font-size="11" font-weight="700" fill="#14140f" font-family="sans-serif">$\Delta L$ = FL(1/$Y_{1}$ + 1/$Y_{2}$) / (2A)</text> </svg>