Elastic potential energy class 11 nbf | Modulus of metallic wire | Work from force extensions graph

Model of Metallic Wire (Young's Modulus)
📌 The Model of Metallic Wire is essentially the Young's Modulus ($Y$), which is the constant ratio between tensile stress (force per unit area causing elongation) and tensile strain (change in length relative to original length, $\Delta L / L_0$).
⚙️ The primary goal is to determine the work done to produce a certain amount of strain in a wire and relate it to the material's elastic properties.
📏 To measure the minute extension ($\Delta L$ or $x$), the experiment requires precise tools like a Vernier scale attached to the test wire and a reference wire with a fixed main scale to cancel out initial setup errors.
📈 A higher Modulus of Metallic Wire value indicates that it is harder to stretch (requires much greater stress for minimal extension).

Work from Force-Extension Graph
📌 The Work Done under a variable or constant force is calculated as the Area Under the Force-Extension (F-x) Graph, analogous to the area under an F-D graph giving work done.
🔺 For the elastic extension of a spring/wire following Hooke's Law ($F=kx$), the resulting graph is a straight line forming a triangle from the origin ($O$) to a point ($A$), where $x$ is the extension and $F$ is the applied force.
📐 The area of this triangle, representing the work done (and stored potential energy), is calculated as $W = \frac{1}{2} \times \text{Base} \times \text{Altitude}$, which translates to $W = \frac{1}{2} Fx$.
⚖️ This calculated elastic work ($W = \frac{1}{2} Fx$) is the foundation for determining the Elastic Potential Energy ($U$).

Elastic Potential Energy (EPE) Derivation
💡 Elastic Potential Energy ($U$) is the energy stored in an elastic material when it is compressed or stretched, arising from the tendency of the material to regain its original configuration.
🔗 The fundamental concept is that Work Done equals Potential Energy Stored ($W = U$); any work done against a restoring force (like stretching a spring) is stored as EPE.
🔗 Using Hooke's Law ($F=kx$), the work done is integrated to yield the EPE formula in terms of the spring constant ($k$) and extension ($x$): $U = \frac{1}{2} kx^2$.
🔗 By substituting the expression for force derived from Young's Modulus ($F = \frac{Y A x}{L_0}$), the EPE is also expressed as $U = \frac{1}{2} \frac{Y A x^2}{L_0}$.

Key Points & Insights
➡️ Potential Energy Concept: True potential energy arises from work done by moving a body against an opposing field force (like gravity or electric force), not merely a change in position.
➡️ Experimental Precision: Using a Vernier scale is critical because the extension ($x$) in a metallic wire under standard loads is extremely small and requires high precision for accurate Young's Modulus calculation.
➡️ Hooke's Law Application: The relationship $F \propto x$ (Hooke's Law) must be obeyed *within the elastic limit* to derive the $U = \frac{1}{2} kx^2$ relationship for stored energy.

📸 Video summarized with SummaryTube.com on Jan 10, 2026, 14:18 UTC