A metal wire of length L, area of cross-section A, and Young's modulus Y is stretched by a varia

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A metal wire of length L, area of cross-section A, and...

A metal wire of length $L$, area of cross-section $\mathrm{A}$, and Young's modulus $Y$ is stretched by a variable force $F$ such that $F$ is always slightly greater than the elastic forces of resistance in the wire. When the elongation of the wire is $\ell$ (A) The work done by $F$ is $\frac{Y A \ell^{2}}{2 L}$ (B) The work done by $F$ is $\frac{Y A \ell^{2}}{L}$ (C) The elastic potential energy stored in the wire is $\frac{Y A \ell^{2}}{2 L}$ (D) No heat is produced during the elongation

A metal wire of length $L$, area of cross-section $\mathrm{A}$, and Young's modulus $Y$ is stretched by a variable force $F$ such that $F$ is always slightly greater than the elastic forces of resistance in the wire. When the elongation of the wire is $\ell$
(A) The work done by $F$ is $\frac{Y A \ell^{2}}{2 L}$
(B) The work done by $F$ is $\frac{Y A \ell^{2}}{L}$
(C) The elastic potential energy stored in the wire is $\frac{Y A \ell^{2}}{2 L}$
(D) No heat is produced during the elongation

Key Concepts

Young's Modulus

Young's modulus is a measure of the stiffness of a material. It is defined as the ratio of stress (force per unit area) to strain (relative deformation) within the elastic limit of the material. This concept is key to understanding how forces cause elongation or compression in materials without permanent deformation.

Elastic Deformation

Elastic deformation refers to the reversible change in shape or size of a material under the influence of an applied force. In this regime, when the load is removed, the material returns to its original form. This concept is crucial in studying how materials store energy and behave under small strains.

Elastic Potential Energy

Elastic potential energy is the energy stored in a material when it is deformed elastically. In the context of a stretched wire, this energy is accumulated as the wire resists the applied force. The amount of stored energy can be calculated by integrating the variable force over the displacement, emphasizing how energy is stored within the material as it is deformed.

Work Done by a Variable Force

When a force that varies with displacement is applied, the work done is determined by integrating the force over the distance of application. In elastic systems, if the force increases linearly with displacement (as per Hooke’s law), the work done becomes the area under the force-displacement curve, which is critical for relating the applied work to stored elastic energy.

Energy Conservation in Elastic Systems

In an ideal elastic deformation, the work done on the material is entirely converted into stored elastic potential energy with no losses due to heat or other forms of energy dissipation. This concept underscores the energy transformations in perfectly elastic materials and is important for determining energy conservation during the process of elastic stretching.

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