Work Done In Stretching A Wire

Edited By Vishal kumar | Updated on Jul 02, 2025 06:36 PM IST

Stretching a wire involves the application of force to elongate it, a concept that intertwines physics with everyday experiences. When you pull on a rubber band or stretch a spring, you are engaging in a practical demonstration of work done in stretching a wire. This phenomenon is governed by principles of material science and mechanics, where energy is transferred to overcome internal resistances and change the shape of the wire. In real-life applications, understanding this concept is crucial, whether it's designing resilient structures, creating durable materials, or even in the mechanics of everyday objects like car springs and bicycle cables. The work done in stretching a wire not only highlights fundamental physical principles but also underscores its importance in engineering and technology. In this article, we will cover the concept of Work Done In Stretching A Wire. This concept falls under the broader category of Properties of Solids and Liquids.

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Work Done in Stretching a Wire
Solved Examples Based on Work Done in Stretching a Wire
Summary

Work Done In Stretching A Wire

Work Done in Stretching a Wire

When a body is in its natural shape, its potential energy corresponding to the molecular forces is minimal. When deformed, internal forces appear and work has to be done against these forces. Thus, the potential energy of the body is increased. This is called the elastic potential energy.

Suppose a wire having natural length L and cross-sectional area A is fixed at one end and is stretched by an external force applied at the other end. When the extension is x,the wire is under a longitudinal stress F/A. The strain is x/L.

If Young's modulus is Y, then

$
\frac{F / A}{x / L}=Y \Rightarrow F=\frac{A Y}{L} x
$

So, the work done for an additional small increase dx in length will be:-
$
d W=F d x \Rightarrow d W=\frac{A Y}{L} x d x
$

The total work done by the external force in increasing the length from 0 to $\Delta L$ will be:-
$
W=\int_0^{\Delta L} \frac{A Y}{L} x d x=\frac{1}{2} \frac{Y A}{L}(\Delta L)^2
$

This work is stored in the wire as its elastic potential energy. So, the elastic potential energy of the stretched wire is:
$
U=\frac{1}{2} \frac{Y A}{L}(\Delta L)^2
$

We can also write,

$
\begin{aligned}
& W=\frac{1}{2}\left[\frac{Y A \Delta L}{L}\right] \Delta L=\frac{1}{2}(\text { maximum stretching force }) \times \text { extension } \\
& W=\frac{1}{2} Y(A L)\left[\frac{\Delta L}{L}\right]^2=\frac{1}{2} \times Y \times \text { Volume } \times(\text { strain })^2 \\
& W=\frac{1}{2}\left[\frac{\Delta L}{L}\right]\left[Y \frac{\Delta L}{L}\right](A L)=\frac{1}{2} \times \text { strain } \times(Y \text { strain }) \times \text { Volume } \\
& W=\frac{1}{2}\left[\frac{\Delta L}{L}\right]\left[Y \frac{\Delta L}{L}\right](A L)=\frac{1}{2} \times \text { strain } \times \text { stress } \times \text { Volume }
\end{aligned}
$

Also, Potential energy per unit volume $=\frac{1}{2} \times \operatorname{strain} \times$ stress

Solved Examples Based on Work Done in Stretching a Wire

Example 1: Two wires of the same materials (Young modulus) and the same length L but radii R and 2R respectively are joined end to end and a weight W is suspended from the combination as shown in the figure.

The elastic PE stored in the system is

1) $\frac{3 w^2 L}{4 \pi^2 R Y}$
2) $\frac{5 w^2 L}{8 \pi R^2 y}$
3) $\frac{3 w^2 L}{8 \pi_R^2 y}$
4) $\frac{\omega^2 L}{8 \pi R^2 y}$

Solution:

Using,

$
U=\frac{1}{2} k x^2=\frac{k}{2}\left(\frac{F}{k}\right)^2=\frac{F^2}{2 k}
$

We have
$
\begin{aligned}
U & =U_1+U_2 \\
& =\frac{F^2}{2 k_1}+\frac{F^2}{2 k_2} \quad(F=W) \\
& =\frac{W^2}{2}\left[\frac{1}{k_1}+\frac{1}{k_2}\right]
\end{aligned}
$
and
$
\begin{aligned}
& k_1=\frac{Y A}{L}=\frac{Y \pi(2 R)^2}{L}, \\
& k_2=\frac{Y \pi R^2}{L}
\end{aligned}
$

Put the values $k_1 \& k_2$ in equation (i)
$
\begin{aligned}
U & =\frac{W^2}{2}\left[\frac{L}{4 \pi R^2 Y}+\frac{L}{\pi R^2 Y}\right] \\
U & =\frac{5 \omega^2 L}{8 \pi R^2 Y}
\end{aligned}
$

Hence, the answer is the option (2).

Example 2: Two wires of the same material and the same length L but radii R and 2R respectively, are joined end to end and a weight w is suspended from the combination as shown in the figure. The elastic potential energy in the system is

1) $\frac{3 w^2 L}{4 \pi R^2 Y}$
2) $\frac{3 w^2 L}{8 \pi R^2 Y}$
3) $\frac{5 w^2 L}{8 \pi R^2 Y}$
4) $\frac{w^2 L}{\pi R^2 Y}$

Solution:

$\begin{aligned} & \Delta l_1=\frac{w L}{\left(4 \pi R^2\right) Y}, \Delta l_2=\frac{w L}{\pi R^2 Y} \\ & \therefore \quad U=\frac{1}{2} k_1\left(\Delta l_1\right)^2+\frac{1}{2} k_2\left(\Delta l_2\right)^2 \\ & U=\frac{1}{2} \times Y\left(\frac{4 \pi R^2}{L}\right) \times\left[\frac{w L}{4 \pi R^2 Y}\right]^2+\frac{1}{2} \times \frac{Y\left(\pi R^2\right)}{L} \times\left[\frac{w L}{\pi R^2 Y}\right]^2 \\ & U=\frac{5 w^2 L}{8 \pi R^2 Y}\end{aligned}$

Hence, the answer is the option (3).

Example 3: An elastic material with Young's modulus y is subjected to a tensile stress S elastic energy stored per unit volume of the material is

1) $\frac{y S}{2}$
2) $y^{-1} S^2$
3) $\frac{S^2}{2 y}$
4) $\frac{S}{2 y}$

Solution:

$\begin{aligned} & \text { strain }=\frac{\text { stress }}{y}=\frac{S}{y} \text { Now, energy stored per unit volume } \\ & =\frac{1}{2} \times \text { stress } \times \text { strain } \\ & =\frac{1}{2} \times S \times \frac{S}{y}=\frac{S^2}{2 y}\end{aligned}$

Hence, the answer is the option (3).

Example 4: A uniform rod is kept at a smooth horizontal surface, and a constant force is applied on the rod in the horizontal direction at end A. Find the ratio of energy stored per unit volume at end A to the energy stored per unit volume in the middle of the rod.

1) 2

2) 4

3) 8

4) 10

Solution:

$\begin{aligned} & U=\frac{1}{2} \frac{\text { stress }}{Y} \\ & \text { stress }=\frac{\text { tension }}{\text { Area }} \\ & \frac{U_A}{U_{\text {middle }}}=\frac{\left(T_A\right)^2}{\left(T_{\text {middle }}\right)^2}=\frac{F^2}{\left(\frac{F}{2}\right)^2}=4\end{aligned}$

Hence, the answer is the option (2).

Example 5:

In an experiment, brass and steel wires of length 1 m each with areas of cross-section 1 mm²are used. The wires are connected in series and one end of the combined wire is connected to a rigid support and the other end is subjected to elongation. The stress (in N/m²) required to produce a net elongation of 0.2mm is, [Given, the Young's Modulus for steel and brass are respectively, $120 \times 10^9 \mathrm{~N} / \mathrm{m}^2$ and $60 \times 10^9 \mathrm{~N} / \mathrm{m}^2$].

1) 1200000

2) 8000000

3) 1800000

4) 200000

Solution:

$\begin{aligned} & k_1=\frac{y_1 A_1}{l_1}=\frac{120 \times 10^9 \times A}{1} \\ & k_2=\frac{y_2 A_2}{l_2}=\frac{60 \times 10^9 \times A}{1} \\ & K_{e q}=\frac{K_1 K_2}{K_1+K_2}=\frac{120 \times 60}{180} \times 10^9 \times A \\ & \Rightarrow K_{e q}=40 \times 10^9 \times A \\ & F=k_{e q}(x) \\ & F=\left(40 \times 10^9\right) A_0\left(0.2 \times 10^{-3}\right) \\ & \Rightarrow \frac{F}{A}=8 \times 10^6 \mathrm{~N} / \mathrm{m}^2\end{aligned}$

Hence, the answer is the option (2).

Summary

The concept of work done in stretching a wire is essential for understanding elastic potential energy and its real-world applications. By analyzing the equations and examples, we see that the work done to stretch a wire can be calculated using Young's modulus and other parameters. The elastic potential energy stored in the wire is determined by its extension and material properties. Examples illustrate practical scenarios involving different materials and configurations, demonstrating how these principles are applied in engineering and everyday objects.

Frequently Asked Questions (FAQs)

1. What happens to the work done if you double the length of a wire while stretching it?

If you double the length of stretch, the work done quadruples. This is because work is proportional to the square of the extension in elastic stretching.

2. How does the concept of strain energy relate to the work done in stretching a wire?

Strain energy is the energy stored in a wire as a result of deformation. The work done in stretching a wire is converted to strain energy, which can be released when the wire returns to its original shape.

3. What is meant by "work done" in stretching a wire?

Work done in stretching a wire refers to the energy transferred to the wire to increase its length. This energy is stored as elastic potential energy in the wire as it is stretched.

4. How does Hooke's Law relate to the work done in stretching a wire?

Hooke's Law states that the force needed to stretch a wire is proportional to its extension. This relationship is crucial in calculating the work done, as work is the product of force and displacement.

5. Why does stretching a wire require work?

Stretching a wire requires work because you need to overcome the internal forces between the atoms in the wire. These forces resist the change in the wire's shape, and energy is needed to overcome this resistance.

6. Is the work done in stretching a wire always positive?

Yes, the work done in stretching a wire is always positive. This is because the force applied and the displacement are in the same direction, resulting in a positive product.

7. How does the cross-sectional area of a wire affect the work done in stretching it?

A wire with a larger cross-sectional area requires more work to stretch to the same length as a thinner wire. This is because there are more atomic bonds to overcome in a thicker wire.

8. How does the material of the wire influence the work required to stretch it?

Different materials have different elastic properties. Stiffer materials (with higher Young's modulus) require more work to stretch to the same length compared to more flexible materials.

9. Can the work done in stretching a wire be recovered?

Yes, if the wire is stretched within its elastic limit, the work done can be recovered when the wire returns to its original length. This is due to the conservation of elastic potential energy.

10. What's the relationship between stress, strain, and work done in stretching a wire?

Work done is related to both stress and strain. It's equal to the area under the stress-strain curve up to the point of extension. In the elastic region, this relationship is linear.

11. How does temperature affect the work required to stretch a wire?

Higher temperatures generally make materials more ductile. This means less work is required to stretch a wire at higher temperatures, as the atomic bonds are easier to overcome.

12. What's the difference between elastic and plastic deformation in wire stretching?

In elastic deformation, the wire returns to its original length when the force is removed, and all work done is recoverable. In plastic deformation, the wire is permanently stretched, and not all work is recoverable.

13. How is the work done in stretching a wire related to its elastic potential energy?

The work done in stretching a wire is equal to the increase in its elastic potential energy. This energy is stored in the wire and can be released when the wire returns to its original length.

14. Why does a real wire eventually break if stretched too much?

A wire breaks when stretched beyond its breaking point because the work done exceeds the strength of the interatomic bonds. At this point, the bonds between atoms break, causing the wire to fail.

15. How does the concept of Young's modulus relate to the work done in stretching a wire?

Young's modulus is a measure of a material's stiffness. Materials with higher Young's modulus require more work to stretch to the same length, as they resist deformation more strongly.

16. Can negative work be done on a wire?

Negative work can be done on a wire when it's allowed to contract back to its original length. This represents the wire doing work on its surroundings, releasing stored elastic energy.

17. How does the work done in stretching a wire relate to the concept of resilience?

Resilience is the ability of a material to absorb energy when deformed elastically and release that energy upon unloading. The work done in stretching a wire within its elastic limit is a measure of its resilience.

18. What's the significance of the yield point in relation to work done on a wire?

The yield point marks the transition from elastic to plastic deformation. Beyond this point, additional work done on the wire results in permanent deformation, and not all energy can be recovered.

19. How does work done in stretching a wire relate to its tensile strength?

Tensile strength is the maximum stress a wire can withstand before breaking. The work done to stretch a wire to its breaking point is related to its tensile strength and ductility.

20. Why isn't all the work done in stretching a wire converted to elastic potential energy?

Some of the work done is converted to heat due to internal friction in the wire. This is why a stretched wire may feel slightly warmer, especially if stretched rapidly.

21. How does the rate of stretching affect the work done on a wire?

The rate of stretching can affect the work done due to internal friction and heat generation. Rapid stretching may require more work as the wire has less time to dissipate heat.

22. What's the relationship between force-extension graph and work done in stretching a wire?

The work done in stretching a wire is equal to the area under the force-extension graph. For a perfectly elastic wire, this area forms a triangle.

23. How does work done in stretching a wire relate to its elastic limit?

The elastic limit is the maximum stress a wire can undergo and still return to its original shape. Work done up to this limit is fully recoverable as elastic potential energy.

24. Why does a prestressed wire require less work to stretch further?

A prestressed wire has already overcome some of the initial resistance to stretching. Additional stretching requires less work as the wire is already partially extended.

25. How does crystal structure of a metal affect the work needed to stretch its wire?

The crystal structure determines how easily atoms can slide past each other. Metals with more slip planes (like face-centered cubic structures) generally require less work to stretch.

26. What's the significance of the area under the stress-strain curve in wire stretching?

The area under the stress-strain curve represents the energy per unit volume required to stretch the wire. It's a measure of the material's toughness.

27. How does work done in stretching relate to a wire's ductility?

Ductility is a measure of a material's ability to deform plastically without fracturing. More ductile materials allow more work to be done in stretching before breaking.

28. Why isn't the work done in stretching a wire always proportional to its length?

Work done is only proportional to length for elastic deformation. In plastic deformation, the relationship becomes non-linear as the wire's properties change with stretching.

29. How does annealing affect the work required to stretch a wire?

Annealing reduces internal stresses in a wire, making it more ductile. This generally reduces the work required to stretch the wire, especially in the plastic deformation region.

30. What's the relationship between work done and the wire's ultimate tensile strength?

The total work done to stretch a wire to its breaking point is related to its ultimate tensile strength. This represents the maximum energy the wire can absorb before failure.

31. How does strain energy density relate to work done in stretching a wire?

Strain energy density is the work done per unit volume in stretching a wire. It's a useful measure for comparing the energy storage capacity of different materials.

32. Why does a wire heat up when stretched?

A wire heats up when stretched due to internal friction and the conversion of some mechanical energy to thermal energy. This effect is more noticeable in rapid stretching.

33. How does work hardening affect the work needed to stretch a wire?

Work hardening increases a wire's resistance to further deformation. As a result, more work is required to continue stretching a work-hardened wire.

34. What's the significance of Poisson's ratio in wire stretching?

Poisson's ratio describes how a wire's cross-section changes as it's stretched. It affects the work done as it influences the wire's overall deformation behavior.

35. How does the concept of resilience modulus relate to work done in stretching a wire?

The resilience modulus is the maximum energy per unit volume that can be elastically stored in a material. It's directly related to the work done in stretching a wire to its elastic limit.

36. Why isn't all the work done in stretching a wire recoverable, even within the elastic limit?

Even within the elastic limit, some energy is lost as heat due to internal friction. This makes the process not perfectly efficient, and not all work is recoverable.

37. How does the presence of defects in a wire affect the work needed to stretch it?

Defects like dislocations can make it easier for a wire to deform, potentially reducing the work needed for initial stretching. However, they can also lead to earlier failure.

38. What's the relationship between a wire's stiffness and the work done in stretching it?

A stiffer wire requires more work to stretch to the same length as a less stiff wire. Stiffness is directly related to the wire's Young's modulus.

39. How does strain rate affect the work done in stretching a wire?

Higher strain rates (faster stretching) generally require more work due to increased internal friction and the material's strain rate sensitivity.

40. Why does a wire's diameter decrease when it's stretched?

As a wire is stretched, its volume remains approximately constant. To maintain this volume while increasing in length, the wire's diameter must decrease. This is described by Poisson's ratio.

41. How does the concept of toughness relate to work done in stretching a wire?

Toughness is the ability of a material to absorb energy before fracturing. It's represented by the total area under the stress-strain curve, which is related to the total work done in stretching the wire to failure.

42. What's the significance of the proportional limit in wire stretching?

The proportional limit is the point beyond which Hooke's Law no longer applies. Beyond this point, the work done in stretching is no longer proportional to the square of the extension.

43. How does cold working affect the work required to stretch a wire?

Cold working typically increases a wire's yield strength and reduces its ductility. This means more work is required for initial stretching, but the wire may fail earlier under large deformations.

44. Why does a stretched wire vibrate when plucked?

A stretched wire vibrates when plucked because some of the work done in stretching is stored as elastic potential energy. This energy is converted to kinetic energy when the wire is disturbed, causing vibration.

45. How does the concept of fatigue relate to repeated stretching of a wire?

Fatigue occurs when a wire is subjected to repeated stress cycles. Over time, this can lead to failure at stress levels below the wire's normal strength, affecting the work required for stretching.

46. What's the relationship between a wire's elasticity and the work done in stretching it?

A more elastic wire can store more energy for a given stretch. This means more work can be done on the wire before it reaches its elastic limit, and more of this work is recoverable.

47. How does the presence of impurities affect the work needed to stretch a wire?

Impurities can affect a wire's mechanical properties. They may increase the work needed for stretching by making the material harder, or decrease it by introducing weak points in the structure.

48. Why does the force required to stretch a wire increase with extension?

As a wire is stretched, more atomic bonds are strained and the resistance to further deformation increases. This results in a greater force required for additional extension, increasing the work done.

49. What's the significance of the ultimate strength in relation to work done on a wire?

The ultimate strength represents the maximum stress a wire can withstand. The work done to stretch a wire to this point is the maximum energy the wire can absorb before failure.

50. How does the microstructure of a wire influence the work required to stretch it?

The microstructure, including grain size and orientation, affects a wire's mechanical properties. Finer grains generally increase strength, requiring more work for stretching, while certain orientations may facilitate easier deformation.