Question

A wire of length \(L\) and radius \(\mathrm{r}\) is rigidly fixed at one end on stretching the other end of the wire a force \(\mathrm{F}\) the increase in its lengths is \(\ell\). If another wire of same material but of length \(2 \mathrm{~L}\) and radius \(2 \mathrm{r}\) is stretched with a force of \(2 \mathrm{~F}\), the increase in its length will be........... (A) \(\varepsilon\) (B) \(2 \ell\) (C) \(\mathrm{C} / 2\) (D) 4

Short answer

The increase in length for the second wire is 2 times the increase in length for the first wire. So, the correct option is \( (B) 2l \).

Step-by-step solution

Understand the given values

The first wire has: - Length: L - Radius: r - Force: F - Increase in length: \(l\) The second wire has: - Length: 2L - Radius: 2r - Force: 2F - Increase in length: We have to find this value

Young's modulus formula

Young's modulus is a measure of the stiffness of a material, and it is defined as the ratio of stress (force per unit area) to strain (relative deformation). In general, it's denoted as Y and is given by the formula: \(Y = \frac{\text{Stress}}{\text{Strain}}\) For wires, we can rewrite this formula using stress and strain in terms of force, cross-sectional area (A), and displacements (∆L): \(Y = \frac{F/A}{\Delta L/L}\) Rearranging the formula for increased length (∆L): \(\Delta L = \frac{FL}{AY}\)

Determine Force and Area relationship

Here, we must determine the relationship between the cross-sectional area (A) and the radius (r). Since the wire has a circular cross-section, the area A can be expressed as: \(A = \pi r^2\) Now, for the second wire, the radius is doubled, meaning: \(A' = \pi (2r)^2 = 4\pi r^2 = 4A\) The force on the second wire is also doubled: \(F' = 2F\)

Calculate the increased length for the second wire

Using the formula \(\Delta L = \frac{FL}{AY}\), we will plug in the values of F', A', and L': \(\Delta L' = \frac{F'L'}{A'Y} = \frac{2F \cdot 2L}{4AY}\) Now, we notice that the given values (F, L, A) are already present in the first wire, which means we can substitute the value of the first wire into the second: \(\Delta L' = \frac{2F \cdot 2L}{4AY} = 2 \cdot \frac{FL}{AY} = 2 \Delta L\) Therefore, the increase in length for the second wire is:

Conclusion

The increase in length for the second wire is 2 times the increase in length for the first wire. So, the correct option is (B) 2l.

Key concepts

Stress and Strain

Understanding the relationship between stress and strain is important when discussing materials under force. Stress is the internal force within a material divided by its cross-sectional area. It is usually measured in Pascals (Pa), and it's an indication of what force is trying to compress or extend the material. Strain, on the other hand, is the measure of deformation representing how much the material has changed in length compared to its original length. It doesn't have a unit and often it's expressed as a fraction or percentage.

In the context of Young's modulus, stress and strain come together in a ratio to describe how a material responds to applied forces. When you stretch a wire, you apply stress on it, and the wire reacts with strain. This ratio is crucial in understanding and calculating how much a material will extend or compress given a certain stress.

Cross-sectional Area

The cross-sectional area of a wire or any object with a similar shape is important to determine its strength and how it will react to forces. The area affects the stress because as per the formula, stress is inversely proportional to the area (stress = force/area).

For a wire with a circular cross-section, the area can be calculated with the formula: \(A = \pi r^2\), where \(r\) is the radius of the cross-section. In the exercise, for the second wire, the radius is doubled turning the area into four times the original, \(A' = 4A\). This significant change in area helps compensate for the increased force, ultimately affecting the material's deformation behavior.

Elasticity

Elasticity is a property that describes how a material returns to its original shape after the removal of a force that makes it deform. Materials that recover quickly and fully are said to have high elasticity. When you stretch a wire and it goes back to the same length once released, it's showing an elastic behavior.

You might have come across the term "elastic limit," which is the maximum stress or force a material can endure before permanently deforming. Young's modulus also falls into this realm, representing the stiffness and elasticity by quantitatively characterizing the material's ability to resist deformation.

Material Stiffness

Material stiffness is often synonymous with the term Young's modulus. It represents how rigid a material is and quantifies its resistance to deformation under stress. Higher stiffness indicates a material is less likely to deform easily, while lower stiffness suggests that it will deform more readily under the same stress.

The stiffness is crucial for applications where maintaining shape and form under loads is important. Engineers look at Young's modulus when selecting materials for constructing buildings, bridges, or gadgets to ensure they will perform as expected throughout their lifespan. The formula \(Y = \frac{F/A}{\Delta L/L}\) gives us a clear understanding of how this stiffness (Young's modulus) is mathematically determined through stress and strain.