The following four wires are made of the same material. Which of these will have the largest extensi

Q: Why does the material being the same mean Young's Modulus is constant?

Young's Modulus is an intrinsic material property, just like density or resistivity. It depends solely on the nature of the atomic bonds of the substance, not its dimensions.

Q: Do I need to convert centimeters and millimeters to standard SI units (meters) before calculating?

Since we are comparing ratios of the same physical quantities across different options, as long as we use consistent units for all options, the relative comparison remains perfectly valid. Keeping them in cm/mm$^2$ saves precious time during the exam!

Q: Why is the elongation inversely proportional to the square of the diameter?

Because the stretching force is distributed over the cross-sectional area. Since area is proportional to the square of the diameter ($A \propto d^2$), doubling the diameter quadruples the area, which dramatically reduces the stress and thus the elongation.

Q: What would happen if the tension applied was different for each wire?

If the tension $F$ varied, elongation would be proportional to $\frac{FL}{d^2}$. We would then need the specific force values to determine the largest extension.

Q: Is Hooke's Law always valid for any amount of stretching?

No, Hooke's Law and the concept of Young's Modulus are only valid within the elastic limit of the material. If the tension is too high, the wire will undergo plastic deformation or break.

Question

The following four wires are made of the same material. Which of these will have the largest extension when the same tension is applied?

Options:

(A) Length = 50 cm, diameter = 0.5 mm

(B) Length = 100 cm, diameter = 1 mm

(C) Length = 200 cm, diameter = 2 mm

(D) Length = 300 cm, diameter = 3 mm

Sigma Percentile · Accepted Answer

### Introduction to Elasticity

When we think of stretching things, we often picture rubber bands or bungee cords.

But in the world of engineering and physics, even the stiffest steel wires behave like microscopic springs.

When a force is applied to a wire, the atomic bonds stretch, leading to a macroscopic elongation.

This behavior is governed by **Hooke's Law** within the elastic limit of the material.

In this problem, we are given four wires made of the **same material** and subjected to the **same tension** (stretching force).

Our task is to determine which of these wires will experience the largest extension.

---

### The Physics of Stretching: Young's Modulus

To analyze this quantitatively, we must define **Young's Modulus** ($Y$), which is an intrinsic property of the material.

It is defined as the ratio of tensile stress to tensile strain:

$$Y = \frac{	ext{Stress}}{	ext{Strain}}$$

Where:
- **Stress** is the restoring force per unit cross-sectional area: $	ext{Stress} = \frac{F}{A}$
- **Strain** is the fractional change in length: $	ext{Strain} = \frac{\Delta L}{L}$

Substituting these definitions into our master equation, we get:

$$Y = \frac{F / A}{\Delta L / L} = \frac{F L}{A \Delta L}$$

Rearranging this equation to solve for the elongation (extension) $\Delta L$ gives:

$$\Delta L = \frac{F L}{A Y}$$

This elegant formula tells us that the extension of a wire is directly proportional to the stretching force $F$ and the original length $L$, and inversely proportional to the cross-sectional area $A$ and Young's Modulus $Y$.

---

### Connecting Area to Diameter

Most wires have a circular cross-section.

Therefore, the area $A$ can be expressed in terms of its diameter $d$ as:

$$A = \pi r^2 = \pi \left(\frac{d}{2}ight)^2 = \frac{\pi d^2}{4}$$

Substituting this expression for $A$ back into our elongation formula yields:

$$\Delta L = \frac{F L}{\left(\frac{\pi d^2}{4}ight) Y} = \frac{4 F L}{\pi d^2 Y}$$

Let's pause and look at this beautiful result.

We are told that:
1. All four wires are made of the **same material**, which means Young's Modulus ($Y$) is constant.
2. The **same tension** ($F$) is applied to all of them.
3. The numbers $4$ and $\pi$ are, of course, universal constants.

By grouping all the constant terms together, we find a powerful proportionality relation:

$$\Delta L \propto \frac{L}{d^2}$$

This means that the wire with the **largest ratio of $\frac{L}{d^2}$** will experience the **largest extension**!

---

### The Battle of the Ratios

Now, let's evaluate the ratio $\frac{L}{d^2}$ for each of the four options.

Note that we do not need to convert the units to standard SI units (meters) because we are only comparing relative ratios.

As long as we keep the units consistent (length in $	ext{cm}$ and diameter in $	ext{mm}$), our comparison will be perfectly valid.

#### Option (a):
- Length $L = 50	ext{ cm}$
- Diameter $d = 0.5	ext{ mm}$

$$\frac{L}{d^2} = \frac{50}{(0.5)^2} = \frac{50}{0.25} = 200	ext{ cm/mm}^2$$

#### Option (b):
- Length $L = 100	ext{ cm}$
- Diameter $d = 1	ext{ mm}$

$$\frac{L}{d^2} = \frac{100}{(1)^2} = 100	ext{ cm/mm}^2$$

#### Option (c):
- Length $L = 200	ext{ cm}$
- Diameter $d = 2	ext{ mm}$

$$\frac{L}{d^2} = \frac{200}{(2)^2} = \frac{200}{4} = 50	ext{ cm/mm}^2$$

#### Option (d):
- Length $L = 300	ext{ cm}$
- Diameter $d = 3	ext{ mm}$

$$\frac{L}{d^2} = \frac{300}{(3)^2} = \frac{300}{9} \approx 33.3	ext{ cm/mm}^2$$

---

### Conclusion

Comparing the calculated ratios:
- Option (a): $200$
- Option (b): $100$
- Option (c): $50$
- Option (d): $33.3$

The ratio $\frac{L}{d^2}$ is clearly maximum for **Option (a)**.

Therefore, the wire with a length of $50	ext{ cm}$ and a diameter of $0.5	ext{ mm}$ will undergo the greatest elongation under the given tension.

**Correct Option: (a)**

Animated Solution for Physics - Properties of Solids and Liquids: The following four wires are made of the same material. Which of these will have the largest extension when the same tension is applied?

Select Answer:

Visualized Solution

Visualizing Wire Elongation under Tension

Hooke's Law and Young's Modulus

Expressing Area in terms of Diameter

Substituting Area into the Elongation Formula

Identifying Proportionality Relations

Evaluating Option (a)

Evaluating Option (b)

Evaluating Option (c)

Evaluating Option (d)

Concluding the Largest Extension

Morals and Extensions

The Sigma Insight: Young's Modulus, Bulk Modulus and Modulus of Rigidity

Introduction to Elasticity

The Physics of Stretching

Connecting Area to Diameter

The Battle of the Ratios

# Option (a)

# Option (b)

# Option (c)

# Option (d)

Conclusion

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