Animated Solution for Physics - Properties of Solids and Liquids: A uniform metallic wire is elongated by $0.04\text{ m}$ when subjected to a linear force $F$. The elongation, if its length and diameter is doubled and subjected to the same force will be ......... $\text{cm}$.

A Tale of Two Wires

Imagine a metallic wire hanging from the ceiling. When we pull it downwards with a constant force $F$, it stretches slightly, acting much like a very stiff spring. We are given that this initial wire, with length $l_1$ and diameter $d_1$, experiences an elongation of $\Delta l_1=0.04\text{ m}$.

Now, we are presented with a second wire. This new wire is made of the exact same metal, but its dimensions have been scaled up. Its length is doubled ($l_2=2l_1$), and its diameter is also doubled ($d_2=2d_1$). We apply the exact same force $F$ to this new wire. The question is: how much will it stretch?

Young's Modulus

To understand how a material responds to stretching, we rely on Young's Modulus ($Y$). It is defined as the ratio of tensile stress to tensile strain.

Mathematically, this gives us our master equation:

Because both wires are made of the same uniform metal, their Young's Modulus is identical ($Y_1=Y_2$). This is the crucial physical constraint that allows us to link the two scenarios together.

Area vs. Diameter

Before we equate the two scenarios, we must be very careful with the cross-sectional area. The area of a circular wire is given by $A=\frac{\pi d^2}{4}$.

Notice that the area depends on the square of the diameter. When the diameter is doubled ($d_2=2d_1$), the new area becomes:

So, the second wire isn't just twice as thick; it has four times the cross-sectional area! This makes it significantly stiffer.

Now, let's equate the Young's Modulus for both wires:

Substitute the relationships we found ($l_2=2l_1$ and $A_2=4A_1$):

This is where the magic happens. The force $F$, the initial length $l_1$, and the initial area $A_1$ are present on both sides of the equation. They cancel out beautifully, leaving us with a pure numerical relationship:

Simplifying the fraction on the right side gives:

Cross-multiplying yields:

The question specifically asks for the answer in centimeters. Multiplying by 100, we get our final answer:

Physical Intuition: Doubling the length of the wire tries to double the elongation (making it $0.08\text{ m}$). However, doubling the diameter increases the area by a factor of 4, which makes the wire four times harder to stretch. The net effect is $\frac{2}{4}=\frac{1}{2}$, meaning the final elongation is exactly half of the original!