Animated Solution for Physics - Properties of Solids and Liquids: The length of metallic wire is $l_1$ when tension in it is $T_1$. It is $l_2$ when the tension is $T_2$. The original length of the wire will be

Imagine you are in a physics lab, looking at a metallic wire suspended from a rigid ceiling. When no external force is applied, the wire hangs peacefully at its natural length, which we will call $l_0$.

Now, we start pulling it. When we apply a tension $T_1$, the wire stretches to a new length $l_1$. If we pull harder with a tension $T_2$, it stretches even further to a length $l_2$. Our mission is to find that elusive original length $l_0$ using only the given tensions and stretched lengths. To do this, we need to assume that the wire has a uniform cross-sectional area $A$ and is made of a material with Young's modulus $Y$.

The secret to solving this lies in Hooke's Law, which governs the elasticity of materials. It tells us that within the elastic limit, the stress applied to a wire is directly proportional to the strain it experiences.

Mathematically, Young's modulus $Y$ is defined as the ratio of tensile stress to tensile strain:

By rearranging this formula, we can express the change in length $\Delta l$ in terms of the applied tension:

This elegant little equation is our master key. It connects the physical stretch of the wire directly to the force we apply.

Let's apply our master equation to the two scenarios described in the problem.

In the first state, the tension is $T_1$ and the final length is $l_1$. The change in length is simply the final length minus the natural length, so $\Delta l_1=l_1-l_0$. Substituting this into our formula gives us our first equation:

Similarly, for the second state, the tension is $T_2$ and the final length is $l_2$. The change in length is $\Delta l_2=l_2-l_0$. This gives us our second equation:

We now have a system of two equations. The variables $A$ and $Y$ are unknown constants that we introduced, so we need to eliminate them. The most elegant way to do this is to divide the first equation by the second equation.

Watch how beautifully the unknown terms cancel out:

Now, we perform a careful cross-multiplication to get rid of the fractions:

Expanding the brackets, we get:

Our goal is to isolate $l_0$. Let's group all the terms containing $l_0$ on the left side of the equation, and move everything else to the right side:

Factoring out $l_0$ on the left side:

Finally, we divide by $(T_1-T_2)$ to solve for $l_0$:

To match the format of the options provided in the question, we can multiply both the numerator and the denominator by $-1$. This doesn't change the value, but it flips the order of the terms:

And there we have it! The original length of the wire is perfectly expressed in terms of the given tensions and stretched lengths.