Question

Two steel wires having same length are suspended from a ceiling under the same load. If the ratio of their energy stored per unit volume is $1:4$. Find the ratio of their diameters.

Answer 1

Step 1: Given data

The ratio of their energy stored per unit volume is $1:4$.

Step 2: Determination of the relation between the energy density and diameter:

1. Energy density is the energy stored in a system per unit volume.
2. The equation of energy density can be written as follows:

$\frac{du}{dV}=\frac{F^2}{2A^{2}y}$

Where, $u$ is the energy, $v$ is the volume, $F$ is the force, $A$ is the area, and $y$ is the length of the wires.

3. Since the length of the wires is the same, and the load applied is also the same, the energy density depends on the square of the area only.

The relationship between the energy density and the diameter can be determined as follows:

$$\begin{gathered} \frac{du}{dv}\propto \frac{1}{A^2} \\ \frac{du}{dv}\propto \frac{1}{{\left({\displaystyle \frac{\pi }{4}}{d}^2\right)}^2} \\ \frac{du}{dv}\propto \frac{1}{d^4} \\ \frac{1}{4}\propto \frac{1}{d^4} \end{gathered}$$

Here, $d$ is the diameter.

$$\begin{gathered} d^4\propto 4 \\ d\propto {\left(4\right)}^{\frac{1}{4}} \\ d\propto \sqrt{2} \end{gathered}$$

Hence, the ratio of diameters of the wires is $\sqrt{2}:1$.