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 $1:4$, the ratio of their diameters is:

• A. $\sqrt{2}:1$
• B. $1:\sqrt{2}$
• C. $1:2$
• D. $2:1$

Answer 1

The correct option is A

$\sqrt{2}:1$

Step 1: Given

Ratio of energy stored per unit volume (Energy density): $\frac{u_1}{u_2}=\frac{1}{4}$

Force applied on both wires: $F_1=F_2=F$

Step 2: Formula Used

$u=\frac{1}{2}\frac{stres{s}^2}{\gamma }$

$Stress=\frac{F}{A}$

$A=\pi r^2=\pi {\left(\frac{D}{2}\right)}^2$

Step 3: Find the ratio of energy densities

Find an expression for energy density of first wire in terms of diameter using the formulas

$\begin{array}{rcl}{u}_1& =& \frac{1}{2}\frac{{\left(stres{s}_1\right)}^2}{\gamma }\\ & =& \frac{1}{2}\frac{{\left({\displaystyle \frac{F_1}{A_1}}\right)}^2}{\gamma }\\ & =& \frac{1}{2}\frac{{\left({\displaystyle \frac{F}{\mathrm{\pi }{\left({\displaystyle \frac{{\mathrm{D}}_1}{2}}\right)}^2}}\right)}^2}{\gamma }\\ & =& \frac{1}{2}\frac{16F^2\gamma }{{\mathrm{\pi }}^2{{\mathrm{D}}_1}^4}\end{array}$

Find an expression for energy density of second wire in terms of diameter using the formulas

$\begin{array}{rcl}{u}_2& =& \frac{1}{2}\frac{{\left(stres{s}_2\right)}^2}{\gamma }\\ & =& \frac{1}{2}\frac{{\left({\displaystyle \frac{F_2}{A_2}}\right)}^2}{\gamma }\\ & =& \frac{1}{2}\frac{{\left({\displaystyle \frac{F}{\mathrm{\pi }{\left({\displaystyle \frac{{\mathrm{D}}_2}{2}}\right)}^2}}\right)}^2}{\gamma }\\ & =& \frac{1}{2}\frac{16F^2\gamma }{{\mathrm{\pi }}^2{{\mathrm{D}}_2}^4}\end{array}$

Calculate the ratios of the energy densities of the wires by dividing energy density of first wire by that of second.

$$\begin{gathered} \frac{u_1}{u_2}=\frac{\frac{1}{2}\frac{16F^2\gamma }{{\mathrm{\pi }}^2{{\mathrm{D}}_1}^4}}{\frac{1}{2}\frac{16F^2\gamma }{{\mathrm{\pi }}^2{{\mathrm{D}}_2}^4}} \\ \Rightarrow \frac{1}{4}=\frac{{{\mathrm{D}}_2}^4}{{{\mathrm{D}}_1}^4} \\ \Rightarrow \frac{{\mathrm{D}}_1}{{\mathrm{D}}_2}=\sqrt[4]{\frac{4}{1}} \\ \Rightarrow \frac{{\mathrm{D}}_1}{{\mathrm{D}}_2}=\frac{\sqrt{2}}{1} \end{gathered}$$

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