Question

The Young's modulus of brass and steel are respectively $1.0\times {10}^{10}N/m^2$ and $2\times {10}^{10}N/m^2$. A brass wire and a steel wire of the same length are extended by $1mm$ under the same force, the radii of brass and steel wires are $R_B$ and $R_S$ respectively. Then

• A. $R_S=\sqrt{2}{R}_B$
• B. $R_S=\frac{R_B}{\sqrt{2}}$
• C. $R_S=4R_B$
• D. $R_S=\frac{R_B}{4}$

Answer 1

Answer: (B)

$R_S=\frac{R_B}{\sqrt{2}}$

$\begin{array}{c}\text{Given, young modulus of brass}={10}^{10}N/m^2\\ \text{Young modulus of steel}=2\times {10}^{10}N/m^2\\ \text{Both wires have same length and}\\ \text{extended by}1mm\text{under same force.}\end{array}$

$\begin{array}{c}F=Y_B\left(\pi R{B}^2\right)\left(\frac{\Delta L}{L}\right)\\ F=Y_S\left(\pi R_S^2\right)\left(\frac{\Delta L}{L}\right)\\ Y_B{\left(R_B\right)}^2=Y_S{\left(R_S\right)}^2\\ R_B=\sqrt{2}{R}_S\text{Hence, option}B\text{is}\\ \text{the correct answer.}\end{array}$