Question

One end of a metal wire is fixed to a ceiling and a load of 2 kg hangs from the other end. A similar wire is attached to the bottom of the load and another load of 1 kg hangs from this lower wire. Then the ratio of longitudinal strain of the upper wire to that of the lower wire will be:\[\text{[Area of cross section of wire = 0.005 cm}^2, \, Y = 2 \times 10^{11} \, \text{N/m}^2, \, g = 10 \, \text{m/s}^2\text{]}\]

Answer 1

Correct Answer: 3

Given: - Upper load mass: \(2 \, \text{kg}\) - Lower load mass: \(1 \, \text{kg}\)

Step 1: Calculate Wire Tensions

Tension in the upper wire (\(T_{\text{upper}}\)) equals the total weight of both loads:

\[ T_{\text{upper}} = (2 \, \text{kg} + 1 \, \text{kg}) \times g = 3 \times 10 = 30 \, \text{N} \]

Tension in the lower wire (\(T_{\text{lower}}\)) equals the weight of the lower load:

\[ T_{\text{lower}} = 1 \times 10 = 10 \, \text{N} \]

Step 2: Calculate Longitudinal Strain in Each Wire

Longitudinal strain (\(\epsilon\)) formula:

\[ \epsilon = \frac{\text{Stress}}{Y} = \frac{T}{A \times Y} \]

Upper wire strain:

\[ \epsilon_{\text{upper}} = \frac{T_{\text{upper}}}{A \times Y} = \frac{30}{5 \times 10^{-7} \times 2 \times 10^{11}} \] \[ \epsilon_{\text{upper}} = \frac{30}{2 \times 10^{6}} = 1.5 \times 10^{-5} \]

Lower wire strain:

\[ \epsilon_{\text{lower}} = \frac{T_{\text{lower}}}{A \times Y} = \frac{10}{5 \times 10^{-7} \times 2 \times 10^{11}} \] \[ \epsilon_{\text{lower}} = \frac{10}{2 \times 10^{6}} = 0.5 \times 10^{-5} \]

Step 3: Calculate Ratio of Longitudinal Strains

Ratio of upper wire strain to lower wire strain:

\[ \text{Ratio} = \frac{\epsilon_{\text{upper}}}{\epsilon_{\text{lower}}} = \frac{1.5 \times 10^{-5}}{0.5 \times 10^{-5}} = 3 \]

Conclusion: The ratio of the upper wire's longitudinal strain to the lower wire's longitudinal strain is \(3\).