Question

A metal rod of cross-sectional area \(3 \times 10^{-6}\,\text{m}^2\) is suspended vertically from one end and has a length of \(0.4\,\text{m}\) at \(100^\circ\text{C}\). Now the rod is cooled upto \(0^\circ\text{C}\), but prevented from contracting by attaching a mass \(m\) at the lower end. The value of \(m\) is
(\(Y = 10^{11}\,\text{N/m}^2\), coefficient of linear expansion \(= 10^{-5}\,\text{K}^{-1}\), \(g = 10\,\text{m/s}^2\))

• A. \(40\,\text{kg}\)
• B. \(20\,\text{kg}\)
• C. \(30\,\text{kg}\)
• D. \(10\,\text{kg}\)

Hint:

If thermal expansion or contraction is prevented, thermal stress develops in the body.

Answer 1

The Correct Option is C

Step 1: Thermal strain in the rod.
If contraction is prevented, thermal strain is \[ \text{strain} = \alpha \Delta T \] \[ \Delta T = 100^\circ\text{C} \]

Step 2: Stress produced in the rod.
\[ \text{stress} = Y \times \text{strain} = Y\alpha\Delta T \] \[ \text{stress} = 10^{11} \times 10^{-5} \times 100 = 10^8\,\text{N/m}^2 \]

Step 3: Force acting on the rod.
\[ F = \text{stress} \times A = 10^8 \times 3\times10^{-6} = 300\,\text{N} \]

Step 4: Calculate mass.
\[ F = mg \Rightarrow m = \frac{300}{10} = 30\,\text{kg} \]