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Current Question (ID: 10370)

Question:

Four identical hollow cylindrical columns of mild steel support a big structure of a mass of $50,000 \text{ kg}$. The inner and outer radii of each column are $30 \text{ cm}$ and $60 \text{ cm}$ respectively. Assuming the load distribution to be uniform, the compressional strain of each column is: (Given, Young's modulus of steel, $Y = 2 \times 10^{11} \text{ Pa}$)

Options:

1. $3.03 \times 10^{-7}$
2. $2.8 \times 10^{-6}$
3. $7.22 \times 10^{-7}$
4. $4.34 \times 10^{-7}$

Solution:

\text{Hint: } Y = \frac{\text{stress}}{\text{strain}} \text{Step 1: Write down the data given in the question.} \text{Inner radius, } r = 30 \text{ cm} = 0.3 \text{ m} \text{Outer radius, } R = 0.6 \text{ m} \text{Young's modulus of steel, } Y = 2 \times 10^{11} \text{ Pa} \text{Total force exerted } = Mg = 50000 \times 9.8 \text{ N} \text{Stress } = \text{force on single column } = 122500 \text{ N} \text{Step 2: Calculate the strain using the above relation.} $Y = \frac{\text{stress}}{\text{strain}}$ $\text{Strain} = \frac{[F/A]}{Y}$ \text{Here,} $\text{Area, } A = \pi [R^2 - r^2] = \pi \{ (0.6)^2 - (0.3)^2 \}$ $\text{Strain} = \frac{122500}{\pi \{ (0.6)^2 - (0.3)^2 \} \times 2 \times 10^{11}} = 7.22 \times 10^{-7}$

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Expected JSON Format:

{
  "question": "The mass of carbon present in 0.5 mole of $\\mathrm{K}_4[\\mathrm{Fe(CN)}_6]$ is:",
  "options": [
    {
      "id": 1,
      "text": "1.8 g"
    },
    {
      "id": 2,
      "text": "18 g"
    },
    {
      "id": 3,
      "text": "3.6 g"
    },
    {
      "id": 4,
      "text": "36 g"
    }
  ],
  "solution": "\\begin{align}\n&\\text{Hint: Mole concept}\\\\\n&1 \\text{ mole of } \\mathrm{K}_4[\\mathrm{Fe(CN)}_6] = 6 \\text{ moles of carbon atom}\\\\\n&0.5 \\text{ mole of } \\mathrm{K}_4[\\mathrm{Fe(CN)}_6] = 6 \\times 0.5 \\text{ mol} = 3 \\text{ mol}\\\\\n&1 \\text{ mol of carbon} = 12 \\text{ g}\\\\\n&3 \\text{ mol carbon} = 12 \\times 3 = 36 \\text{ g}\\\\\n&\\text{Hence, 36 g mass of carbon present in 0.5 mole of } \\mathrm{K}_4[\\mathrm{Fe(CN)}_6].\n\\end{align}",
  "correct_answer": 4
}