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Current Question (ID: 19983)

Question:

$\text{A transverse wave travels along a uniform wire with a length of } 50 \text{ cm}$ $\text{and a mass of } 10 \text{ grams at a speed of } 60 \text{ m/s. If the wire has a}$ $\text{cross-sectional area of } 2.0 \text{ mm}^2 \text{ and a Young's modulus of }$ $1.2 \times 10^{11} \text{ N/m}^2, \text{ the extension of the wire over its natural length}$ $\text{due to its tension will be:}$

Options:

1. $0.12 \text{ mm}$
2. $0.15 \text{ mm}$
3. $0.20 \text{ mm}$
4. $0.25 \text{ mm}$

Solution:

$\text{Hint: } Y = \frac{FL}{A \Delta L}$ $\text{Step 1: Find the tension in the given string.}$ $v = \sqrt{\frac{T}{\mu}}$ $\Rightarrow 60 = \sqrt{\frac{T}{0.2 \times 10^{-1}}}$ $\Rightarrow T = 72 \text{ N}$ $\text{Step 2: Find the extension of the wire.}$ $\Delta L = \frac{TL}{AY}$ $\Rightarrow \Delta L = \frac{72 \times 0.5}{2 \times 10^{-6} \times 1.2 \times 10^{11}}$ $\Rightarrow \Delta L = 150 \times 10^{-5} \text{ m} = 0.15 \text{ mm}$ $\text{Hence, option (2) is the correct answer.}$

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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
}