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Current Question (ID: 20459)

Question:

$\text{A magnetic needle of magnetic moment } 6.7 \times 10^{-2} \text{ A m}^2 \text{ and moment of inertia } 7.5 \times 10^{-6} \text{ kg m}^2 \text{ is performing simple harmonic oscillation in a magnetic field of } 0.01 \text{ T. Time taken for 10 complete oscillations is:}$ $1.\ 6.65\ \text{s}$ $2.\ 8.89\ \text{s}$ $3.\ 6.98\ \text{s}$ $4.\ 8.76\ \text{s}$

Options:

1. $6.65\ \text{s}$
2. $8.89\ \text{s}$
3. $6.98\ \text{s}$
4. $8.76\ \text{s}$

Solution:

$\text{Hint: } T = 2\pi \sqrt{\frac{I}{MB}}$ $\text{Step: Find the time taken for 10 complete oscillations.}$ $\text{The time period of a magnetic needle performing simple harmonic motion in a magnetic field is given by:}$ $T = 2\pi \sqrt{\frac{I}{MB}}$ $\text{Substitute the known values we get:}$ $\Rightarrow T = 2\pi \sqrt{\frac{7.5 \times 10^{-6}}{(6.7 \times 10^{-2}) \cdot 0.01}}$ $\Rightarrow T = 2\pi \sqrt{\frac{7.5 \times 10^{-6}}{6.7 \times 10^{-4}}}$ $\Rightarrow T = 2\pi \times 1.12 \times 10^{-2}$ $\Rightarrow T = 2\pi \times 0.1059$ $\Rightarrow T \approx 0.665 \text{ s}$ $T_{10} = 10T = 10 \times 0.665 \approx 6.65 \text{ s}$ $\text{Hence, option (1) is the correct answer.}$

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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
}