Import Question JSON

Current Question (ID: 19449)

Question:

$\text{Two stars of masses } m \text{ and } 2m \text{ at a distance } d \text{ rotate about their common centre of mass in free space. The period of revolution is:}$

Options:

1. $\frac{1}{2\pi} \sqrt{\frac{d^3}{3Gm}}$
2. $2\pi \sqrt{\frac{d^3}{3Gm}}$
3. $\frac{1}{2\pi} \sqrt{\frac{3Gm}{d^3}}$
4. $2\pi \sqrt{\frac{3Gm}{d^3}}$

Solution:

$\text{Hint: } F = \frac{GM_1M_2}{R^2} = m\omega^2r$ $F = \frac{G(2m)m}{d^2} = \left(2m\right)e^2\left(\frac{d}{3}\right)$ $\frac{Gm}{d^2} = \omega^2 \frac{d}{3}$ $\Rightarrow \omega^2 = \frac{3Gm}{d^3} \Rightarrow \omega = \sqrt{\frac{3Gm}{d^3}}$ $\Rightarrow T = \frac{2\pi}{\omega} = 2\pi \sqrt{\frac{d^3}{3Gm}}$

Import JSON File

Upload a JSON file containing LaTeX/MathJax formatted question, options, and solution.

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
}