Import Question JSON

Current Question (ID: 19433)

Question:

$\text{Four particles, each of mass } M \text{ and equidistant from each other, move along a circle of radius } R \text{ under the action of their mutual gravitational attraction. The speed of each particle is:}$

Options:

1. $\sqrt{2\sqrt{2} \frac{GM}{R}}$
2. $\sqrt{\frac{GM}{R} \left(1 + 2\sqrt{2}\right)}$
3. $\frac{1}{2} \sqrt{\frac{GM}{R} \left(1 + 2\sqrt{2}\right)}$
4. $\sqrt{\frac{GM}{R}}$

Solution:

$\text{Hint: } F_{\text{net}} = \frac{mv^2}{R}$ $F_R = \sqrt{2}F + F_1 = \sqrt{2} \times \frac{GM^2}{(\sqrt{2}R)^2} + \frac{GM^2}{(2R)^2}$ $F_R = \frac{GM^2}{\sqrt{2}R^2} + \frac{GM^2}{4R^2}$ $\text{If they are moving in circle then}$ $\frac{Mv^2}{R} = \frac{GM^2}{R^2} \left(\frac{1}{\sqrt{2}} + \frac{1}{4}\right)$ $v = \sqrt{\frac{GM}{R} \left(\frac{1}{\sqrt{2}} + \frac{1}{4}\right)} = \frac{1}{2} \sqrt{\frac{GM}{R} \left(2\sqrt{2} + 1\right)}$

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
}