Import Question JSON

Current Question (ID: 19470)

Question:

$\text{Two point masses (mass } m \text{ each) are moving in a circle of radius } R \text{ under mutual gravitational attraction. What is the speed of each mass?}$

Options:

1. $\sqrt{\frac{GM}{4R}}$
2. $\sqrt{\frac{GM}{2R}}$
3. $\sqrt{\frac{GM}{8R}}$
4. $\sqrt{\frac{GM}{R}}$

Solution:

$\text{Hint: Mutual gravitational force provides the centripetal force.}$ $\text{Step: Find the speed of each mass.}$ $\Rightarrow \frac{G \times m \times m}{(2R)^2} = \frac{mv^2}{r}$ $\Rightarrow v = \sqrt{\frac{Gm}{4R}}$ $\text{Therefore, the speed of each particle is } v = \sqrt{\frac{Gm}{4R}}.$ $\text{Hence, option (1) is the correct answer.}$

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
}