Import Question JSON

Current Question (ID: 19436)

Question:

$\text{A test particle is moving in a circular orbit in the gravitational field produced by a mass density } \rho(r) = \frac{K}{r^2}. \text{ Identify the correct relation between the radius } R \text{ of the particle's orbit and its period } T:$

Options:

1. $\frac{T}{R^2} \text{ is a constant}$
2. $\frac{T}{R} \text{ is a constant}$
3. $\frac{T^2}{R^3} \text{ is a constant}$
4. $TR \text{ is a constant}$

Solution:

$\text{Hint:}$ $\int dm = \int 4\pi r^2 dr \cdot \frac{K}{r^2}$ $m = 4\pi Kr$ $V = \sqrt{\frac{G4\pi Kr}{r}}$ $V = \sqrt{4\pi KG}$ $T = \frac{2\pi R}{V}$ $\frac{T}{R} \rightarrow \text{const.}$

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
}