Import Question JSON

Current Question (ID: 19131)

Question:

$\text{A particle is moving with a uniform speed in a circular orbit of radius } R \text{ in a central force inversely proportional to the } n^{\text{th}} \text{ power of } R. \text{ If the period of rotation of the particle is } T, \text{ then:}$

Options:

1. $T \propto R^{3/2} \text{ for any } n$
2. $T \propto R^{n/2 + 1}$
3. $T \propto R^{(n+1)/2}$
4. $T \propto R^{n/2}$

Solution:

$\text{Hint: } \frac{mv^2}{R} = \frac{C}{R^n}$ $F \propto \frac{1}{R^n}$ $F = \frac{C}{R^n}$ $\frac{mv^2}{R} = \frac{C}{R^n}$ $v^2 = \frac{C}{m} R^{1-n}$ $v = \sqrt{\frac{C}{m}} R^{\frac{1-n}{2}}$ $T = \frac{2\pi R}{v}$ $T = \frac{2\pi R}{\sqrt{\frac{C}{m} R^{\frac{1-n}{2}}}}$ $T \propto R^{1 - \frac{1-n}{2}} \propto R^{\frac{2-1+n}{2}} \propto R^{\frac{n+1}{2}}$

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
}