Import Question JSON

Current Question (ID: 19921)

Question:

$\text{A particle is dropped inside a tunnel of the earth about any diameter.}$ $\text{Particle starts oscillating, with time period=}$ $(R = \text{Radius of earth}, g = \text{acceleration due to gravity on earth's surface}).$

Options:

1. $2\pi \sqrt{\frac{R}{g}}$
2. $\pi \sqrt{\frac{R}{g}}$
3. $2\pi \sqrt{\frac{2R}{g}}$
4. $2\pi \sqrt{\frac{3R}{g}}$

Solution:

$\text{Hint: } F = -kr$ $\text{Step: Find the time period of the particle.}$ $\text{The restoring force acting on the particle is given by:}$ $F = -\left(\frac{GMm}{R^3}\right)r$ $\Rightarrow m \frac{dv}{dt} = -\left(\frac{GMm}{R^3}\right)r$ $\Rightarrow \frac{dv}{dt} = \frac{d^2r}{dt^2} = -\left(\frac{g}{R}\right)r = -\omega^2 r$ $\Rightarrow \omega = \sqrt{\frac{g}{R}}$ $\text{The time period of the particle is given by:}$ $T = \frac{2\pi}{\omega} = 2\pi \sqrt{\frac{R}{g}}$ $\text{Hence, option (1) is the correct answer.}$

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
}