Import Question JSON

Current Question (ID: 19451)

Question:

$\text{Consider two satellites, } S_1 \text{ and } S_2, \text{ with periods of revolution } 1 \text{ hr and } 8 \text{ hr, respectively revolving around a planet in circular orbits.}$ $\text{The ratio of the angular velocity of the satellite } S_1 \text{ to the angular velocity of the satellite } S_2 \text{ is:}$

Options:

1. $8 : 1$
2. $1 : 4$
3. $2 : 1$
4. $1 : 8$

Solution:

$\text{Hint: } T = \frac{2\pi}{\omega}$ $\text{Step: Find the ratio of the angular velocity of satellite } S_1 \text{ to the angular velocity of a satellite } S_2.$ $\text{The time period of the satellite is given by:}$ $T = \frac{2\pi}{\omega}$ $\Rightarrow \omega = \frac{2\pi}{T}$ $\text{The ratio of the angular velocity of satellite } S_1 \text{ to the angular velocity of a satellite } S_2 \text{ is given by:}$ $\frac{\omega_1}{\omega_2} = \frac{\frac{2\pi}{T_1}}{\frac{2\pi}{T_2}} = \frac{T_2}{T_1}$ $\Rightarrow \frac{\omega_1}{\omega_2} = \frac{8}{1} \left[ \frac{T_1}{T_2} = \frac{1}{8} \right]$ $\text{Hence option (1) is the correct answer.}$

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
}