Import Question JSON

Current Question (ID: 19395)

Question:

$\text{Solid sphere rolls on horizontal plane. Ratio of angular momentum about COM to total energy is } \frac{\pi}{22}. \text{ The angular velocity of the solid sphere in (rad/s) is:}$ $1.\ 4$ $2.\ 3$ $3.\ 2$ $4.\ 1$

Options:

1. $4$
2. $3$
3. $2$
4. $1$

Solution:

$L = I_{\text{COM}} \omega \text{ and } K = \frac{1}{2} I_{\text{COM}} \omega^2 + \frac{1}{2} M v_0^2$ $L = \frac{2}{5} M R^2 \frac{v_0}{R} \quad K = \frac{1}{2} \left( \frac{2}{5} M R^2 \right) \frac{v_0^2}{R^2} + \frac{1}{2} M v_0^2$ $L = \frac{2 M R v_0}{5} \quad K = \frac{7}{10} M v_0^2$ $\text{Ratio } \frac{L}{K} = \frac{4}{7} \frac{R}{v_0} = \frac{\pi}{22} \Rightarrow \omega = \frac{4}{7} \times \frac{22}{22} \times 7 = 4$

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
}