Import Question JSON

Current Question (ID: 10293)

Question:

$\text{A boy is standing on a disc rotating about the vertical axis passing through its centre. He pulls his arms towards himself, reducing his moment of inertia by a factor of m. The new angular speed of the disc becomes double its initial value. If the moment of inertia of the boy is } I_0\text{, then the moment of inertia of the disc will be:}$

Options:

1. $2I_0m$
2. $I_0\left(1 - \frac{2}{m}\right)$
3. $I_0\left(1 - \frac{1}{m}\right)$
4. $\left(\frac{I_0}{2m}\right)$

Solution:

$\text{Hint: Angular momentum remains the same}$ $\text{Step 1: Find initial angular momentum}$ $L_i = I\omega + I_0\omega$ $\text{Step 2: Find final angular momentum}$ $L_f = \left(\frac{I_0}{m}\right)2\omega + I(2\omega)$ $\text{Step 3: Apply conservation of angular momentum}$ $L_i = L_f$ $\left(I + I_0\right)\omega = 2\omega\left[\frac{I_0}{m} + I\right]$ $I = I_0\left[1 - \frac{2}{m}\right]$

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
}