Import Question JSON

Current Question (ID: 10213)

Question:

$\text{A thin uniform circular disc of mass } M \text{ and radius } R \text{ is rotating in a horizontal plane about an axis passing through its center and perpendicular to its plane with an angular velocity } \omega. \text{ Another disc of the same dimensions but of mass } \frac{1}{4}M \text{ is placed gently on the first disc co-axially. The angular velocity of the system will be:}$

Options:

1. $\frac{2}{5}\omega$
2. $\frac{4}{5}\omega$
3. $\frac{3}{4}\omega$
4. $\frac{1}{3}\omega$

Solution:

$\text{Using conservation of angular momentum:}$ $I_1\omega_1 = I_2\omega_2 \text{ or, } \omega_2 = \frac{I_1}{I_2}\omega_1$ $\text{For the initial disc: } I_1 = \frac{1}{2}MR^2$ $\text{For the system after placing the second disc:}$ $I_2 = \frac{1}{2}MR^2 + \frac{1}{2}(\frac{1}{4}M)R^2 = \frac{1}{2}MR^2 + \frac{1}{8}MR^2 = \frac{5}{8}MR^2$ $\text{Therefore:}$ $\omega_2 = \frac{\frac{1}{2}MR^2}{\frac{5}{8}MR^2}\omega = \frac{4}{5}\omega$

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
}