Import Question JSON

Current Question (ID: 19347)

Question:

$\text{A thin rod of mass } 0.9 \text{ kg and length } 1 \text{ m is suspended, at rest, from one end so that it can freely oscillate in the vertical plane.}$ $\text{A particle of mass } 0.1 \text{ kg moving in a straight line with velocity } 80 \text{ m/s hits the rod at its bottom most point and sticks to it (see figure).}$ $\text{The angular speed (in rad/s) of the rod immediately after the collision will be:}$

Options:

1. $20$
2. $40$
3. $60$
4. $80$

Solution:

$\text{Hint: Applying conservation of angular momentum of particle + rod about the hinge.}$ $\vec{L}_i = \vec{L}_f$ $mvL = I\omega$ $mvL = \left( \frac{ML^2}{3} + mL^2 \right) \omega$ $0.1 \times 80 \times 1 = \left( \frac{0.9 \times 1^2}{3} + 0.1 \times 1^2 \right) \omega$ $8 = \left( \frac{3}{10} + \frac{1}{10} \right) \omega$ $8 = \frac{4}{10} \omega$ $\omega = 20 \text{ rad/sec}$

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
}