Import Question JSON

Current Question (ID: 10304)

Question:

$\text{A uniform rod on a smooth horizontal table is moving with uniform horizontal speed } v. \text{ Suddenly rod is hinged at the center of the rod. The angular velocity of the rod, now, will be:}$

Options:

1. $\frac{2v}{3l}$
2. $\frac{3v}{2l}$
3. $\frac{v}{l}$
4. $\text{zero}$

Solution:

$\text{Hint: Angular momentum about the hinge point will be conserved.}$ $\text{Step 1: Find angular momentum about the center.}$ $\vec{L}_C = m\vec{v} \times \vec{r}$ $\vec{L}_C = 0 \text{ ... (1)}$ $\text{Step 2: Find the new angular velocity of the rod.}$ $L_f = I\omega$ $L_f = \left(\frac{ML^2}{12}\right) \times \omega \text{ ... (2)}$ $\text{Angular momentum about the hinge point will be conserved, therefore compare the equation (1) and (2) we will get } L_C = L_f$ $\left(\frac{ML^2}{12}\right) \times \omega = 0$ $\Rightarrow \omega = 0$ $\text{Hence, option (4) is the correct answer.}$

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
}