Import Question JSON

Current Question (ID: 19135)

Question:

$\text{A smooth wire of length } 2\pi r \text{ is bent into a circle and kept in a vertical plane.}$ $\text{A bead can slide smoothly on the wire.}$ $\text{When the circle is rotating with angular speed } \omega \text{ about the vertical diameter } AB,$ $\text{as shown in figure, the bead is at rest with respect to the circular ring at position } P \text{ as shown.}$ $\text{Then the value of } \omega^2 \text{ is equal to:}$

Options:

1. $\frac{g\sqrt{3}}{r}$
2. $\frac{2g}{r}$
3. $\frac{g\sqrt{3}}{2r}$
4. $\frac{2g}{\sqrt{3}r}$

Solution:

$\text{Hint: Balance forces along tangential and normal direction.}$ $m\omega^2 \frac{r}{2} \sin \theta = mg \cos \theta$ $\omega^2 = \frac{2g}{r \tan \theta}$ $\tan \theta = \frac{\sqrt{r^2 - r^2/4}}{r/2} = \frac{\sqrt{3}r/2}{r/2} = \sqrt{3}$ $\omega^2 = \frac{2}{\sqrt{3}}$

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
}