Import Question JSON

Current Question (ID: 19329)

Question:

$\text{A thin disc of mass } M \text{ and radius } R \text{ has mass per unit area } \sigma(r) = kr^2 \text{ where } r \text{ is the distance from its centre.}$ $\text{Its moment of inertia about an axis going through its centre of mass and perpendicular to its plane is:}$

Options:

1. $\frac{2MR^2}{3}$
2. $\frac{MR^2}{6}$
3. $\frac{MR^2}{3}$
4. $\frac{MR^2}{2}$

Solution:

$I = \int_0^R r^2 dm$ $= \int_0^R \sigma (2\pi r) dr \ r^2$ $= 2\pi \int_0^R r^5 dr$ $= 2\pi \left( \frac{R^6}{6} \right)$ $= \frac{\pi k R^6}{3}$ $\text{Mass of disc } M = k 2\pi \int_0^R r^3 dr$ $M = k 2\pi \left( \frac{R^4}{4} \right)$ $k = \frac{4M}{2\pi R^4}$ $\text{put the value}$ $I = \frac{\pi}{3} \left( \frac{4M}{2\pi R^4} \right) R^6$ $= \frac{2MR^2}{3}$

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
}