Import Question JSON

Current Question (ID: 19354)

Question:

$\text{A uniform thin bar of mass } 6 \text{ kg and length } 2.4 \text{ meter is bent to make an equilateral hexagon. The moment of inertia about an axis passing through the centre of mass and perpendicular to the plane of hexagon is:}$

Options:

1. $0.8 \text{ kg m}^2$
2. $0.4 \text{ kg m}^2$
3. $0.1 \text{ kg m}^2$
4. $8 \text{ kg m}^2$

Solution:

$\text{Apply the parallel axis theorem.}$ $m = \text{mass of one side of hexagon} = 1 \text{ kg}$ $6\ell = 2.4, \ell = 0.4 \text{ m}$ $\sin 60^\circ = \frac{r}{\ell}$ $r = \ell \sin 60^\circ = \frac{\ell \sqrt{3}}{2}$ $\text{MOI, } I = \left[ \frac{m \ell^2}{12} + mr^2 \right] 6$ $\left[ \frac{m \ell^2}{12} + m \left( \frac{\ell \sqrt{3}}{2} \right)^2 \right] 6$ $= 5 m \ell^2$ $= 5 \times 1 \times 0.16$ $= 0.8$ $I = 8 \times 10^{-1} \text{ kg m}^2$

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
}