Import Question JSON

Current Question (ID: 10208)

Question:

$\text{The moment of inertia of a thin uniform circular disc about one of its diameter is } I. \text{ Its moment of inertia about an axis perpendicular to the circular surface and passing through its center will be:}$

Options:

1. $\sqrt{2I}$
2. $2I$
3. $\frac{I}{2}$
4. $\frac{I}{\sqrt{2}}$

Solution:

$\text{Using the Perpendicular Axis Theorem for a thin uniform circular disc:}$ $\text{Let } I_x \text{ and } I_y \text{ be the moments of inertia about two perpendicular diameters, and } I_z \text{ be the moment of inertia about an axis perpendicular to the disc and passing through its center.}$ $\text{For a thin uniform circular disc, by symmetry, the moment of inertia about any diameter is the same. So, if the moment of inertia about one of its diameters is } I, \text{ then } I_x = I_y = I.$ $\text{According to the Perpendicular Axis Theorem:}$ $I_z = I_x + I_y$ $I_z = I + I$ $I_z = 2I$ $\text{Therefore, the moment of inertia about an axis perpendicular to the circular surface and passing through its center will be } 2I.$ $\text{The correct option is 2. } 2I.$

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
}