Import Question JSON

Current Question (ID: 10141)

Question:

$\text{Four thin rods, each of mass } m \text{ and the length } L, \text{ form a square. The moment of inertia on any side of the square is:}$

Options:

1. $\frac{5}{3}mL^2$
2. $4mL^2$
3. $\frac{1}{4}mL^2$
4. $\frac{2}{3}mL^2$

Solution:

$\text{For the moment of inertia (I) about one side of the square:}$ $\text{• The rod forming the axis has } I_1 = 0.$ $\text{• The opposite parallel rod has } I_2 = mL^2.$ $\text{• Each of the two perpendicular rods (top and bottom) has } I = \frac{1}{3}mL^2 \text{ since the axis is at their end.}$ $\text{Total moment of inertia:}$ $I_{\text{total}} = I_1 + I_2 + I_3 + I_4 = 0 + mL^2 + \frac{1}{3}mL^2 + \frac{1}{3}mL^2 = mL^2 + \frac{2}{3}mL^2 = \frac{5}{3}mL^2.$ $\text{The final answer is } \frac{5}{3}mL^2.$

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
}