Import Question JSON

Current Question (ID: 10194)

Question:

$\text{Two rotating bodies A and B of masses } m \text{ and } 2m \text{ with moments of inertia } I_A \text{ and } I_B \text{ (}I_B > I_A\text{) have equal kinetic energy of rotation. If } L_A \text{ and } L_B \text{ be their angular momenta respectively, then:}$

Options:

1. $L_A = \frac{L_B}{2}$
2. $L_A = 2L_B$
3. $L_B > L_A$
4. $L_A > L_B$

Solution:

$\text{As we know that, the kinetic energy of a rotating body,}$ $KE = \frac{1}{2}I\omega^2 = \frac{1}{2} \frac{I^2\omega^2}{I} = \frac{L^2}{2I}$ $\text{Also, angular momentum, } L = I\omega$ $\text{Thus, } K_A = K_B$ $\Rightarrow \frac{1}{2} \frac{L_A^2}{I_A} = \frac{1}{2} \frac{L_B^2}{I_B}$ $\Rightarrow \left(\frac{L_A}{L_B}\right)^2 = \frac{I_A}{I_B} \Rightarrow \frac{L_A}{L_B} = \sqrt{\frac{I_A}{I_B}}$ $L \propto \sqrt{I}$ $\therefore L_A < L_B$ $\text{[} \because I_B > I_A \text{]}$

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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
}