Import Question JSON

Current Question (ID: 10178)

Question:

$\text{A solid sphere of mass } m \text{ is in pure rolling motion with the centre of mass moving with velocity } v_0\text{. It's angular momentum about the point } P \text{ shown in the diagram is:}$

Options:

1. $\frac{2}{5}mv_0R$
2. $\frac{1}{5}mv_0R$
3. $\frac{7}{5}mv_0R$
4. $mv_0R$

Solution:

$\text{Hint: } L = r \times p$ $\text{Step 1: Draw a diagram.}$ $\text{Step 2: Find the angular momentum about point } P\text{.}$ $L = L_T + L_{Rot}$ $\Rightarrow L = (mv_0R) + I\omega$ $\Rightarrow L = mv_0R + \frac{2}{5}mR^2\omega$ $\Rightarrow L = mv_0R + \frac{2}{5}mv_0R \text{ } [v_0 = R\omega]$ $\Rightarrow L = \frac{7}{5}mv_0R$ $\text{Hence, option (3) is the correct answer.}$

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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
}