Import Question JSON

Current Question (ID: 19355)

Question:

$\text{A sphere of radius } 'a' \text{ and mass } 'm' \text{ rolls along a horizontal plane with constant speed } v_0. \text{ It encounters an inclined plane at angle } \theta \text{ and climbs upward. Assuming that it rolls without slipping, how far up the sphere will travel?}$

Options:

1. $\frac{7v_0^2}{10g \sin \theta}$
2. $\frac{v_0^2}{5g \sin \theta}$
3. $\frac{2}{5} \frac{v_0^2}{g \sin \theta}$
4. $\frac{v_0^2}{2g \sin \theta}$

Solution:

$\text{Hint: Apply conservation of angular momentum and energy.}$ $\text{Angular momentum conservation about A}$ $mv_0 \cdot a \cos \theta + \frac{2}{5} ma^2 \omega = mva + \frac{2}{5} ma^2 \omega'$ $mv_0 \cdot a \left[ \frac{2}{5} + \cos \theta \right] = \frac{7}{5} mva$ $v = \frac{5}{7} v_0 \left[ \frac{2}{5} + \cos \theta \right]$ $\frac{1}{2} mv^2 + \frac{1}{2} I \omega^2 = \frac{7}{10} mv^2 = mgh$ $\text{No option Matching}$

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
}