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Current Question (ID: 11266)

Question:

$\text{The period of oscillation of a simple pendulum of length } L \text{ suspended from the roof of a vehicle which moves without friction down an inclined plane of inclination } \theta, \text{ is given by:}$

Options:

1. $2\pi \sqrt{\frac{L}{g \cos \theta}}$ (Correct)
2. $2\pi \sqrt{\frac{L}{g \sin \theta}}$
3. $2\pi \sqrt{\frac{L}{g}}$
4. $2\pi \sqrt{\frac{L}{g \tan \theta}}$

Solution:

$\text{See the following force diagram.}$ $\text{The vehicle is moving down the frictionless inclined surface so, it's acceleration is } g \sin \theta. \text{ Since the vehicle is accelerating, a pseudo force } m(g \sin \theta) \text{ will act on bob of pendulum which cancels the } \sin \theta \text{ component of the weight of the bob.}$ $\text{Hence net force on the bob is } F = mg \cos \theta \text{ or net acceleration of the bob is}$ $g_{eff} = g \cos \theta$ $\text{So Time period } T = 2\pi \sqrt{\frac{l}{g_{eff}}} = 2\pi \sqrt{\frac{L}{g \cos \theta}}$

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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
}