Import Question JSON

Current Question (ID: 19918)

Question:

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

Options:

1. $2\pi \sqrt{\frac{L}{g \cos \alpha}}$
2. $2\pi \sqrt{\frac{L}{g \sin \alpha}}$
3. $2\pi \sqrt{\frac{L}{g}}$
4. $2\pi \sqrt{\frac{L}{g \tan \alpha}}$

Solution:

$\text{Hint: } T = 2\pi \sqrt{\frac{L}{g_{\text{eff}}}}$ $\text{Step 1: Draw the free-body diagram of the system.}$ $\text{Step 2: Find the time period of the given simple pendulum.}$ $\text{The time period of a simple pendulum is given by:}$ $T = 2\pi \sqrt{\frac{L}{g_{\text{eff}}}}$ $\text{The effective acceleration of the simple pendulum is given by:}$ $g_{\text{eff}} = \frac{F_{\text{net}}}{m} = g \cos \alpha$ $\text{Therefore, the time period of the given pendulum is:}$ $T = 2\pi \sqrt{\frac{L}{g \cos \alpha}}$ $\text{Hence, option (1) is the correct answer.}$

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
}