Import Question JSON

Current Question (ID: 19932)

Question:

$\text{The phasor of a particle performing SHM is as shown in the diagram.}$ $\text{The SHM has an angular frequency } \omega \text{ and at } t = 0, \text{ the phasor lies along } OP. \text{ At any time } t \text{ further, the projection of the phasor along the } y\text{-axis is given by:}$

Options:

1. $R \sin \left( \omega t + \frac{\pi}{6} \right)$
2. $R \cos \left( \omega t + \frac{\pi}{6} \right)$
3. $R \sin \left( \omega t - \frac{\pi}{6} \right)$
4. $R \cos \left( \omega t - \frac{\pi}{6} \right)$

Solution:

$\text{Hint: } y\text{-projection} = r \sin \theta$ $\text{Step: Find the projection of the phasor along the } y\text{-axis.}$ $\text{The general equation of the SHM is given by;}$ $y = A \sin(\omega t + \theta)$ $\text{The angle made by the particle with the } x\text{-axis at the time } t \text{ is given by;}$ $\theta = \omega t + \frac{\pi}{6}$ $\text{The projection of phasor along the } y\text{-axis at the time } t \text{ is given by;}$ $y = R \sin \left( \omega t + \frac{\pi}{6} \right)$ $\text{Hence, option (1) is the correct answer.}$

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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
}