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Current Question (ID: 19928)

Question:

$\text{The equations of two simple harmonic motions is given by}$ $y_1 = 10 \sin \left( \omega t + \frac{\pi}{3} \right)$ $\text{and}$ $y_2 = 5[\sin(\omega t) + \sqrt{3} \cos(\omega t)].$ $\text{The amplitude of the resultant SHM is:}$ $1. \ 10 \ \text{m}$ $2. \ 20 \ \text{m}$ $3. \ 5 \ \text{m}$ $4. \ 15 \ \text{m}$

Options:

1. $10 \ \text{m}$
2. $20 \ \text{m}$
3. $5 \ \text{m}$
4. $15 \ \text{m}$

Solution:

$y_1 = 10 \sin \left( \omega t + \frac{\pi}{3} \right)$ $y_2 = 5 \left[ \sin(\omega t) + \sqrt{3} \cos(\omega t) \right] = 10 \sin \left( \omega t + \frac{\pi}{3} \right)$ $\text{Resultant of the SHM}$ $= 10 \sin \left( \omega t + \frac{\pi}{3} \right) + 10 \sin \left( \omega t + \frac{\pi}{3} \right)$ $y_{\text{resultant}} = y_1 + y_2 = 20 \sin \left( \omega t + \frac{\pi}{3} \right)$ $\text{Amplitude} = 20 \ \text{m}$

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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
}