Import Question JSON

Current Question (ID: 15946)

Question:

$\text{Two SHMs have equations:}$ $x_1 = a \sin(\omega t + \phi_1)$ $\text{and}$ $x_2 = a \sin(\omega t + \phi_2).$ $\text{If the amplitude of the resultant SHM is equal to amplitude of superimposing SHM(s), the phase difference between them is:}$

Options:

1. $\frac{\pi}{6}$
2. $\frac{2\pi}{3}$
3. $\frac{\pi}{3}$
4. $\frac{4\pi}{3}$

Solution:

$\text{Hint: Treat amplitude as vectors.}$ $\text{Step 1: Draw the diagram.}$ $\text{Step 2: Find the phase difference.}$ $R = \sqrt{A^2 + B^2 + 2AB \cos \theta}$ $A^2 = A^2 + A^2 + 2A^2 \cos \theta$ $\cos \theta = \frac{-1}{2}$ $\theta = \frac{2\pi}{3}$ $\text{Option (2) is correct.}$

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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
}