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Current Question (ID: 16020)

Question:

$\text{Two waves are propagating to the point } P \text{ along a straight line produced by two sources, } A \text{ and } B, \text{ of simple harmonic and equal frequency. The amplitude of every wave at } P \text{ is } a \text{ and the phase of } A \text{ is ahead by } \frac{\pi}{3} \text{ than that of } B, \text{ and the distance } AP \text{ is greater than } BP \text{ by } 50 \text{ cm. If the wavelength is } 1 \text{ m, then the resultant amplitude at point } P \text{ will be:}$

Options:

1. $2a$
2. $a\sqrt{3}$
3. $a\sqrt{2}$
4. $a$

Solution:

$\text{Hint: } \Delta \phi = \frac{2\pi}{\lambda} (\Delta x)$ $\text{Step: Find the resultant amplitude of the wave at the point } P.$ $\text{The path difference between the two waves is } 50 \text{ cm } = \frac{1}{2} \text{ m}$ $\text{The phase difference between the waves is given by:}$ $\Delta \phi = \frac{2\pi}{\lambda} \times \Delta x = \frac{2\pi}{1} \times \frac{1}{2} = \pi$ $\text{Total phase difference } = \pi - \frac{\pi}{3} = \frac{2\pi}{3}$ $\Rightarrow A = \sqrt{a^2 + a^2 + 2a^2 \cos(2\pi/3)} = a$ $\text{Therefore, the resultant amplitude of the waves at the point } P \text{ is } a.$ $\text{Hence, option (4) is the correct answer.}$

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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
}