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Current Question (ID: 15945)

Question:

$\text{Two progressive waves are represented by, } y_1 = 5 \sin(200t - 3.14x) \text{ and } y_2 = 10 \sin\left(200t - 3.14x + \frac{\pi}{3}\right) \text{ (} x \text{ is in metres, and } t \text{ is in seconds). Path difference between the two waves is:}$

Options:

1. $\frac{100}{\pi} \text{ m}$
2. $\frac{1}{3} \text{ m}$
3. $3.14 \times \frac{\pi}{3} \text{ m}$
4. $\frac{\pi^2}{9} \text{ m}$

Solution:

$\text{Hint: Path difference depends on the phase difference.}$ $\text{Step 1: Write the equation for path difference.}$ $\Delta \phi = \frac{2\pi}{\lambda} \left(\Delta x\right)$ $\Delta x = \frac{\lambda}{2\pi} \left(\Delta \phi\right)$ $\text{Step 2: Find the wavelength.}$ $k = 3.14 = \frac{2\pi}{\lambda} \Rightarrow \lambda = 2 \text{ m}$ $\text{Step 3: Find the path difference.}$ $\Delta \phi = \frac{\pi}{3}$ $\Delta x = \frac{2}{2\pi} \times \frac{\pi}{3} = \frac{1}{3} \text{ m}$ $\text{Therefore, the path difference between the two waves is } \frac{1}{3} \text{ m.}$ $\text{Hence, option (2) is the correct answer.}$

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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
}