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Current Question (ID: 15954)

Question:

$\text{The phase difference between two waves, represented by}$ $y_1 = 10^{-6} \sin \left\{ 100t + \left( \frac{x}{50} \right) + 0.5 \right\} \text{ m}$ $y_2 = 10^{-6} \cos \left\{ 100t + \left( \frac{x}{50} \right) \right\} \text{ m}$ $\text{where } x \text{ is expressed in metres and } t \text{ is expressed in seconds, is approximate:}$

Options:

1. $2.07 \text{ radians}$
2. $0.5 \text{ radians}$
3. $1.5 \text{ radians}$
4. $1.07 \text{ radians}$

Solution:

$\text{Hint: } y = a \sin(\omega t - kx)$ $\text{Step: Find the phase difference between two waves.}$ $\text{The given two waves are:}$ $y_1 = 10^{-6} \sin \left\{ 100t + \frac{x}{50} + 0.5 \right\}$ $y_2 = 10^{-6} \cos \left\{ 100t + \frac{x}{50} \right\} = 10^{-6} \sin \left\{ \frac{\pi}{2} + 100t + \frac{x}{50} \right\}$ $\text{Phase difference between}$ $y_1 \text{ and } y_2 = \frac{\pi}{2} - 0.5 = 1.57 - 0.5 = 1.07 \text{ radians.}$ $\text{Therefore, the phase difference between the two waves is } 1.07 \text{ radians.}$ $\text{Hence, option (4) is the correct answer.}$

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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
}