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Current Question (ID: 15952)

Question:

$\text{Two waves are represented by the equations } y_1 = a \sin(\omega t + kx + 0.57) \text{ m}$ $\text{and } y_2 = a \cos(\omega t + kx) \text{ m, where } x \text{ is in meters and } t \text{ in seconds. The phase difference between them is:}$

Options:

1. $1.25 \text{ rad}$
2. $1.57 \text{ rad}$
3. $0.57 \text{ rad}$
4. $1.0 \text{ rad}$

Solution:

$\text{Hint: } y = a \sin(\omega t - kx + \phi)$ $\text{Step: Find the phase difference between the waves.}$ $\text{Given,}$ $\text{Equations of two waves as below,}$ $y_1 = a \sin(\omega t + kx + 0.57) \text{ m and } y_2 = a \cos(\omega t + kx) \text{ m,}$ $\text{or,}$ $y_2 = a \sin(\omega t + kx + \frac{\pi}{2}) \text{ m}$ $\text{on comparing with the standard equation } y = a \sin(\omega t - kx + \phi) \text{ we get,}$ $\phi_1 = 0.57 \text{ rad and } \phi_2 = \frac{\pi}{2} \text{ rad}$ $\text{The phase difference of the waves is given by;}$ $\Delta \phi = \phi_2 - \phi_1$ $\Rightarrow \Delta \phi = \frac{\pi}{2} - 0.57$ $\Rightarrow \Delta \phi = 1 \text{ rad}$ $\text{Therefore, the phase difference between the waves is 1 rad.}$ $\text{Hence, option (4) is the correct answer.}$

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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
}