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Current Question (ID: 15950)

Question:

$\text{A transverse harmonic wave on a string is described by,}$ $y(x, t) = 3.0 \sin\left(36t + 0.018x + \frac{\pi}{4}\right)$ $\text{where } x \text{ and } y \text{ are in cm and } t \text{ in sec.}$ $\text{The positive direction of } x \text{ is from left to right.}$ $\text{What is the shortest distance between two successive crests in the wave?}$

Options:

1. $1.3 \text{ m}$
2. $3.0 \text{ m}$
3. $2.5 \text{ m}$
4. $3.5 \text{ m}$

Solution:

$\text{Hint: The distance between two consecutive crests is equal to the wavelength.}$ $\text{Step 1: Find the value of } k.$ $y(x, t) = 3 \sin\left(36t + 0.018x + \frac{\pi}{4}\right)$ $y = A\sin(\omega t + kx + \phi)$ $k = 0.018$ $\text{Step 2: Find the wavelength of the wave.}$ $\lambda = \frac{2\pi}{k} = \frac{2\pi}{0.018} \approx 348.9 \text{ cm} = 3.5 \text{ m}$ $\text{Therefore, the shortest distance between two successive crests in the wave is } 3.5 \text{ m.}$ $\text{Hence, option (4) is the correct answer.}$

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Upload a JSON file containing LaTeX/MathJax formatted question, options, and solution.

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
}