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Current Question (ID: 15985)

Question:

$\text{A standing wave is represented by } y = A \sin(100t) \cos(0.01x) \text{ where } y \text{ and } A \text{ are in millimetres, } t \text{ is in seconds and } x \text{ is in metres. The velocity of the wave is:}$

Options:

1. $10^4 \text{ m/s}$
2. $1 \text{ m/s}$
3. $10^{-4} \text{ m/s}$
4. $\text{Not derivable from the above data}$

Solution:

$\text{Hint: Compare with the standard wave equation.}$ $\text{Step 1: Find the angular frequency and the wave number.}$ $\text{By comparing the general equation of standing wave } y = 2A \sin(2\omega t) \cos(2kx) \text{ with the given equation } y = A \sin(100t) \cos(0.01x) \text{ we get;}$ $2\omega = 100, \ 2k = 0.01$ $\text{Step 2: Find the wave speed.}$ $\text{The speed of the wave is given by;}$ $v = \frac{\omega}{k} = 10^4 \text{ m/s.}$ $\text{Hence, option (1) is the correct answer.}$

Import JSON File

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
}