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Current Question (ID: 15983)

Question:

$\text{The equation of a stationary wave is given as } y = A \sin(0.5\pi t) \cos(0.2\pi x) \text{ where } t \text{ is in seconds and } x \text{ in centimetres. Which of the following is correct?}$

Options:

1. $\text{Wavelength of the component waves is } 10 \text{ cm.}$
2. $\text{The separation between a node and the nearest antinode is } 2.5 \text{ cm.}$
3. $\text{Frequency of the component wave is } 0.25 \text{ Hz.}$
4. $\text{All of these}$

Solution:

$\text{Hint: Compare the given equation with the standard equation of stationary wave.}$ $\text{Step 1: Find the wavelength of a stationary wave.}$ $\text{Use the standard equation of the stationary wave.}$ $y = A \sin \omega t \cos kx$ $\text{The given equation in the question is;}$ $y = A \sin(0.5\pi t) \cos(0.2\pi x)$ $\text{On comparing two equations we get;}$ $k = \frac{2\pi}{\lambda} = 0.2\pi \text{ cm}^{-1}$ $\lambda = 10 \text{ cm}$ $\text{Therefore, the separation is given by;}$ $\frac{\lambda}{4} = 2.5 \text{ cm}$ $\text{Step 2: Find the frequency of the wave.}$ $\omega = 0.5\pi \Rightarrow 2\pi f = 0.5\pi \Rightarrow f = 0.25 \text{ Hz}$ $\text{Hence, option (4) is the correct answer.}$

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
}