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Current Question (ID: 15938)

Question:

$\text{If a wave is travelling in a positive } x\text{-direction with } A = 0.2 \text{ m,}$ $v = 360 \text{ m/s, and } \lambda = 60 \text{ m, then the correct expression for the wave will be:}$

Options:

1. $y = 0.2 \sin \left[ 2\pi \left( 6t + \frac{x}{60} \right) \right]$
2. $y = 0.2 \sin \left[ \pi \left( 6t + \frac{x}{60} \right) \right]$
3. $y = 0.2 \sin \left[ 2\pi \left( 6t - \frac{x}{60} \right) \right]$
4. $y = 0.2 \sin \left[ \pi \left( 6t - \frac{x}{60} \right) \right]$

Solution:

$\text{Hint: } y = a \sin(\omega t - kx)$ $\text{Step: Find the correct expression for the wave.}$ $\text{Given: } A = 0.2 \text{ m, } v = 360 \text{ m/s, } \lambda = 60 \text{ m}$ $\text{The wave vector is given by: } k = \frac{2\pi}{\lambda} = \frac{2\pi}{60}$ $\text{The velocity of the wave is given as:}$ $v = \frac{\omega}{k} \Rightarrow \omega = vk = 360 \times \frac{2\pi}{60} = 12\pi \text{ rad/s}$ $\text{So, the equation of the wave moving along the positive } x\text{-axis is given by:}$ $y = 0.2 \left[ 2\pi \left( 6t - \frac{x}{60} \right) \right]$ $\text{Hence, option (3) is the correct answer.}$

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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
}