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Current Question (ID: 15984)

Question:

$\text{The equation of vibration of a taut string, fixed at both ends, is given by:}$ $y = (3 \ \text{mm}) \cos\left(\frac{\pi x}{10 \ \text{cm}}\right) \sin(800\pi \ \text{s}^{-1} \ t)$ $\text{The speed of waves on the string is:}$

Options:

1. $20 \ \text{m/s}$
2. $40 \ \text{m/s}$
3. $80 \ \text{m/s}$
4. $160 \ \text{m/s}$

Solution:

$\text{Hint: } v = \frac{\omega}{k}$ $\text{Step: Find the speed of waves on the string.}$ $\text{The equation of vibration of a taut string is given as;}$ $y = (3 \ \text{mm}) \cos\left(\frac{\pi x}{10 \ \text{cm}}\right) \sin(800\pi \ \text{s}^{-1} \ t) \ldots (1)$ $\text{The general equation of the wave is given by;}$ $y = A \cos(kx) \sin(\omega t) \ldots (2)$ $\text{By comparing equations (1) and (2) we get;}$ $k = \frac{\pi}{10 \ \text{cm}}, \omega = 800\pi \ \text{s}^{-1}$ $\text{The speed of the wave on the string is given by;}$ $v = \frac{\omega}{k}$ $\Rightarrow v = \frac{800\pi \ \text{s}^{-1}}{\frac{\pi}{10 \ \text{cm}}} = 8000 \ \text{cm/s}$ $\Rightarrow v = 80 \ \text{m/s}$ $\text{Hence, option (3) is the correct answer.}$

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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
}