Import Question JSON

Current Question (ID: 15940)

Question:

$\text{Given the equation for a wave on the string, } y = 0.5 \sin(5x - 3t) \text{ where } y \text{ and } x \text{ are in metres and } t \text{ in seconds, the ratio of the maximum speed of particle to the speed of wave is:}$

Options:

1. $1 : 1$
2. $5 : 2$
3. $3 : 2$
4. $4 : 5$

Solution:

$\text{Hint: The maximum velocity of the particle } = A\omega.$ $\text{Step 1: Find the maximum particle speed.}$ $v_{\text{P, max}} = A\omega$ $= 0.5 \times 3$ $= 1.5 \text{ m/s}$ $\text{Step 2: Find the wave speed.}$ $v = \frac{\omega}{k} = \frac{3}{5} \text{ m/s}$ $\text{Step 3: Find the ratio of the maximum speed of the particle to the speed of the wave.}$ $\Rightarrow \frac{v_{\text{max}}}{v} = \frac{3 \times 5}{2 \times 3} = \frac{5}{2}$ $\text{Therefore, the ratio of the maximum speed of the particle to the speed of wave is } 5 : 2.$ $\text{Hence, option (2) is the correct answer.}$

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
}