Import Question JSON

Current Question (ID: 19922)

Question:

$\text{A particle performing simple harmonic motion with amplitude, } A \text{ starts from } x = 0 \text{ and reaches } x = A/2 \text{ in } 2 \text{ s.}$ $\text{The time required for the particle to go from } x = A/2 \text{ to } x = A \text{ is:}$

Options:

1. $1.5 \text{ s}$
2. $4 \text{ s}$
3. $6 \text{ s}$
4. $1 \text{ s}$

Solution:

$\text{Hint: Equation of SHM be: } x = A \sin \left( \frac{2\pi}{T} t \right)$ $\text{Step 1: Find the time to go } x = 0 \text{ to } x = A/2.$ $x = A \sin \left( \frac{2\pi}{T} t \right)$ $\text{Time to go from } x = 0 \text{ to } x = A/2$ $t_1 = \frac{T}{12}$ $\text{Step 2: Find the time to go } x = A/2 \text{ to } x = A.$ $x = A \sin \left( \frac{2\pi}{T} t \right)$ $\text{Time to go from } x = A/2 \text{ to } x = A.$ $t_2 = \frac{T}{4} - \frac{T}{12} = \frac{T}{6}$ $\Rightarrow \frac{t_2}{t_1} = 2 \Rightarrow t_2 = 2 \times 2 \text{ s} = 4 \text{ s}$ $\text{Hence, option (2) is the correct answer.}$

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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
}