Import Question JSON

Current Question (ID: 19915)

Question:

$\text{The motion of a simple pendulum executing simple harmonic motion is represented by the equation;}$ $y = A \sin(\pi t + \phi), \text{ where time is measured in seconds. The length of the pendulum is:}$

Options:

1. $97.23 \text{ cm}$
2. $25.3 \text{ cm}$
3. $99.4 \text{ cm}$
4. $406.1 \text{ cm}$

Solution:

$\text{Hint: } \omega = \sqrt{\frac{g}{l}}$ $\text{Step 1: Find the angular frequency.}$ $\text{The simple harmonic motion is represented by the equation;} \Rightarrow y = A \sin(\pi t + \phi)$ $\text{Compare the above equation with the standard equation } y = A \sin(\omega t + \phi), \text{ we get;} \text{The angular frequency is;} \omega = \pi$ $\text{Step 2: Find the length of the pendulum.}$ $\text{As, } \Rightarrow \omega = \pi = \sqrt{\frac{g}{l}}$ $\text{The length of the pendulum is given by;} \Rightarrow l = \frac{g}{\pi^2} \Rightarrow l = 99.4 \text{ cm}$ $\text{Hence, option (3) is the correct answer.}$

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
}