Import Question JSON

Current Question (ID: 11260)

Question:

$\text{In a simple pendulum, the period of oscillation } T \text{ is related to the length of the pendulum } L \text{ as:}$

Options:

1. $\frac{L}{T} = \text{constant}$
2. $\frac{L^2}{T} = \text{constant}$
3. $\frac{L}{T^2} = \text{constant}$ (Correct)
4. $\frac{L^2}{T^2} = \text{constant}$

Solution:

$\text{Hint: } T = 2\pi \sqrt{\frac{L}{g}}$ $\text{Step: Find the relationship between } L \text{ and } T \text{ in a simple pendulum.}$ $\text{The time period } T \text{ of a simple pendulum is related to the length } L \text{ by the following formula:}$ $T = 2\pi \sqrt{\frac{L}{g}} \ldots (1)$ $\text{Square both sides of equation (1) we get:}$ $T^2 = 4\pi^2 \frac{L}{g}$ $\Rightarrow \frac{L}{T^2} = \frac{g}{4\pi^2} = \text{constant}$ $\text{Therefore, the correct relationship between } L \text{ and } T \text{ is: } \frac{L}{T^2} = \text{constant}$ $\text{Hence, option (3) is the correct answer.}$

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
}