Import Question JSON

Current Question (ID: 11242)

Question:

$\text{Kinetic energy of a particle executing simple harmonic motion in straight line is } pv^2 \text{ and potential energy is } qx^2, \text{ where } v \text{ is speed at distance } x \text{ from the mean position. The time period of the SHM is given by the expression:}$

Options:

1. $2\pi\sqrt{\frac{q}{p}}$
2. $2\pi\sqrt{\frac{p}{q}}$ (Correct)
3. $2\pi\sqrt{\frac{q}{p+q}}$
4. $2\pi\sqrt{\frac{p}{p+q}}$

Solution:

$\text{Hint: Recall the general equation for KE and PE}$ $\text{Step 1: Find the angular frequency of the SHM.}$ $K = \frac{1}{2}mv^2 = pv^2 \Rightarrow p = \frac{m}{2}$ $U = \frac{1}{2}m\omega^2x^2 = qx^2 \Rightarrow q = \frac{m\omega^2}{2}$ $\Rightarrow \frac{q}{p} = \omega^2$ $\Rightarrow \omega = \sqrt{\frac{q}{p}}$ $\text{Step 2: Find time period in the SHM.}$ $T = \frac{2\pi}{\omega} = 2\pi\sqrt{\frac{p}{q}}$ $\text{Hence, option (2) is the correct answer.}$

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
}