Import Question JSON

Current Question (ID: 9882)

Question:

$\text{A particle of mass 'm' is moving in a horizontal circle of radius 'r' under a centripetal force equal to } -\frac{K}{r^2}, \text{ where K is a constant. The total energy of the particle will be:}$

Options:

1. $\frac{K}{2r}$
2. $-\frac{K}{2r}$
3. $-\frac{K}{r}$
4. $\frac{K}{r}$

Solution:

\text{Here } \frac{mv^2}{r} = \frac{K}{r^2} \text{Therefore, K.E.} = \frac{1}{2}mv^2 = \frac{K}{2r} U = -\int_{\infty}^{r} F \cdot dr = -\int_{\infty}^{r} \left(-\frac{K}{r^2}\right) dr = -\frac{K}{r} \text{Total energy } E = \text{K.E.} + \text{P.E.} = \frac{K}{2r} - \frac{K}{r} = -\frac{K}{2r}

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
}