Import Question JSON

Current Question (ID: 19134)

Question:

$\text{A uniform rod of length } l \text{ is being rotated in a horizontal plane with a constant angular speed about an axis passing through one of its ends.}$ $\text{If the tension generated in the rod due to rotation is } T(x) \text{ at a distance } x \text{ from the axis, then which of the following graphs depicts it most closely?}$

Options:

1. $\text{Graph 1}$
2. $\text{Graph 2}$
3. $\text{Graph 3}$
4. $\text{Graph 4}$

Solution:

$\text{Hint: Tension decreases from axis to other end.}$ $T(x) = \int_x^l (dm) \omega^2 L$ $= \int_x^l \left[ \frac{m}{l} dL \right] \omega^2 L$ $= \frac{m}{l} \omega^2 \left[ \frac{L^2}{2} \right]_x^l$ $T(x) = A - Bx^2$ $= \frac{m \omega^2}{2l} \left( l^2 - x^2 \right)$ $\text{(Parabola, mouth down)}$

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
}