Import Question JSON

Current Question (ID: 19174)

Question:

$\text{A train moves along a circular track of radius } R \text{ with speed } v. \text{ The track has a width } w \,(w \ll R). \text{ To ensure safe motion without relying on friction, the outer rail is elevated above the inner rail. What should be the required elevation of the outer track?}$

Options:

1. $\frac{v^2 w}{R g}$
2. $\frac{v^2 w}{2 R g}$
3. $\frac{g w v^2}{R}$
4. $\frac{R}{g w v^2}$

Solution:

$\text{Hint: } \tan \theta = \frac{v^2}{r g}$ $\text{Step 1: Draw the diagram.}$ $\text{Step 2: Find the elevation of the outer track.}$ $N \sin \theta = \frac{m v^2}{r} \quad \ldots (1)$ $N \cos \theta = m g \quad \ldots (2)$ $\text{By using above two equations, we get:}$ $\Rightarrow \tan \theta = \frac{v^2}{r g}$ $\Rightarrow \frac{h}{w} = \frac{v^2}{r g} \left[ \therefore \tan \theta = \frac{h}{w} \right]$ $\Rightarrow h = \frac{v^2 w}{r g}$ $\text{Hence, option (1) is the correct answer.}$

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
}