Import Question JSON

Current Question (ID: 9892)

Question:

$\text{A chain of length L and mass m is placed upon a smooth surface. The length of BA is (L-b). What will be the velocity of the chain when its end A reaches B?}$

Options:

1. $\sqrt{\frac{2g\sin\theta}{L}(L^2 - b^2)}$
2. $\sqrt{\frac{g\sin\theta}{2L}(L^2 - b^2)}$
3. $\sqrt{\frac{g\sin\theta}{L}(L^2 - b^2)}$
4. $\text{None of these}$

Solution:

$\text{Hint: For extended objects, centre of mass is used for potential energy calculation}$ $\text{Step 1:}$ $\text{Assume reference for P.E.}$ $\text{Step 2:}$ $\text{Find P.E.}_i \text{ & P.E.}_f$ $m_i = \lambda b$ $\text{P.E.}_i = -(\lambda b)g\frac{b}{2}\sin\theta$ $\text{P.E.}_f = -(\lambda L)g\frac{L}{2}\sin\theta$ $\text{Step 3:}$ $\text{Conserving mechanical energy:}$ $\text{P.E.}_i = \text{P.E.}_f + \text{K.E.}$ $\frac{-\lambda gb^2}{2}\sin\theta = \frac{-\lambda gL^2}{2}\sin\theta + \frac{(\lambda L)}{2}v^2$ $v = \sqrt{\frac{g(L^2 - b^2)}{L}\sin\theta}$

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
}