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Current Question (ID: 11281)

Question:

$\text{An ideal spring with spring-constant } K \text{ is hung from the ceiling and a block of mass } M \text{ is attached to its lower end. The mass is released with the spring initially un-stretched. Then the maximum extension in the spring will be:}$

Options:

1. $4Mg/K$
2. $2Mg/K$ (Correct)
3. $Mg/K$
4. $Mg/2K$

Solution:

$\text{Let } x \text{ be the maximum extension of the spring.}$ \\ $\text{We can use the principle of energy conservation.}$ \\ $\text{Loss in gravitational potential energy} = \text{Gain in potential energy of spring}$ \\ $\text{The gravitational potential energy lost is the weight of the block } (Mg) \text{ times the distance it falls } (x)\text{.}$ \\ $\text{Loss in G.P.E.} = Mgx$ \\ $\text{The potential energy stored in the spring is given by the formula } \frac{1}{2}Kx^2\text{, where } K \text{ is the spring constant and } x \text{ is the extension.}$ \\ $\text{Gain in P.E. of spring} = \frac{1}{2}Kx^2$ \\ $\text{Equating the two energies:}$ \\ $Mgx = \frac{1}{2}Kx^2$ \\ $\text{To solve for } x\text{, we can cancel } x \text{ from both sides (assuming } x \neq 0\text{):}$ \\ $Mg = \frac{1}{2}Kx$ \\ $\text{Rearranging the equation to find } x\text{:}$ \\ $x = \frac{2Mg}{K}$ \\ $\text{Therefore, the maximum extension in the spring is } \frac{2Mg}{K}\text{.}$

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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
}