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Current Question (ID: 19262)

Question:

$\text{A block of mass } m \text{ (as shown in the figure) moving with kinetic energy } E \text{ compresses a spring through a distance } 25 \text{ cm when, its speed is halved. The value of the spring constant of the used spring is } nE \text{ Nm}^{-1}. \text{ The value of } n \text{ will be:}$

Options:

1. $32$
2. $24$
3. $20$
4. $16$

Solution:

$\text{Hint: Apply the conservation of energy.}$ $\text{Step 1: Find the value of the spring constant.}$ $\text{By using the conservation of mechanical energy i.e., loss in kinetic energy of block = gain in potential energy of spring}$ $\Rightarrow \frac{1}{2}mv^2 - \frac{1}{2}m\left(\frac{v}{2}\right)^2 = \frac{1}{2}kx^2$ $\Rightarrow \frac{3mv^2}{8} = \frac{1}{2}kx^2$ $\Rightarrow k = \frac{3mv^2}{4x^2}$ $\text{Step 2: Find the value of } n.$ $\text{Given: } k = nE$ $\text{By substituting the values we get;}$ $\frac{3}{4}\frac{mv^2}{x^2} = nE$ $\Rightarrow nE = \frac{3E}{2x^2} \quad \left[E = \frac{1}{2}mv^2\right]$ $\Rightarrow n = \frac{3}{2\times(25\times10^{-2})^2}$ $\Rightarrow n = \frac{3\times100\times100}{2\times25\times25}$ $\Rightarrow n = 24$ $\text{Hence, option (2) is the correct answer.}$

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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
}