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Current Question (ID: 11292)

Question:

$\text{All the surfaces are smooth and the system, given below, is oscillating with an amplitude } A. \text{ What is the extension of spring having spring constant } k_1, \text{ when the block is at the extreme position?}$

Options:

1. $\frac{k_1}{k_1 + k_2} A$
2. $\frac{k_2 A}{k_1 + k_2}$ (Correct)
3. $A$
4. $\frac{A}{2}$

Solution:

$\text{Hint: } x_1 + x_2 = A.\\ \text{Step 1: Find the extension of spring having spring constant } k_1.\\ \text{Let the extension in the springs be } x_1 \text{ and } x_2. \\ \text{The total extension is the sum of the individual extensions, which is equal to the amplitude, so } x_1 + x_2 = A. \\ \text{Also, the springs are in series, so the force exerted by each spring is equal. } F_1 = F_2 \Rightarrow k_1x_1 = k_2x_2. \\ \text{From this, we can express one extension in terms of the other: } \frac{x_1}{x_2} = \frac{k_2}{k_1} \Rightarrow x_2 = \frac{k_1}{k_2}x_1. \\ \text{Substitute this into the total extension equation: } x_1 + \frac{k_1x_1}{k_2} = A. \\ \text{Factor out } x_1: x_1(1 + \frac{k_1}{k_2}) = A \Rightarrow x_1(\frac{k_2 + k_1}{k_2}) = A. \\ \text{Solving for } x_1: x_1 = \frac{k_2 A}{k_1 + k_2}. \\ \text{Hence, option (2) is the correct answer.}$

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
}