Import Question JSON

Current Question (ID: 11295)

Question:

$\text{A spring has a spring constant } k \text{. It is cut into two parts } A \text{ and } B \text{ whose lengths are in the ratio of } m : 1 \text{. The spring constant of the part } A \text{ will be:}$

Options:

1. $\frac{k}{m}$
2. $\frac{k}{m+1}$
3. $k$
4. $\frac{k(m+1)}{m}$ (Correct)

Solution:

$\text{Hint: For a spring, } kx \text{ is a constant. The spring constant } k \text{ is inversely proportional to the length of the spring. This is because: } k_{\text{new}} = k \cdot \frac{L_{\text{original}}}{L_{\text{new}}}. \text{If the original spring is divided into two parts in the ratio } m : 1 \text{, then: Length of part } A = \frac{m}{m+1} L_{\text{original}} \text{ and Length of part } B = \frac{1}{m+1} L_{\text{original}}. \text{Let } k_A \text{ be the spring constant of part } A \text{. From the inverse proportionality:} k_A = k \cdot \frac{L_{\text{original}}}{L_A}. \text{Substitute } L_A = \frac{m}{m+1} L_{\text{original}}: k_A = k \cdot \frac{L_{\text{original}}}{\frac{m}{m+1}L_{\text{original}}} \Rightarrow k_A = k \cdot \frac{m+1}{m}. \text{Hence, option (4) is the correct answer.}$

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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
}