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Current Question (ID: 19911)

Question:

$\text{If two identical springs, each with a spring constant } k, \text{ are connected in series, the new spring constant and time period will change by a factor of:}$

Options:

1. $\frac{1}{2}, \sqrt{2}$
2. $\frac{1}{4}, \sqrt{2}$
3. $\frac{1}{4}, 2\sqrt{2}$
4. $\frac{1}{2}, 2\sqrt{2}$

Solution:

$\text{Hint: } T = 2\pi \sqrt{\frac{m}{k}}$ $\text{Step 1: Find the new spring constant.}$ $\frac{1}{k'} = \frac{1}{k_1} + \frac{1}{k_2}$ $\Rightarrow \frac{1}{k'} = \frac{1}{k} + \frac{1}{k}$ $\Rightarrow \frac{1}{k'} = \frac{2}{k}$ $\Rightarrow k' = \frac{k}{2}$ $\text{Therefore, the new spring constant is changed by a factor of } \frac{1}{2}.$ $\text{Step 2: Find the new time period.}$ $T \propto \frac{1}{\sqrt{k}}$ $\Rightarrow \frac{T'}{T} = \sqrt{\frac{k}{k'}}$ $\Rightarrow \frac{T'}{T} = \sqrt{\frac{k}{k/2}}$ $\Rightarrow \frac{T'}{T} = \sqrt{2}$ $\text{Therefore, the new time period changed by a factor of } \sqrt{2}.$ $\text{Hence, option (1) is the correct answer.}$

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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
}