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Current Question (ID: 19917)

Question:

$\text{As per the given figures, two springs of spring constants } k \text{ and } 2k \text{ are connected to mass } m. \text{ If the period of oscillation in figure } (a) \text{ is } 3 \text{ s, then the period of oscillation in figure } (b) \text{ is } \sqrt{x} \text{ s. The value of } x \text{ is:}$

Options:

1. $3$
2. $4$
3. $2$
4. $1$

Solution:

$\text{Hint: } T = 2\pi \sqrt{\frac{m}{k}}.$ $\text{Step: Find the time period of the oscillation.}$ $\text{For Figure (a):}$ $\text{The effective spring constant is given by:}$ $\Rightarrow K_{eq} = \frac{K \times 2K}{K + 2K} = \frac{2K}{3}$ $\text{The time period of the oscillation is given by:}$ $T = 2\pi \sqrt{\frac{m}{K_{eq}}}$ $\Rightarrow T = 2\pi \sqrt{\frac{m}{2K/3}} = 2\pi \sqrt{\frac{3m}{2K}}$ $\text{For Figure (b):}$ $\text{The effective spring constant is given by:}$ $K_{eq} = K_1 + K_2 = 3K,$ $\text{The time period of the oscillation is given by:}$ $T = 2\pi \sqrt{\frac{m}{K_{eq}}}$ $\Rightarrow T' = 2\pi \sqrt{\frac{m}{3K}}$ $\frac{T'}{T} = \sqrt{\frac{m \times 2K}{3K \times 3m}} = \frac{\sqrt{2}}{3}$ $\Rightarrow T' = \sqrt{2} \text{ s}$ $\Rightarrow x = 2$ $\text{Hence, option (3) is the correct answer.}$

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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
}