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Current Question (ID: 19919)

Question:

$\text{The metallic bob of a simple pendulum has a relative density equal to } 5. \text{ The time period of this pendulum is } 10 \text{ s. If the metallic bob is immersed in water, then the new time period becomes } 5\sqrt{x} \text{ s. The value of } x \text{ will be:}$ $1. \ 5$ $2. \ 7$ $3. \ 9$ $4. \ 2$

Options:

1. $5$
2. $7$
3. $9$
4. $2$

Solution:

$\text{Hint: } T = 2\pi \sqrt{\frac{l}{g}}$ $\text{Step 1: Find the effective acceleration due to gravity say } g' \text{ when a metallic bob is immersed in water.}$ $\Rightarrow g' = \frac{mg - F_B}{m}$ $\Rightarrow g' = \frac{mg - \frac{mg}{5}}{m}$ $\Rightarrow g' = \frac{4g}{5}$ $\text{Step 2: Find the new time period of the metallic bob.}$ $\Rightarrow T = 2\pi \sqrt{\frac{l}{g}} = 10$ $\Rightarrow T' = 2\pi \sqrt{\frac{5l}{4g}} = 10\sqrt{\frac{5}{4}}$ $\Rightarrow 5\sqrt{x} = 10\sqrt{\frac{5}{4}}$ $\Rightarrow x = 5$ $\text{Therefore, the value of } x \text{ in new time period is } 5. \text{ Hence, option (1) is the correct answer.}$

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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
}