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Current Question (ID: 11287)

Question:

$\text{In a spring pendulum, in place of mass, a liquid is used. If liquid leaks out continuously, then the time period of the spring pendulum:}$

Options:

1. $\text{decreases continuously}$ (Correct)
2. $\text{increases continuously}$
3. $\text{first increases and then decreases}$
4. $\text{first decreases and then increases}$

Solution:

$\text{The time period of a spring pendulum is given by the formula:}$ \\ $T = 2\pi \sqrt{\frac{m}{k}}$ \\ $\text{where } T \text{ is the time period, } m \text{ is the mass attached to the spring, and } k \text{ is the spring constant.}$ \\ $\text{From this formula, we can see that the time period } T \text{ is directly proportional to the square root of the mass } (\sqrt{m})\text{.}$ \\ $\text{In the given problem, the mass is a liquid that leaks out continuously. This means the total mass of the system } (m) \text{ is continuously decreasing.}$ \\ $\text{Since } T \propto \sqrt{m}\text{, as the mass } m \text{ decreases continuously, the time period } T \text{ must also decrease continuously.}$ \\ $\text{Therefore, the time period of the spring pendulum decreases continuously.}$

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
}