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Current Question (ID: 19914)

Question:

$\text{A body is performing simple harmonic motion with an amplitude of } 10 \text{ cm. The velocity of the body was tripled by air jet when it is at } 5 \text{ cm from its mean position. The new amplitude of vibration is } \sqrt{x} \text{ cm. The value of } x \text{ is:}$

Options:

1. $500$
2. $600$
3. $700$
4. $800$

Solution:

$\text{Hint: Use conservation of energy.}$ $\text{Step 1: Find the velocity at } x = 5 \text{ cm.}$ $\text{By using energy conservation we get;} \Rightarrow \frac{1}{2} k (10)^2 = \frac{1}{2} k (5)^2 + \frac{1}{2} mv^2$ $\Rightarrow v = \sqrt{\frac{75k}{m}}$ $\text{Step 2: Find the new amplitude of vibration.}$ $\text{Now, velocity is tripled through external mean so the amplitude of SHM will change and so the total energy, (but potential) energy at this moment will remain same)}$ $\Rightarrow \frac{1}{2} k (5)^2 + \frac{1}{2} m \left(3 \sqrt{\frac{75k}{m}}\right)^2 = \frac{1}{2} k A^2$ $\Rightarrow 25k + 675k = kA^2$ $\Rightarrow A = \sqrt{700}$ $\Rightarrow \sqrt{x} = \sqrt{700} \Rightarrow x = 700$ $\text{Hence, option (3) is the correct answer.}$

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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
}