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Current Question (ID: 19416)

Question:

$\text{Two uniform circular discs are rotating independently in the same direction around their common axis passing through their centres.}$ $\text{The moment of inertia and angular velocity of the first disc are } 0.1 \text{ kg-m}^2 \text{ and } 10 \text{ rad s}^{-1} \text{ respectively}$ $\text{while those for the second one are } 0.2 \text{ kg-m}^2 \text{ and } 5 \text{ rad s}^{-1} \text{ respectively.}$ $\text{At some instant, they get stuck together and start rotating as a single system about their common axis with some angular speed.}$ $\text{The Kinetic energy of the combined system is:}$

Options:

1. $\frac{10}{3} \text{ J}$
2. $\frac{2}{3} \text{ J}$
3. $\frac{5}{3} \text{ J}$
4. $\frac{20}{3} \text{ J}$

Solution:

$\text{Apply the conservation of angular momentum.}$ $I_1 \omega_1 + I_2 \omega_2 = (I_1 + I_2) \omega$ $0.1 \times 10 + 0.2 \times 5 = (0.1 + 0.2) \omega$ $1 + 1 = 0.3 \omega$ $\omega = \frac{2}{0.3} = \frac{20}{3} \text{ rad s}^{-1}$ $\text{Kinetic energy } = \frac{1}{2} (I_1 + I_2) \omega^2 = \frac{1}{2} \times 0.3 \times \left(\frac{20}{3}\right)^2 = \frac{20}{3} \text{ J}$

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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
}