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Current Question (ID: 19333)

Question:

$\text{A circular disc of radius } b \text{ has a hole of radius } a \text{ at its centre (see figure).}$ $\text{If the mass per unit area of the disc varies as } \left( \frac{\sigma_0}{r} \right), \text{ then the radius of gyration of the disc about its axis passing through the centre is:}$

Options:

1. $\frac{a+b}{3}$
2. $\sqrt{\frac{a^2+b^2+ab}{3}}$
3. $\sqrt{\frac{a^2+b^2+ab}{2}}$
4. $\frac{a+b}{2}$

Solution:

$\text{Hint: } I = mK^2$ $\text{dm} = \left( \frac{\sigma_0}{r} \right) (2\pi r \text{d}r) = 2\pi \sigma_0 \text{d}r$ $I = mk^2 : k = \text{radius of gyration}$ $\int_a^b (\text{dm}) r^2 = k^2 \int_a^b 2\pi \sigma_0 \text{d}r$ $= \int_a^b (2\pi \sigma_0 \text{d}r) r^2 = k^2 \int_a^b 2\pi \sigma_0 \text{d}r$ $= 2\pi \sigma_0 \left[ \frac{b^3}{3} - \frac{a^3}{3} \right] = k^2 2\pi \sigma_0 (b-a)$ $= (b-a) \left( \frac{b^2 + a^2 + ab}{3} \right) = k^2 \left( b-a \right)$ $k = \sqrt{\frac{a^2 + b^2 + ab}{3}}$

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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
}