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Current Question (ID: 19386)

Question:

$\text{A solid sphere and a ring have equal masses and equal radii of gyration. If the sphere is rotating about its diameter and the ring about an axis passing through and perpendicular to its plane, then the ratio of the radii is } \sqrt{\frac{x}{2}}. \text{ The value of } x \text{ is:}$

Options:

1. $5$
2. $2$
3. $1$
4. $4$

Solution:

$\text{Hint: } mk_1^2 = \frac{2}{5}mR_1^2 \text{ and } mk_2^2 = mR_2^2$ $\text{Step: Find the value of } x.$ $\text{As the solid sphere and ring have equal masses and radius of gyration i.e., } I_1 = I_2 \text{ or } K_1 = K_2 \ldots (1).$ $\text{The moment of inertia of a solid sphere about its diameter is given by:}$ $\frac{2}{5}mR_1^2 = mK_1^2 \Rightarrow K_1 = \sqrt{\frac{2}{5}} R_1 \ldots (2)$ $\text{The moment of inertia of a ring about an axis passing through and perpendicular to its plane is given by:}$ $mR_2^2 = mK_2^2 \Rightarrow K_2 = R_2 \ldots (3)$ $\text{From the equations } (1), (2) \text{ and } (3) \text{ we get:}$ $\Rightarrow R_2 = \sqrt{\frac{2}{5}} R_1 \quad [K_1 = K_2]$ $\Rightarrow \frac{R_1}{R_2} = \sqrt{\frac{5}{2}} = \sqrt{\frac{x}{2}}$ $\text{Therefore, the value of } x \text{ is } 5.$ $\text{Hence, option } (1) \text{ is the correct answer.}$

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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
}