Import Question JSON

Current Question (ID: 19397)

Question:

$\text{What is the ratio of the radius of gyration of a solid sphere to that of a solid cylinder, given that both objects have the same mass and radius?}$

Options:

1. $\frac{2}{\sqrt{5}}$
2. $\frac{\sqrt{5}}{2}$
3. $\sqrt{\frac{2}{5}}$
4. $\frac{2}{\sqrt{3}}$

Solution:

$\text{Hint: For a solid sphere, } I = \frac{2}{5} mR^2; \text{ For a solid cylinder, } I = \frac{1}{2} mR^2$ $\text{Step 1: Find the radius of gyration of a solid sphere and a solid cylinder.}$ $\text{The radius of gyration is given by:}$ $\Rightarrow k = \sqrt{\frac{I}{m}}$ $\text{The radius of gyration of the solid sphere is given by:}$ $\Rightarrow k = \sqrt{\frac{2}{5}} R \quad \ldots (1)$ $\text{The radius of gyration of the solid cylinder is given by:}$ $\Rightarrow k' = \frac{R}{\sqrt{2}} \quad \ldots (2)$ $\text{Step 2: Find the ratio of the radius of gyration of a solid sphere and a solid cylinder.}$ $\text{From equations (1) and (2), we get:}$ $\Rightarrow \frac{k}{k'} = \frac{2}{\sqrt{5}}$ $\text{Hence, option (1) is the correct answer.}$

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
}