Import Question JSON

Current Question (ID: 19362)

Question:

$\text{A solid spherical ball is rolling on a frictionless horizontal plane surface about its axis of symmetry. The ratio of the rotational kinetic energy of the ball to its total kinetic energy is:}$

Options:

1. $\frac{2}{5}$
2. $\frac{2}{7}$
3. $\frac{1}{5}$
4. $\frac{7}{10}$

Solution:

$\text{Hint: Total kinetic energy} = \frac{1}{2}mv^2 \left[ 1 + \frac{K^2}{R^2} \right]$ $\text{Step 1: Find the rotational kinetic energy and total kinetic energy of the spherical ball.}$ $\text{Rotational kinetic energy} = \frac{1}{2}mv^2 \beta$ $\text{Given that } \beta = \frac{I}{MR^2} = \frac{2}{5}$ $\text{Total kinetic energy} = \frac{1}{2}mv^2(1 + \beta)$ $\text{Step 2: Find the ratio of the rotational kinetic energy of the ball to its total kinetic energy.}$ $\frac{\text{Rotational K.E}}{\text{Total K.E}} = \frac{\frac{1}{2}mv^2 \beta}{\frac{1}{2}mv^2(1+\beta)} = \frac{\beta}{1+\beta} = \frac{\frac{2}{5}}{1+\frac{2}{5}} = \frac{2}{7}$

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
}