Import Question JSON

Current Question (ID: 10187)

Question:

$\text{A billiard ball of mass m and radius r, when hit in a horizontal direction by a cue at a height h above its centre, acquires a linear velocity } v_0 \text{. The angular velocity } \omega_0 \text{ acquired by the ball will be:}$

Options:

1. $\frac{5v_0r^2}{2h}$
2. $\frac{2v_0r^2}{5h}$
3. $\frac{2v_0h}{5r^2}$
4. $\frac{5v_0h}{2r^2}$

Solution:

\text{When the ball is hit by a cue, the linear impulse imparted to the ball = change in momentum = } mv_0 \text{As is clear from figure, the ball experiences both linear and angular motion.} \text{Angular momentum = Moment of momentum} I\omega_0 = (mv_0)h \text{For a solid sphere: } \frac{2}{5}mr^2\omega_0 = mv_0h \text{Therefore: } \omega_0 = \frac{5v_0h}{2r^2}

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
}