Import Question JSON

Current Question (ID: 10316)

Question:

$\text{A solid cylinder of mass } 50 \text{ kg and radius } 0.5 \text{ m is free to rotate about the horizontal axis. A massless string is wound around the cylinder with one end attached to it and the other end hanging freely. The tension in the string required to produce an angular acceleration of } 2 \text{ rev/s}^2 \text{ will be:}$

Options:

1. $25 \text{ N}$
2. $50 \text{ N}$
3. $78.5 \text{ N}$
4. $157 \text{ N}$

Solution:

\text{Hint: } \tau = I\alpha \text{Step: Find the tension in the string.} \text{Given: } M = 50 \text{ kg}, \, R = 0.5 \text{ m}, \, \alpha = 2 \text{ rev/s}^2 \text{Torque: } \tau = TR \text{Moment of inertia for solid cylinder: } I = \frac{1}{2}MR^2 \text{Angular acceleration of the cylinder: } \alpha = \frac{\tau}{I} = \frac{TR}{\frac{1}{2}MR^2} \text{Solving for tension: } T = \frac{MR\alpha}{2} = \frac{50 \times 0.5 \times 4\pi}{2} T = \frac{50 \times 0.5 \times 4\pi}{2} = 157 \text{ N} \text{Hence, option (4) 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
}