Import Question JSON

Current Question (ID: 10275)

Question:

$\text{A rope of negligible mass is wound around a hollow cylinder of mass 3 kg and radius 40 cm. What is the angular acceleration of the cylinder if the rope is pulled with a force of 30 N? (Assume that there is no slipping.)}$

Options:

1. $21 \text{ rad/s}^2$
2. $24 \text{ rad/s}^2$
3. $20 \text{ rad/s}^2$
4. $25 \text{ rad/s}^2$

Solution:

$\text{Hint: } \tau = I\alpha$ $\text{Step 1: Find the moment of inertia of the given cylinder.}$ $\text{The moment of inertia of the hollow cylinder about its geometric axis is given by;}$ $I = mr^2$ $\Rightarrow I = 3 \times (0.4)^2 = 0.48 \text{ kg m}^2$ $\text{Step 2: Find the torque acting on the cylinder.}$ $\text{The torque acting on the cylinder is given by;}$ $\tau = rF$ $\Rightarrow \tau = 0.4 \times 30 = 12 \text{ N-m}$ $\text{Step 3: Find the angular acceleration of the cylinder.}$ $\text{By applying the relation between torque and angular acceleration;}$ $\tau = I\alpha$ $\Rightarrow \alpha = \frac{\tau}{I} = \frac{12}{0.48} = 25 \text{ rad s}^{-2}$ $\text{Hence, option (4) is the correct answer.}$

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
}