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Current Question (ID: 10129)

Question:

\text{A wheel is subjected to uniform angular acceleration about its axis. Initially, its angular velocity is zero. In the first 2 s, it rotates through an angle } \theta_1\text{. In the next 2 s, it rotates through an additional angle } \theta_2\text{.} \text{The ratio of } \frac{\theta_2}{\theta_1} \text{ is:}

Options:

1. $1$
2. $2$
3. $3$
4. $4$

Solution:

\text{Using relation, } \theta = \omega_0 t + \frac{1}{2}\alpha t^2 \theta_1 = \frac{1}{2}(\alpha)(2)^2 = 2\alpha \text{ ...(i)} \text{(As } \omega_0 = 0\text{, } t = 2 \text{ sec)} \text{Now using same equation for } t = 4 \text{ sec, } \omega_0 = 0 \theta_1 + \theta_2 = \frac{1}{2}\alpha(4)^2 = 8\alpha \text{ ...(ii)} \text{From (i) and (ii), } \theta_1 = 2\alpha \text{ and } \theta_2 = 6\alpha \therefore \frac{\theta_2}{\theta_1} = 3

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
}