Import Question JSON

Current Question (ID: 9648)

Question:

$\text{Two particles } A \text{ and } B \text{ are moving in a uniform circular motion in concentric circles of radii } r_A \text{ and } r_B \text{ with speeds } v_A \text{ and } v_B \text{ respectively. Their time periods of rotation are the same. The ratio of the angular speed of } A \text{ to that of } B \text{ will be:}$

Options:

1. $1 : 1$
2. $r_A : r_B$
3. $v_A : v_B$
4. $r_B : r_A$

Solution:

$\text{Hint: The angular speed depends on the time period.}$ $\text{Step 1: Find the relation between their time periods.}$ $T_A = T_B$ $\text{Step 2: Find the ratio of angular speed of the of } A \text{ to } B.$ $\Rightarrow \frac{2\pi}{\omega_A} = \frac{2\pi}{\omega_B}$ $\Rightarrow \omega_A = \omega_B$ $\text{Therefore, the ratio of angular speed of the of } A \text{ to } B \text{ is } 1 : 1.$ $\text{Hence, option (1) is the correct answer.}$

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
}