Import Question JSON

Current Question (ID: 9657)

Question:

$\text{Certain neutron stars are believed to be rotating at about 1 rev/s. If such a star has a radius of 20 km, the acceleration of an object on the equator of the star will be:}$

Options:

1. $20 \times 10^8 \text{ m/s}^2$
2. $8 \times 10^5 \text{ m/s}^2$
3. $120 \times 10^5 \text{ m/s}^2$
4. $4 \times 10^8 \text{ m/s}^2$

Solution:

$\text{Hint: } a_c = \omega^2 r$ $\text{Step: Find the acceleration of an object on the equator of the star.}$ $\text{For the centripetal acceleration of an object on the equator of a rotating neutron star, we can use the following formula:}$ $a_c = \omega^2 r$ $\text{Since 1 rev } = 2\pi \text{ radians, an angular velocity of 1 rev/s corresponds to:}$ $\Rightarrow \omega = 1 \text{ rev/s} \times 2\pi \text{ rad/rev} = 2\pi \text{ rad/s}$ $\text{Substituting the known values we get;}$ $a_c = (2\pi)^2 \times (20 \times 10^3)$ $a_c = 4\pi^2 \times 20 \times 10^3$ $\text{Using } \pi^2 \approx 9.87:$ $a_c = 4 \times 9.87 \times 20 \times 10^3 = 789,600 \text{ m/s}^2$ $a_c \approx 8 \times 10^5 \text{ m/s}^2$ $\text{Hence, option (2) is the correct answer.}$

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
}