Import Question JSON

Current Question (ID: 10211)

Question:

$\text{Given below are two statements:}$ $\text{Assertion (A): If there is no external torque on a body about its centre of mass, then the velocity of the centre of mass remains constant.}$ $\text{Reason (R): The linear momentum of an isolated system remains constant.}$

Options:

1. $\text{Both (A) and (R) are True and (R) is the correct explanation of (A).}$
2. $\text{Both (A) and (R) are True but (R) is not the correct explanation of (A).}$
3. $\text{(A) is True but (R) is False.}$
4. $\text{(A) is False but (R) is True.}$

Solution:

$\text{Assertion (A) is False: If there is no external torque on a body about its center of mass, it means its angular momentum about the center of mass is conserved. This does not necessarily mean the velocity of the center of mass remains constant.}$ $\text{For the velocity of the center of mass to remain constant, the net external force on the body must be zero (by Newton's first law or the impulse-momentum theorem, } F_{net} = \frac{dp}{dt} \text{).}$ $\text{A body can have external forces acting on it that produce no net torque about its center of mass (e.g., forces acting through the center of mass), and these forces would change the velocity of the center of mass.}$ $\text{Reason (R) is True: The linear momentum of an isolated system (a system with no external forces acting on it) remains constant. This is a fundamental principle of physics, the law of conservation of linear momentum.}$ $\text{Therefore, the correct answer is option 4: (A) is False but (R) is True.}$

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
}