Import Question JSON

Current Question (ID: 10121)

Question:

$\text{Given the following statements:}$ $\text{(a) The centre of gravity (C.G.) of a body is the point at which the weight of the body acts.}$ $\text{(b) If the earth is assumed to have an infinitely large radius, the centre of mass coincides with the centre of gravity.}$ $\text{(c) To evaluate the gravitational field intensity due to any body at an external point, the entire mass of the body can be considered to be concentrated at its C.G.}$ $\text{(d) The radius of gyration of any body rotating about an axis is the length of the perpendicular dropped from the C.G. of the body to the axis.}$ $\text{Which one of the following pairs of statements is correct?}$

Options:

1. $\text{(a) and (b)}$
2. $\text{(b) and (c)}$
3. $\text{(c) and (d)}$
4. $\text{(d) and (a)}$

Solution:

\text{Hint: } X_{\text{c.o.m}} = \frac{\sum_{i=1}^{n} m_i r_i}{\sum_{i=1}^{n} m_i} \text{Explanation: The centre of gravity (C.G.) is the point at which } \tau_{\text{net}} \text{ due to gravitation force should be zero.} \text{The centre of mass is the weighted mean of the position of the system of particles.} \text{The mathematical expression for the centre of mass is given by: } X_{\text{c.o.m}} = \frac{\sum_{i=1}^{n} m_i r_i}{\sum_{i=1}^{n} m_i} \text{If } g \text{ is uniform everywhere or } g \text{ is constant then } X_{\text{c.o.m}} = \text{Centre of gravity (C.G.)} \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
}