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Current Question (ID: 10238)

Question:

$\text{Four particles of mass } m_1 = 2m, m_2 = 4m, m_3 = m, \text{ and } m_4 \text{ are placed at the four corners of a square. What should be the value of } m_4 \text{ so that the centre of mass of all the four particles is exactly at the centre of the square?}$

Options:

1. $2m$
2. $8m$
3. $6m$
4. $\text{None of these}$

Solution:

$\text{Setting up coordinates with the center of the square at the origin:}$ $\text{Let the side length be } 2a, \text{ so the particles are at coordinates:}$ $m_1 = 2m \text{ at } (-a, -a)$ $m_2 = 4m \text{ at } (a, -a)$ $m_3 = m \text{ at } (a, a)$ $m_4 \text{ at } (-a, a)$ $\text{For the center of mass to be at the origin:}$ $X_{CM} = 0$ $\frac{m_1 x_1 + m_2 x_2 + m_3 x_3 + m_4 x_4}{m_1 + m_2 + m_3 + m_4} = 0$ $(2m)(-a) + (4m)(a) + (m)(a) + m_4(-a) = 0$ $-2ma + 4ma + ma - m_4 a = 0$ $m_4 = 3m$ $\text{Similarly for } Y_{CM} = 0:$ $(2m)(-a) + (4m)(-a) + (m)(a) + m_4(a) = 0$ $-2ma - 4ma + ma + m_4 a = 0$ $m_4 = 5m$ $\text{Since the values of } m_4 \text{ are different to satisfy both } X_{CM} = 0 \text{ and } Y_{CM} = 0, \text{ it is not possible.}$ $\text{Hence, the answer is None of these.}$

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
}