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Current Question (ID: 10109)

Question:

$\text{Three identical spheres, each of mass } M, \text{ are placed at the corners of a right-angle triangle with mutually perpendicular sides equal to 2 m (see figure). Taking the point of intersection of the two mutually perpendicular sides as the origin, find the position vector of the centre of mass.}$

Options:

1. $2(\hat{i} + \hat{j})$
2. $(\hat{i} + \hat{j})$
3. $\frac{2}{3}(\hat{i} + \hat{j})$
4. $\frac{4}{3}(\hat{i} + \hat{j})$

Solution:

$\text{Given:}$ $\text{Three identical spheres, each of mass } M.$ $\text{Coordinates: } (0, 0), (2, 0), \text{ and } (0, 2).$ $\text{The position vector of the center of mass } R_{CM} \text{ is given by:}$ $R_{CM} = \frac{\sum m_i r_i}{\sum m_i}$ $R_{CM} = \frac{M(0\hat{i} + 0\hat{j}) + M(2\hat{i} + 0\hat{j}) + M(0\hat{i} + 2\hat{j})}{M + M + M}$ $R_{CM} = \frac{0 + 2M\hat{i} + 2M\hat{j}}{3M}$ $R_{CM} = \frac{2M(\hat{i} + \hat{j})}{3M}$ $R_{CM} = \frac{2}{3}(\hat{i} + \hat{j})$ $\text{The correct option is 3.}$

Import JSON File

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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
}