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Current Question (ID: 19374)

Question:

$\text{Two bodies of masses } 1 \text{ kg and } 3 \text{ kg have position vectors } (\hat{i} + 2\hat{j} + \hat{k}) \text{ and } (-3\hat{i} - 2\hat{j} + \hat{k}) \text{ respectively. The magnitude of the position vector of the centre-of-mass of this system will be similar to the magnitude of the vector:}$

Options:

1. $\hat{i} - 2\hat{j} + \hat{k}$
2. $-3\hat{i} - 2\hat{j} + \hat{k}$
3. $-2\hat{i} + 2\hat{k}$
4. $-2\hat{i} - \hat{j} + 2\hat{k}$

Solution:

$\text{Hint: Use the formula of the centre of mass.}$ $\text{Step: Find the position vector of the center of mass.}$ $\text{The centre of mass is given by:}$ $X = \frac{m_1r_1 + m_2r_2}{m_1 + m_2}$ $x_{cm} = \frac{1 \times (\hat{i} + 2\hat{j} + \hat{k}) + 3 \times (-3\hat{i} - 2\hat{j} + \hat{k})}{1 + 3}$ $x_{cm} = \frac{\hat{i}(1 - 9) + \hat{j}(2 - 6) + \hat{k}(1 + 3)}{4}$ $|x_{cm}| = |-2\hat{i} - \hat{j} + \hat{k}| = \sqrt{4 + 1 + 1}$ $\Rightarrow |x_{cm}| = \sqrt{6}$ $\text{The magnitude of } |\hat{i} - 2\hat{j} + \hat{k}| = \sqrt{6}, \text{ similar to the magnitude of the position vector of centre-of-mass of this system.}$ $\text{Hence, option (1) is the correct answer.}$

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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
}