Import Question JSON

Current Question (ID: 10104)

Question:

$\text{The centre of the mass of 3 particles, 10 kg, 20 kg, and 30 kg, is at }(0, 0, 0)\text{. Where should a particle with a mass of 40 kg be placed so that its combined centre of mass is }(3, 3, 3)\text{?}$

Options:

1. $(0, 0, 0)$
2. $(7.5, 7.5, 7.5)$
3. $(1, 2, 3)$
4. $(4, 4, 4)$

Solution:

$\text{Using the center of mass formula:}$ $\vec{r}_{cm} = \frac{m_1\vec{r}_1 + m_4\vec{r}_2}{m_1 + m_2}$ $\text{Where } m_1 = 60 \text{ kg (combined mass of first 3 particles at origin)}$ $\text{and } m_4 = 40 \text{ kg (mass of the fourth particle)}$ $\vec{r}_{cm} = \frac{60 \times (0\hat{i} + 0\hat{j} + 0\hat{k}) + 40\vec{r}_2}{100}$ $\text{Setting this equal to the desired center of mass }(3, 3, 3)\text{:}$ $\vec{r}_2 = \frac{5}{2}(3\hat{i} + 3\hat{j} + 3\hat{k}) = 7.5\hat{i} + 7.5\hat{j} + 7.5\hat{k}$ $\text{Therefore, the 40 kg particle should be placed at }(7.5, 7.5, 7.5)$

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
}