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Current Question (ID: 10159)

Question:

\text{Which of the following is the value of the torque of force } \vec{F} \text{ about origin } O\text{?} \text{Given: } \vec{F} = (5\sqrt{3} \hat{i} + 5\hat{j}) \text{ N and position vector from origin to point of application is } \vec{r} = (\hat{i} + \hat{j})

Options:

1. $\vec{\tau} = 5(1 - \sqrt{3})\hat{k} \text{ N-m}$
2. $\vec{\tau} = 5(1 - \sqrt{3})\hat{j} \text{ N-m}$
3. $\vec{\tau} = 5(\sqrt{3} - 1)\hat{i} \text{ N-m}$
4. $\vec{\tau} = \sqrt{3}\hat{j} \text{ N-m}$

Solution:

\text{Hint: Torque depends on cross product of force and position vector about which torque is to be calculated.} \text{Step 1: Write the formula for torque} \vec{\tau} = \vec{r} \times \vec{F} \text{Step 2: Find position vector about O and } \vec{F} \vec{r} = \hat{i} + \hat{j} \vec{F} = 5\sqrt{3}\hat{i} + 5\hat{j} \text{Step 3: Calculate } \vec{\tau} \text{ using formula} \vec{\tau} = (\hat{i} + \hat{j}) \times (5\sqrt{3}\hat{i} + 5\hat{j}) = 5(1 - \sqrt{3})\hat{k} \text{ N-m}

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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
}