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Current Question (ID: 10179)

Question:

$\text{A force } \vec{F} = \alpha\hat{i} + 3\hat{j} + 6\hat{k} \text{ is acting at a point } \vec{r} = 2\hat{i} - 6\hat{j} - 12\hat{k}\text{. The value of } \alpha \text{ for which angular momentum is conserved about the origin is:}$

Options:

1. $-1$
2. $2$
3. $\text{zero}$
4. $1$

Solution:

\text{Key Concept: When the resultant external torque acting on a system is zero, the total angular momentum of a system remains constant. This is the principle of the conservation of angular momentum.} \text{Given, force } \vec{F} = \alpha\hat{i} + 3\hat{j} + 6\hat{k} \text{ is acting at a point } \vec{r} = 2\hat{i} - 6\hat{j} - 12\hat{k} \text{As, angular momentum about origin is conserved i.e. } \tau = 0 \Rightarrow \text{Torque, } \tau = 0 \Rightarrow \vec{r} \times \vec{F} = 0 \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 2 & -6 & -12 \\ \alpha & 3 & 6 \end{vmatrix} = 0 \Rightarrow (-36 + 36)\hat{i} - (12 + 12\alpha)\hat{j} + (6 + 6\alpha)\hat{k} = 0 \Rightarrow 0\hat{i} - 12(1 + \alpha)\hat{j} + 6(1 + \alpha)\hat{k} = 0 \Rightarrow 6(1 + \alpha) = 0 \Rightarrow \alpha = -1 \text{So, value of } \alpha \text{ for angular momentum about origin is conserved, } \alpha = -1

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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
}