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Current Question (ID: 19273)

Question:

$\text{A particle is constrained to move along the } x\text{-axis under the action of a force:}$ $\vec{F} = (2 + 3x)\hat{i}.$ $\text{What is the work done by this force when the particle is displaced from } x = 0 \text{ to } x = 4 \text{ m?}$

Options:

1. $16 \text{ J}$
2. $32 \text{ J}$
3. $4 \text{ J}$
4. $8 \text{ J}$

Solution:

$\text{Hint: } W = \int \vec{F} \cdot \vec{dx}$ $\text{Step: Find the work done by a given force.}$ $\text{The work done by a variable force is given by:}$ $W = \int \vec{F} \cdot \vec{dx}$ $\Rightarrow W = \int_{0}^{4} (2 + 3x)\hat{i} \cdot (dx)\hat{i}$ $\Rightarrow W = \left[ 2x + \frac{3}{2} x^2 \right]_{0}^{4}$ $\Rightarrow W = 32 \text{ J}$ $\text{Therefore, work done by the force is } 32 \text{ J.}$ $\text{Hence, option (2) is the correct answer.}$

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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
}