Import Question JSON

Current Question (ID: 19220)

Question:

$\text{When a rubber band is stretched by a distance } x, \text{ it exerts a restoring}$ $\text{force given by: } F = (ax + bx^2), \text{ where } a \text{ and } b \text{ are constants.}$ $\text{What is the total work done in stretching the rubber band from its natural}$ $\text{(unstretched) length to a length } L?$

Options:

1. $\frac{1}{2}(aL^2 + bL^3)$
2. $\left(\frac{aL^2}{2} + \frac{bL^3}{3}\right)$
3. $\frac{1}{2}\left(\frac{aL^2}{2} + \frac{bL^3}{3}\right)$
4. $(aL^2 + bL^3)$

Solution:

$\text{Hint: } W = \int F \cdot dx$ $\text{Step: Find the work done in stretching the unstretched rubber band.}$ $W = \int_0^L F \cdot dx$ $\Rightarrow W = \int_0^L (ax + bx^2)dx$ $\Rightarrow W = \left[\frac{ax^2}{2} + \frac{bx^3}{3}\right]_0^L$ $\Rightarrow W = \left[\frac{aL^2}{2} + \frac{bL^3}{3}\right]$ $\text{Hence, option (2) is the correct answer.}$

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
}