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Current Question (ID: 9880)

Question:

$\text{The figure shows the potential energy function U(x) for a system in which a particle is in a one-dimensional motion. What is the direction of the force when the particle is in region AB? (symbols have their usual meanings)}$

Options:

1. $\text{The positive direction of x}$
2. $\text{The negative direction of x}$
3. $\text{Force is zero, so direction not defined}$
4. $\text{The negative direction of y}$

Solution:

The relationship between force F(x) and potential energy U(x) in one dimension is given by: $$F(x) = -\frac{dU}{dx}$$ In region AB, the potential energy U(x) is decreasing as x increases. This means that the slope $\frac{dU}{dx}$ in region AB is negative. Since $F(x) = -\frac{dU}{dx}$ and $\frac{dU}{dx}$ is negative in region AB, then $F(x)$ will be positive. A positive force in one-dimensional motion indicates that the force is in the positive direction of x. Therefore, the correct answer is "1. The positive direction of x".

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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
}