Import Question JSON

Current Question (ID: 9668)

Question:

$\text{Figure below shows a body of mass } M \text{ moving with a uniform speed } v \text{ on a circular path of radius, } R\text{. What is the change in acceleration in going from } P_1 \text{ to } P_2\text{?}$

Options:

1. $\text{zero}$
2. $\frac{v^2}{2R}$
3. $\frac{2v^2}{R}$
4. $\frac{v^2}{R} \times \sqrt{2}$

Solution:

\text{In uniform circular motion, the magnitude of centripetal acceleration remains constant at } \frac{v^2}{R}\text{, but its direction changes continuously.} \text{From the figure, } P_1 \text{ is at the rightmost position and } P_2 \text{ is at the topmost position, making an angle of } 90°\text{ between them.} \text{At } P_1\text{: acceleration is directed toward center (leftward)} \text{At } P_2\text{: acceleration is directed toward center (downward)} \text{The change in acceleration vector is found using vector subtraction:} \Delta \vec{a} = \vec{a}_2 - \vec{a}_1 \text{Since the two acceleration vectors are perpendicular and have equal magnitudes } \frac{v^2}{R}\text{:} |\Delta \vec{a}| = 2a \sin\left(\frac{\theta}{2}\right) = 2 \times \frac{v^2}{R} \times \sin 45° = \sqrt{2} \times \frac{v^2}{R}

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
}