Import Question JSON

Current Question (ID: 9811)

Question:

$\text{A particle of mass } m \text{ having speed } v \text{ goes in a vertical circular motion such that its centre is at its origin, as shown in the figure. If at any instant the angle made by the string with a negative } y\text{-axis is } \theta \text{ then the tension in the string is:}$ $\text{[Take radius = } R\text{]}$

Options:

1. $mg \sin \theta + \frac{mv^2}{R}$
2. $mg \cos \theta - \frac{mv^2}{R}$
3. $mg \cos \theta + \frac{mv^2}{R}$
4. $mg \sin \theta - \frac{mv^2}{R}$

Solution:

$\text{Hint: The net force towards the center of the circle is centripetal force.}$ $\text{Step 1: Find the tension in the string.}$ $\text{From the force analysis diagram, we can see that the forces acting on the particle are:}$ $\text{- Tension } T \text{ along the string (radially inward)}$ $\text{- Weight } mg \text{ vertically downward}$ $\text{- Component of weight along the radial direction: } mg \cos \theta$ $\text{Applying Newton's second law in the radial direction:}$ $T - mg \cos \theta = \frac{mv^2}{R}$ $\text{Therefore:}$ $T = mg \cos \theta + \frac{mv^2}{R}$

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
}