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Current Question (ID: 19615)

Question:

$\text{A small ball of mass } m \text{ and density } \rho \text{ is dropped in a viscous liquid of density } \rho_0. \text{ After sometime, the ball falls with constant velocity. The viscous force on the ball is:}$

Options:

1. $mg \left( \frac{\rho_0}{\rho} - 1 \right)$
2. $mg \left( 1 - \rho \rho_0 \right)$
3. $mg \left( 1 + \frac{\rho}{\rho_0} \right)$
4. $mg \left( 1 - \frac{\rho_0}{\rho} \right)$

Solution:

$\text{The viscous force on the ball is given by the equation:}$ $F_v = mg - V \rho_0 g$ $\text{where } V \text{ is the volume of the ball.}$ $\text{Since the ball falls with constant velocity, the net force is zero:}$ $mg - V \rho_0 g = 0$ $\text{Thus, } V = \frac{m}{\rho}$ $\text{Substituting, we get:}$ $F_v = mg - \frac{m}{\rho} \rho_0 g = mg \left( 1 - \frac{\rho_0}{\rho} \right)$

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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
}