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Current Question (ID: 19584)

Question:

$\text{The velocity of a small ball of mass } m \text{ and density } d_1, \text{ when dropped in a container filled with glycerine, becomes constant after some time. If the density of glycerine is } d_2, \text{ then the viscous force acting on the ball will be:}$

Options:

1. $mg \left( 1 - \frac{d_1}{d_2} \right)$
2. $mg \left( 1 - \frac{d_2}{d_1} \right)$
3. $mg \left( \frac{d_1}{d_2} - 1 \right)$
4. $mg \left( \frac{d_2}{d_1} - 1 \right)$

Solution:

$\text{Hint: Apply Archimedes principle.}$ $\text{Step: Find the viscous force acting on the ball.}$ $\text{The viscous force acting on the ball is given by:}$ $F = 6 \pi \eta r v_T$ $\Rightarrow F = 6 \pi \eta r \left( \frac{2gr^2(d_1-d_2)}{9 \eta} \right)$ $\left[ v_T = \frac{2gr^2(d_1-d_2)}{9 \eta} \right]$ $\Rightarrow F = \frac{4}{3} \pi r^3 g d_1 \left( 1 - \frac{d_2}{d_1} \right)$ $\Rightarrow F = mg \left( 1 - \frac{d_2}{d_1} \right)$ $\left[ mg = \frac{4}{3} \pi r^3 d_1 g \right]$ $\text{Therefore, the viscous force acting on the ball is } mg \left( 1 - \frac{d_2}{d_1} \right).$ $\text{Hence, option (2) is the correct answer.}$

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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
}