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Current Question (ID: 19568)

Question:

$\text{Water from a tap emerges vertically downwards with an initial speed of } 1.0 \text{ m/s.}$ $\text{The cross-sectional area of the tap is } 10^{-4} \text{ m}^2.$ $\text{Assume that the pressure is constant throughout the stream of water and that the flow is streamlined.}$ $\text{The cross-sectional area of the stream, } 0.15 \text{ m below the tap would be:}$ $\text{(Take } g = 10 \text{ m/s}^2)$

Options:

1. $5 \times 10^{-4} \text{ m}^2$
2. $2 \times 10^{-5} \text{ m}^2$
3. $5 \times 10^{-5} \text{ m}^2$
4. $1 \times 10^{-5} \text{ m}^2$

Solution:

$\text{Hint: Use the equation of continuity.}$ $\text{Step 1: Draw the diagram.}$ $\text{Step 2: Apply Bernoulli's equation.}$ $P_0 + \rho gh_1 + \frac{1}{2} \rho v_1^2 = P_0 + 0 + \frac{1}{2} \rho v_2^2$ $\Rightarrow \rho gh_1 + \frac{1}{2} \rho v_1^2 = \frac{1}{2} \rho v_2^2$ $\Rightarrow 1000 \times 10 \times 0.15 + 0.5 \times 1000 \times 1^2 = 0.5 \times 1000 v_2^2$ $\Rightarrow v_2 = 2 \text{ m/s}$ $\text{Step 3: Use equation of continuity.}$ $A_1 v_1 = A_2 v_2$ $\Rightarrow 10^{-4} \times 1 = A_2 \times 2$ $\Rightarrow A_2 = 5 \times 10^{-5} \text{ m}^2$ $\text{Hence, option (3) is the correct answer.}$

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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
}