Import Question JSON

Current Question (ID: 10434)

Question:
A liquid is filled in a container as shown in the figure. The force exerted by the liquid on the base of the container must be:
Options:
  • 1. Equal to the weight of the liquid
  • 2. More than the weight of the liquid
  • 3. Less than the weight of the liquid
  • 4. More or less than the weight of the liquid
Solution:
\text{Hint: Use the relation between force and pressure.} \text{Step 1: Find the pressure at the bottom.} P = \rho gh \text{Step 2: Find force due to pressure.} F = P \times A = \rho ghA \text{This is the force exerted on the base of the container, where } A \text{ is the area of the base and } h \text{ is the total height of the liquid from the base to the top surface.} \text{Step 3: Convert this force into weight form.} \text{Let's consider a hypothetical cylinder of liquid with base area } A \text{ and height } h \text{. The volume of this hypothetical liquid would be } V = Ah \text{. The mass of this liquid would be } m = \rho V = \rho Ah \text{. The weight of this hypothetical liquid would be } W = mg = (\rho Ah)g \text{.} \text{From Step 2, we have the force exerted on the base as } F = \rho ghA \text{.} \text{From Step 3, we have the weight of a hypothetical column of liquid of the same height and base area as } W = \rho ghA \text{.} \text{So, the force on the base is equal to the weight of a hypothetical column of liquid of height } h \text{ and area } A \text{. The actual volume of the liquid in the container is less than } Ah \text{ (due to the narrow upper section). Hence, the actual weight of the liquid is less than the force on the base.} \text{Step 4: Compare this force with the actual weight of the liquid.} \text{The actual volume occupied by the fluid is less than the volume of a cylinder with area } A \text{ and height } h \text{ (i.e., less than } Ah \text{). The force exerted by the liquid on the base, which is } F = \rho ghA \text{, is equal to the weight of a liquid column with volume } Ah \text{. Since the actual volume and thus the actual weight of the liquid in the container is less than the weight of this hypothetical column, the force exerted by the liquid at the bottom is more than the weight of the liquid.} \text{Therefore, the correct answer is that the force exerted by the liquid on the base of the container is more than the weight of the liquid.}

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
}