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Current Question (ID: 10373)

Question:

$\text{The bulk modulus of a spherical object is } B. \text{ If it is subjected to uniform pressure } P, \text{ the fractional decrease in radius will be:}$

Options:

1. $\frac{P}{B}$
2. $\frac{B}{3P}$
3. $\frac{3P}{B}$
4. $\frac{P}{3B}$

Solution:

The object is spherical and the bulk modulus is represented by $B$. It is the ratio of normal stress to the volumetric strain. Hence $B = \frac{F/A}{\Delta V/V}$ $\Rightarrow \frac{\Delta V}{V} = \frac{P}{B} \Rightarrow \left|\frac{\Delta V}{V}\right| = \frac{P}{B}$ Hence $P$ is applied pressure on the object and $\frac{\Delta V}{V}$ is volume strain. Fractional decrease in volume is: $\frac{\Delta V}{V} = 3\frac{\Delta R}{R} \quad [\text{So, } V = \frac{4}{3}\pi R^3]$ Volume of the sphere decreases due to the decrease in its radius. Hence, $\frac{\Delta V}{V} = \frac{3\Delta R}{R} = \frac{P}{B} \Rightarrow \frac{\Delta R}{R} = \frac{P}{3B}$

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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
}