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Current Question (ID: 18952)

Question:

$\text{The velocity } v \text{ is expressed in terms of wavelength } \lambda, \text{ acceleration due to gravity } g, \text{ and density of water } \rho \text{ as:}$ $v = \lambda^a g^b \rho^c.$ $\text{If this equation is dimensionally consistent, what are the values of } a, b, \text{ and } c?$

Options:

1. $\left( \frac{1}{2}, \frac{1}{2}, 0 \right)$
2. $\left( 1, \frac{1}{2}, 0 \right)$
3. $\left( \frac{1}{2}, 1, 0 \right)$
4. $(1, 1, 0)$

Solution:

$\text{Hint: Use the principle of dimensions homogeneity.}$ $\text{Step: Find the values of } a, b, \text{ and } c.$ $\text{As } v = \lambda^a g^b \rho^c$ $\text{Write the dimensions of physical quantities on both sides of the equation:}$ $[LT^{-1}] = [L]^a [LT^{-2}]^b [ML^{-3}]^c$ $\Rightarrow [M^0 LT^{-1}] = [M]^c [L]^{a+b-3c} [T]^{-2b}$ $\text{Now, compare the powers:}$ $c = 0; \ a + b - 3c = 1 \text{ and } -2b = -1$ $\text{By solving above equations:}$ $(a, b, c) \equiv \left( \frac{1}{2}, \frac{1}{2}, 0 \right)$ $\text{Hence, option (1) is the correct answer.}$

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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
}