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Current Question (ID: 8716)

Question:

$\text{Identify the pair whose dimensions are equal:}$

Options:

1. $\text{stress and energy}$
2. $\text{force and work}$
3. $\text{torque and work}$
4. $\text{velocity gradient and time}$

Solution:

\text{Hint: Find the dimensional formula of given quantities.} \text{Explanation:} \text{Let's find the dimensional formula for each quantity:} \text{Option 1: Stress and Energy} \text{Stress} = \frac{\text{Force}}{\text{Area}} = \frac{[MLT^{-2}]}{[L^2]} = [ML^{-1}T^{-2}] \text{Energy} = \text{Work} = \text{Force} \times \text{distance} = [MLT^{-2}] \times [L] = [ML^2T^{-2}] \text{These are different.} \text{Option 2: Force and Work} \text{Force} = [MLT^{-2}] \text{Work} = [ML^2T^{-2}] \text{These are different.} \text{Option 3: Torque and Work} \text{Torque} = \text{Force} \times \text{perpendicular distance} = [MLT^{-2}] \times [L] = [ML^2T^{-2}] \text{Work} = \text{Force} \times \text{distance} = [MLT^{-2}] \times [L] = [ML^2T^{-2}] \text{These are equal!} \text{Option 4: Velocity gradient and Time} \text{Velocity gradient} = \frac{\text{velocity}}{\text{distance}} = \frac{[LT^{-1}]}{[L]} = [T^{-1}] \text{Time} = [T] \text{These are different.} \text{Therefore, torque and work have equal dimensions } [ML^2T^{-2}]

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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
}