Import Question JSON

Current Question (ID: 10730)

Question:

\text{A large steel wheel is to be fitted onto a shaft of the same material. At 27°C, the outer diameter of the shaft is 8.70 cm and the wheel's central hole has a diameter of 8.69 cm. The shaft is cooled using 'dry ice'. At what temperature of the shaft does the wheel slip on the shaft?} \text{(Assume the coefficient of linear expansion of the steel to be constant over the required temperature range and } \alpha_{\text{steel}} = 1.20 \times 10^{-5} \text{ K}^{-1}\text{)}

Options:

1. $\text{68°C}$
2. $\text{-70°C}$
3. $\text{-69°C}$
4. $\text{-67°C}$

Solution:

\textbf{Hint:} \text{Apply the concept of thermal expansion.} \text{The given temperature, } T = 27°\text{C} = 27 + 273 = 300 \text{ K} \textbf{Step 1:} \text{Use the formula of linear expansion.} \text{We know that, } \Delta d = -d_1\alpha_{\text{steel}}(T_1 - T) \text{where } T_1 \text{ is temperature after cooling.} \textbf{Step 2:} \text{Find the final temperature.} 0.01 = -8.7 \times 1.20 \times 10^{-5} \times (T_1 - 300) (T_1 - 300) = -95.78 T_1 = 204.21 \text{ K} = -68.95°\text{C} \text{Therefore the wheel will slip on the shaft when the temperature is } -69°\text{C}

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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
}