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Current Question (ID: 10343)

Question:

$\text{The breaking stress of a wire going over a smooth pulley in the following question is } 2 \times 10^9 \text{ N/m}^2\text{. What would be the minimum radius of the wire used if it is not to break?}$

Options:

1. $0.46 \times 10^{-6} \text{ m}$
2. $0.46 \times 10^{-4} \text{ m}$
3. $0.46 \times 10^3 \text{ m}$
4. $0.46 \times 10^{-11} \text{ m}$

Solution:

\text{Hint: The breaking stress of the wire is given as } \sigma_b = T/A_{\text{min}} \text{Step 1: Find the tension in the wire.} \text{For a pulley system with masses } m_1 = 1 \text{ kg and } m_2 = 2 \text{ kg:} T = (2m_1 m_2)/(m_1 + m_2) \cdot g = (2 \times 1 \times 2 \times g)/3 = 4g/3 \text{Taking } g = 10 \text{ m/s}^2: T = (4 \times 10)/3 = 40/3 \text{ N} \text{Step 2: Apply the breaking stress formula.} \sigma_b = T/A_{\text{min}} \text{Where } A_{\text{min}} = \pi r^2 \text{Step 3: Put all values and find the radius.} 2 \times 10^9 = (40/3)/(\pi r^2) r^2 = (40/3)/(\pi \times 2 \times 10^9) = 40/(6\pi \times 10^9) r^2 = 40/(6 \times 3.14 \times 10^9) = 40/(18.84 \times 10^9) = 2.12 \times 10^{-9} r = \sqrt{2.12 \times 10^{-9}} = 0.46 \times 10^{-4} \text{ m}

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
}