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Current Question (ID: 9794)

Question:

$\text{A massless string of length } 1 \text{ m fixed at one end carries a mass of } 2 \text{ kg at the other end. The string makes } \frac{2}{\pi} \text{ rev/s around the axis through the fixed end as shown in the figure. The tension on the string will be:}$

Options:

1. $32 \text{ N}$
2. $3 \text{ N}$
3. $16 \text{ N}$
4. $4 \text{ N}$

Solution:

\text{Hint: Horizontal component of } T = mr\omega^2 \text{Step 1: Find angular velocity in SI units} \omega = \frac{2\pi}{T} = \frac{2\pi}{1} = 2\pi \text{ rad/s} \text{Step 2: Apply force equation for circular motion} T \sin \theta = m\omega^2 r \text{Step 3: Find } \sin \theta \text{ from the diagram} \sin \theta = \frac{r}{l} \text{Thus, } T = \frac{m\omega^2 r}{\sin \theta} = \frac{m\omega^2 r \cdot l}{r} = m\omega^2 l \text{Substituting values: } T = 2 \times (2\pi)^2 \times l \text{ N}

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
}