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Current Question (ID: 19279)

Question:

$\text{A particle attached to a light, inextensible string undergoes vertical circular motion. The speed at the lowest point } B \text{ is just sufficient for the particle to complete the full circle. Ignoring air resistance, what is the ratio of kinetic energy at the highest point } A \text{ to that at the lowest point } B?$

Options:

1. $1 : 5$
2. $5 : 1$
3. $1 : 7\sqrt{2}$
4. $1 : 5\sqrt{2}$

Solution:

$\text{Hint: } v_A = \sqrt{rg} \text{ and } v_B = \sqrt{5rg}$ $\text{Step 1: Find the speed of the particle at the points } A \text{ and } B.$ $\text{For just completing the circle, the tension at the top must be zero; hence,}$ $\frac{mv_A^2}{R} = mg$ $\Rightarrow v_A = \sqrt{Rg}$ $\text{Apply conservation of energy between points } A \text{ and } B :$ $\frac{1}{2}mv_A^2 + mg(2R) = \frac{1}{2}mv_B^2$ $\Rightarrow \frac{1}{2}mRg + 2mRg = \frac{1}{2}mv_B^2$ $\Rightarrow v_B = \sqrt{5Rg}$ $\text{Step 2: Find the ratio of kinetic energy.}$ $\frac{K_A}{K_B} = \frac{\frac{1}{2}mv_A^2}{\frac{1}{2}mv_B^2}$ $\Rightarrow \frac{K_A}{K_B} = \frac{Rg}{5Rg}$ $\Rightarrow \frac{K_A}{K_B} = \frac{1}{5}$ $\text{Hence, option (1) is the correct answer.}$

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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
}