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Current Question (ID: 10308)

Question:

$\text{A child is standing on the edge of a merry-go-round that has the shape of a disk, as shown in the figure. The mass of the child is 40 kilograms. The merry-go-round has a mass of 200 kilograms and a radius of 2.5 meters, and it is rotating with an angular velocity of } \omega = 2.0 \text{ radians per second. The child then walks slowly towards the center of the merry-go-round. When the child reaches the center, what is the angular velocity of the disc? (The size of the child can be neglected.)}$

Options:

1. $2.0 \text{ rad/s}$
2. $2.2 \text{ rad/s}$
3. $2.4 \text{ rad/s}$
4. $2.8 \text{ rad/s}$

Solution:

$\text{Hint: Apply conservation of angular momentum.}$ $\text{Step 1: Write the given values.}$ $\text{Given,}$ $m = 40 \text{ kg}, M = 200 \text{ kg}, R = 2.5 \text{ m}$ $\text{and let } \omega \text{ be the final angular velocity.}$ $\text{Step 2: Find } \omega.$ $\text{By conservation of angular momentum,}$ $\left(\frac{1}{2}MR^2 + mR^2\right)\omega_0 = \left(0 + \frac{1}{2}MR^2\right)\omega$ $\Rightarrow 2\left[200 \times \frac{(2.5)^2}{2} + 40 \times (2.5)^2\right] = \omega\left[200 \times \frac{(2.5)^2}{2}\right]$ $\Rightarrow \omega = 2.8 \text{ rad/s}$

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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
}