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Current Question (ID: 10198)

Question:

$\text{Two particles of mass, 2 kg and 4 kg, are projected from the top of a tower simultaneously, such that 2 kg of mass is projected with a speed 20 m/s at an angle 30° above horizontal and 4 kg is projected at 40 m/s horizontally. The acceleration of the centre of mass of the system of two particles will be:}$

Options:

1. $\frac{g}{2}$
2. $\frac{g}{4}$
3. $g$
4. $2g$

Solution:

$\text{The acceleration of the center of mass of the system will be } g.$ $\text{Explanation: For a system of particles under the influence of gravity, the acceleration of the center of mass is independent of the initial velocities and directions of the individual particles. Since both particles are subject only to gravitational force after being projected, the acceleration of the center of mass of the system equals the acceleration due to gravity, which is } g \text{ downward.}$ $\text{This is a fundamental principle: the center of mass of a system moves as if all the mass were concentrated at that point and all external forces were applied there. Since the only external force acting on both particles is gravity, the center of mass accelerates downward at } g.$

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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
}