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Current Question (ID: 9902)

Question:

$\text{A particle of mass } m \text{ is driven by a machine that delivers a constant power of } k \text{ watts. If the particle starts from rest, the force on the particle at time } t \text{ is:}$

Options:

1. $\sqrt{\frac{mk}{2}} t^{-1/2}$
2. $\sqrt{mkt} t^{-1/2}$
3. $\sqrt{2mkt} t^{-1/2}$
4. $\frac{1}{2}\sqrt{mkt} t^{-1/2}$

Solution:

$\text{Hint: } P = F \cdot v$ $\text{Given:}$ $\text{Mass of particle } = m$ $\text{Constant power delivered } = k \text{ watts}$ $\text{Particle starts from rest}$ $\text{Step 1: Find the velocity at time } t$ $\text{Since the machine delivers constant power:}$ $P = F \cdot v = k \text{ (constant)}$ $\text{Using } F = m\frac{dv}{dt}\text{:}$ $m\frac{dv}{dt} \cdot v = k$ $\text{Rearranging:}$ $\int v\,dv = \frac{k}{m}\int dt$ $\frac{v^2}{2} = \frac{k}{m}t$ $\text{Therefore:}$ $v = \sqrt{\frac{2kt}{m}}$ $\text{Step 2: Find the force at time } t$ $\text{Force is given by:}$ $F = m\frac{dv}{dt} = m\frac{d}{dt}\left(\sqrt{\frac{2kt}{m}}\right)$ $F = m \times \frac{1}{2}\sqrt{\frac{2k}{m}} \times t^{-1/2}$ $F = \sqrt{\frac{mk}{2}} \times t^{-1/2}$ $\text{Hence, option (1) is the correct answer.}$

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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
}