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Current Question (ID: 19225)

Question:

$\text{A time dependent force } F = 6t \text{ acts on a particle of mass } 1 \text{ kg. If the particle starts from rest, the work done by the force during the first } 1 \text{ s will be:}$ $1. \ 4.5 \text{ J}$ $2. \ 22 \text{ J}$ $3. \ 9 \text{ J}$ $4. \ 18 \text{ J}$

Options:

1. $4.5 \text{ J}$
2. $22 \text{ J}$
3. $9 \text{ J}$
4. $18 \text{ J}$

Solution:

$\text{Hint: Apply work energy theorem.}$ $\text{Step 1: Find the velocity after } 1 \text{ s.}$ $\text{We have given that;}$ $F = ma = 6t$ $\text{Here; } m = 1 \text{ kg}$ $\text{Then,}$ $\Rightarrow a = \frac{F}{m} = \frac{6t}{1}$ $\Rightarrow \frac{dv}{dt} = 6t$ $\Rightarrow \int_0^v dv = \int_0^1 6t \cdot dt$ $\Rightarrow v = \frac{6 \times (1)^2}{2} = 3 \text{ ms}^{-1}$ $\text{Step 2: Find the work done by force after } 1 \text{ s.}$ $W = \Delta K = \frac{1}{2} mv^2 - \frac{1}{2} mu^2$ $\text{Here, initial velocity } u = 0.$ $\Rightarrow W = \frac{1}{2} mv^2 = \frac{1}{2} \times 1 \times 3^2$ $\Rightarrow W = 4.5 \text{ J}$ $\text{Therefore, the work done by the force during the first } 1 \text{ s will be } 4.5 \text{ J.}$ $\text{Hence, option (1) is the correct answer.}$

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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
}