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Current Question (ID: 19366)

Question:

$\text{A ball experiences an angular acceleration given by:}$ $\alpha = (6t^2 - 2t),$ $\text{where } t \text{ is in seconds.}$ $\text{At } t = 0, \text{ the ball has an angular velocity of } 10 \text{ rad/s and an angular}$ $\text{position of } 4 \text{ rad. Which of the following expressions correctly}$ $\text{represents the angular position } \theta(t) \text{ of the ball?}$

Options:

1. $\frac{3}{2}t^4 - t^2 + 10t$
2. $\frac{t^4}{2} - \frac{t^3}{3} + 10t + 4$
3. $2t^4 - \frac{t^3}{6} + 10t + 12$
4. $2t^4 - \frac{t^3}{2} + 5t + 4$

Solution:

$\text{Hint: } \alpha = \frac{d^2\theta}{dt^2}$ $\text{Step 1: Find the expression for the angular velocity of the ball.}$ $\int_{10}^{\omega} d\omega = \int_{0}^{t} \alpha dt$ $\Rightarrow \omega - 10 = \int_{0}^{t} (6t^2 - 2t) dt$ $\Rightarrow \omega - 10 = 2t^3 - t^2$ $\Rightarrow \omega = 2t^3 - t^2 + 10$ $\text{Step 2: Find the expression for the angular position of the ball.}$ $\int_{4}^{\theta} d\theta = \int_{0}^{t} \omega dt$ $\Rightarrow \theta - 4 = \int_{0}^{t} \omega dt$ $\Rightarrow \theta - 4 = \int_{0}^{t} (2t^3 - t^2 + 10) dt$ $\Rightarrow \theta = \frac{t^4}{2} - \frac{t^3}{3} + 10t + 4$ $\text{Hence, option (2) is the correct answer.}$

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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
}