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Current Question (ID: 9665)

Question:

$\text{A particle rotates in a circular path starting from rest. If angular acceleration is } 4 \text{ rad/s}^2\text{, then the time after which angle between net acceleration and tangential acceleration becomes } 45^\circ \text{ is?}$

Options:

1. $0.5 \text{ s}$
2. $0.25 \text{ s}$
3. $2 \text{ s}$
4. $4 \text{ s}$

Solution:

$\text{Hint: If the angle between } a_N \text{ and } a_T \text{ will be } 45^\circ \text{ then } a_C = a_T\text{.}$ $\text{Step 1: Find angular velocity as a function of time.}$ $\omega_0 = 0$ $\omega = \omega_0 + \alpha t$ $\omega = 4t$ $\text{Step 2: Find radial and tangential accelerations}$ $a_R = \omega^2 R = (4t)^2 R$ $a_T = R\alpha = 4R$ $\text{Step 3: Find the net acceleration}$ $\text{From the geometry, when the angle between net acceleration and tangential acceleration is } 45^\circ\text{:}$ $\tan 45^\circ = \frac{a_T}{a_R} = 1$ $\text{Step 4: Equate radial and tangential accelerations}$ $a_R = a_T$ $\omega^2 R = R\alpha$ $(4t)^2 = 4$ $t = 0.5 \text{ s}$

Import JSON File

Upload a JSON file containing LaTeX/MathJax formatted question, options, and solution.

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
}