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Current Question (ID: 19900)

Question:

$\text{The displacement-time (} S-t \text{) graph of a particle executing simple harmonic motion (SHM) is provided (the sketch is schematic and not to scale).}$ $\text{Which of the following statements are true for this motion?}$

Options:

1. $\text{(A) The force is zero at } t = \frac{3T}{4}.$
2. $\text{(B) The acceleration is maximum at } t = T.$
3. $\text{(C) The speed is maximum at } t = \frac{T}{4}.$
4. $\text{(D) The potential energy is equal to the kinetic energy of the oscillation at } t = \frac{T}{2}.$

Solution:

$\text{Hint: } F = m\omega^2x$ $\text{Step: Analyse each statement one by one.}$ $\text{The displacement of the particle is zero at times } \frac{T}{4}, \frac{3T}{4}, \frac{5T}{4}.$ $\text{The acceleration of the particle performing SHM is given by } a = -\omega^2x.$ $\text{When the displacement of the particle is zero, then the particle's acceleration is also zero i.e., at } t = \frac{T}{4}, \frac{3T}{4}, \frac{5T}{4}, \text{ the acceleration of the particle is zero.}$ $\text{According to Newton's second law of motion, the force acting on the particle is given by:}$ $F = ma$ $\Rightarrow F = 0 \text{ if } a = 0$ $\text{So, the force is zero at } t = \frac{3T}{4}.$ $\text{The displacement of the particle is maximum at times } t = 0, \frac{2T}{4}, T.$ $\text{So, the acceleration of the particle is maximum at } t = T.$ $\text{The velocity of the particle is given by:}$ $v = \omega\sqrt{A^2 - x^2}$ $\text{The velocity of the particle is maximum } (v_{\text{max}} = A\omega) \text{ when the displacement of the particle is zero i.e., at } t = \frac{T}{4}, \frac{3T}{4}, \frac{5T}{4}.$ $\text{The potential energy is equal to the kinetic energy of the oscillation i.e., } \frac{1}{2}m\omega^2x^2 = \frac{1}{2}m\omega^2(A^2 - x^2)$ $\Rightarrow x = \frac{A}{\sqrt{2}}$ $\text{At } t = \frac{T}{2}, \text{ the particle is at an extreme position means potential energy is maximum and kinetic energy is zero.}$ $\text{Therefore, the correct statements are (A), (B), and (C) only.}$ $\text{Hence, option (4) is the correct answer.}$

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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
}