The Question is from neet-physics , and it was created by the author VENKATESWARARAO

Question

A particle executes linear simple harmonic motion with an amplitude of $3\;{\rm{ }}cm.$ When the particle is at ${\rm{2 \;}}cm$ from the mean position, the magnitude of its velocity is equal to that of its acceleration. Then its time period in seconds is

VENKATESWARARAO · Accepted Answer

## CORE INFORMATION\n✓ **Answer**: D\n## CONCEPT FRAMEWORK\n**Primary Concept**: Simple Harmonic Motion (SHM)\n**Related Topics**: Velocity and Acceleration in SHM, Time Period of SHM\n**Prerequisites**: Understanding of SHM equations for displacement, velocity, and acceleration.\n## SOLUTION PATHWAY\n**Given Information**:\n    * Amplitude (A) = 3 cm\n    * Displacement (x) = 2 cm\n    * Magnitude of velocity (v) = Magnitude of acceleration (a)\n**Method & Steps**:\n    1. **Write the equations for velocity and acceleration in SHM:**\n        * Velocity:  $v = \omega \sqrt{A^2 - x^2}$\n        * Acceleration:  $a = \omega^2 x$\n        Reasoning: These are the standard formulas for velocity and acceleration in SHM at a given displacement x.\n    2. **Set the magnitudes of velocity and acceleration equal to each other:**\n        * $|v| = |a|$\n        * $\omega \sqrt{A^2 - x^2} = \omega^2 x$\n        Reasoning:  The problem states that the magnitudes are equal. \n    3. **Solve for $\omega$:**\n        * $\sqrt{A^2 - x^2} = \omega x$\n        * $\omega = \frac{\sqrt{A^2 - x^2}}{x}$\n        * Substitute given values $A=3$ cm and $x=2$ cm: $\omega = \frac{\sqrt{3^2 - 2^2}}{2} = \frac{\sqrt{9-4}}{2} = \frac{\sqrt{5}}{2}$\n        Key Point: We have found angular frequency $\omega$.\n    4.  **Calculate the time period (T):**\n         * $T = \frac{2\pi}{\omega}$\n         * $T = \frac{2\pi}{\frac{\sqrt{5}}{2}} = \frac{4\pi}{\sqrt{5}}$\n        Result: The time period is $\frac{4\pi}{\sqrt{5}}$ seconds.\n ## OPTION ANALYSIS\n[A] :x: Incorrect time period.\n[B] :x: Incorrect time period.\n[C] :x: Incorrect time period.\n[D] ✓: Correct time period.\n ## LEARNING AIDS\n**Common Mistakes**:\n    1. **Incorrect use of SHM formulas**: Ensure the correct formulas for velocity and acceleration are used, paying attention to the signs.\n    2. **Algebra errors**: Be careful with simplifications and calculations.\n**Quick Tips**:\n    1. **Memorize SHM formulas**:  Knowing the basic equations for displacement, velocity, and acceleration will save time.\n    2. **Simplify equations**: Before substituting values, simplify the equation as much as possible to avoid errors.

VENKATESWARARAO · Answer

$\frac{{2\pi }}{{\sqrt 3 }}$

VENKATESWARARAO · Answer

$\frac{{\sqrt 5 }}{\pi }$

VENKATESWARARAO · Answer

$\frac{{\sqrt 5 }}{{2\pi }}$

VENKATESWARARAO · Answer

The correct Option is "$\frac{{4\pi }}{{\sqrt 5 }}$"

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