The Question is from neet-physics , and it was created by the author P.NARASIMHA

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

P.NARASIMHA · Accepted Answer

## CORE INFORMATION
✓ **Answer**: D
## CONCEPT FRAMEWORK
	**Primary Concept**: Simple Harmonic Motion (SHM)
	**Related Topics**: Velocity and Acceleration in SHM, Time Period of SHM
	**Prerequisites**: Understanding of basic SHM equations and their relationships
## SOLUTION PATHWAY
**Given Information**:
	* Amplitude (A) = 3 cm
	* Position (x) = 2 cm
	* Magnitude of velocity = Magnitude of acceleration
**Method & Steps**:
	1. Write the expressions for velocity and acceleration in SHM:
		* Velocity (v) = ω√(A² - x²)
		* Acceleration (a) = ω²x
	2. Set the magnitudes of velocity and acceleration equal to each other: |v| = |a|
		* ω√(A² - x²) = ω²x
	3. Substitute the given values of A and x:
		* ω√((3²) - (2²)) = ω²(2)
		* ω√(9 - 4) = 2ω²
		* ω√5 = 2ω²
	4. Simplify and solve for ω:
		* √5 = 2ω
		* ω = √5 / 2
	5. Calculate the time period T using the formula T = 2π/ω:
		* T = 2π / (√5 / 2)
		* T = 4π / √5
## OPTION ANALYSIS
[A] :x: Incorrect time period.
[B] :x: Incorrect time period.
[C] :x: Incorrect time period.
[D] ✓: Correct time period derived.
## LEARNING AIDS
	**Common Mistakes**:
		1. Forgetting to take the square root in the velocity formula: Make sure to use v = ω√(A² - x²) and not v = ω(A² - x²).
		2. Confusing the formulas for velocity and acceleration: Remember v = ω√(A² - x²) and a = ω²x
	**Quick Tips**:
		1. Always write down the known values first before starting the solution.
		2. Remember the relationship between angular frequency (ω) and time period (T) : T = 2π/ω

P.NARASIMHA · Answer

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

P.NARASIMHA · Answer

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

P.NARASIMHA · Answer

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

P.NARASIMHA · Answer

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

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