The Question is from neet-physics oscillations-waves , and it was created by the author Mr. ANIL KUMAR.G

Question

A body of mass $m$ is attached to the lower end of a spring whose upper end is fixed. The spring has negligible mass. When the mass $m$ is slightly pulled down and released, it oscillates with a time period of 3 s. When the mass $m$ is increased by 1 kg, the time period of oscillations becomes 5 s. The value of $m$ in kg is :

Mr. ANIL KUMAR.G · Accepted Answer

## CORE INFORMATION
✓ **Answer**: A
## CONCEPT FRAMEWORK
	**Primary Concept**: Time period of a spring-mass system
	**Related Topics**: Simple Harmonic Motion, Oscillations
	**Prerequisites**: Understanding of SHM, formula for time period of spring-mass system
## SOLUTION PATHWAY
**Given Information**:
	* Time period $T_1 = 3$ s when mass is $m$
	* Time period $T_2 = 5$ s when mass is $m+1$
**Method & Steps**:
	1. Write the formula for the time period of a spring-mass system: $T = 2\pi \sqrt{\frac{m}{k}}$, where $k$ is the spring constant.
		Reasoning: This is the fundamental formula relating time period, mass and spring constant.
		Formula/Rule: $T = 2\pi \sqrt{\frac{m}{k}}$
	2. Write the equations for the given conditions:
		$T_1 = 3 = 2\pi \sqrt{\frac{m}{k}}$ and $T_2 = 5 = 2\pi \sqrt{\frac{m+1}{k}}$
		Working: Substituting given values in time period formula.
		Key Point: The spring constant $k$ is the same in both cases.
	3. Divide the second equation by the first equation:
		$\frac{5}{3} = \sqrt{\frac{m+1}{m}}$
		Working: Dividing two equations to eliminate k
	4. Square both sides:
		$\frac{25}{9} = \frac{m+1}{m}$
		Working: Squaring to remove square root.
	5. Solve for $m$:
		$25m = 9m + 9$
		$16m = 9$
		$m = \frac{9}{16}$
		Result: Final answer for m.
		Verification: The answer seems reasonable.
 ## OPTION ANALYSIS
[A] ✓: This is the correct solution as derived above.
[B] :x: This is an incorrect value for m.
[C] :x: This is an incorrect value for m.
[D] :x: This is an incorrect value for m.
 ## LEARNING AIDS
	**Common Mistakes**:
		1. Forgetting to square both sides when eliminating the square root: This will lead to wrong answer.
		2. Incorrectly dividing the equations: Always divide the second equation by the first to simplify.
	**Quick Tips**:
		1. Remember the formula for the time period of a spring mass system $T = 2\pi \sqrt{\frac{m}{k}}$.
		2. When dealing with ratios, try to eliminate common terms by dividing equations.

Mr. ANIL KUMAR.G · Answer

The correct Option is "$\frac{9}{{16}}$"

Mr. ANIL KUMAR.G · Answer

$\frac{3}{{4}}$

Mr. ANIL KUMAR.G · Answer

$\frac{4}{{3}}$

Mr. ANIL KUMAR.G · Answer

$\frac{16}{{9}}$

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