The Question is from neet-physics units-and-measurement priniciple-of-homoginity, and it was created by the author Mr. ANIL KUMAR.G

Question

The period $T$ of oscillation of a spring-mass system depends on the mass $m$ and the spring constant $k$. Using dimensional analysis and the principle of homogeneity, determine the relationship between $T$, $m$, and $k$.

Mr. ANIL KUMAR.G · Accepted Answer

## CORE INFORMATION
✓ **Answer**: Option B
## CONCEPT FRAMEWORK
	**Primary Concept**: Dimensional Analysis
	**Related Topics**: Principle of Homogeneity, Simple Harmonic Motion
	**Prerequisites**: Understanding of dimensions (mass [M], length [L], time [T]), basic algebra
## SOLUTION PATHWAY
**Given Information**:
	* Period of oscillation, $T$, depends on mass $m$ and spring constant $k$.
	* We need to find the relationship $T = f(m, k)$ using dimensional analysis.

**Method & Steps**:
	1. **Express the relationship with unknown exponents:**
		Reasoning: We assume the relationship is of the form $T \propto m^a k^b$ where $a$ and $b$ are unknown exponents.
		Formula/Rule: $T = C m^a k^b$ , where C is a dimensionless constant.
	2. **Write the dimensions of each quantity:**
		Working:  
            * [T] = T
            * [m] = M
            * [k] =  Force/Length = MLT$^{-2}$/L = MT$^{-2}$
		Key Point:  The spring constant has dimensions of mass per time squared.
	3. **Substitute dimensions into the equation and apply the principle of homogeneity:**
		Working: $[T] = [M]^a [MT^{-2}]^b$
		Key Point:  Principle of homogeneity states that dimensions must be the same on both sides of the equation.
	4. **Equate the exponents of each dimension:**
		Working: $T^1 = M^{a+b} T^{-2b}$
			* For M: 0 = a + b
			* For T: 1 = -2b
	5. **Solve the equations for a and b:**
		Working:  From 1 = -2b, we get b = -1/2. Substituting this into 0 = a + b, we get a = 1/2
	6. **Substitute the values of a and b back into the equation:**
		Result: $T = C m^{1/2} k^{-1/2} = C \sqrt{\frac{m}{k}}$
		Verification: We know from SHM that $C = 2\pi$, so $T = 2\pi\sqrt{\frac{m}{k}}$
 ## OPTION ANALYSIS
[A] :x: Incorrect. The formula is inverted.
[B] ✓: Correct. Dimensional analysis and the principle of homogeneity lead to this relationship.
[C] :x: Incorrect. The formula is inverted.
[D] :x: Incorrect.  The dimensions of m and k are not correct for time.
 ## LEARNING AIDS
	**Common Mistakes**:
		1.[Error 1]: Incorrectly determining the dimensions of the spring constant. [How to avoid]: Remember that k is Force/Length, and Force = mass * acceleration.
		2.[Error 2]: Incorrectly applying the principle of homogeneity. [How to avoid]: Ensure that the exponents of each dimension are properly equated on both sides of the equation.
	**Quick Tips**:
		1.[Helpful shortcut]: Always start by writing the relationship in terms of unknown exponents.
		2.[Memory aid]: Remember the dimensions of basic quantities like mass, length, and time.

Mr. ANIL KUMAR.G · Answer

$T = 2\pi\sqrt{mk}$

Mr. ANIL KUMAR.G · Answer

The correct Option is "$T = 2\pi\sqrt{\frac{m}{k}}$"

Mr. ANIL KUMAR.G · Answer

$T = 2\pi\sqrt{\frac{k}{m}}$

Mr. ANIL KUMAR.G · Answer

$T = 2\pi\frac{m}{k}$

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