A body of mass `m` hung at one end of the spring executes simple harmonic motion . The force constant of a spring is `k` while its period of vibration is `T`. Prove by dimensional method that the equation `T=2pisqrt(m//k)` is correct. Dervive the correct equation , assuming that they are related by a power law.

Explanation

To prove the equation \( T = 2\pi \sqrt{\frac{m}{k}} \) using the dimensional method and derive the correct equation assuming a power law relationship, we can follow these steps:

### Step 1: Identify the quantities involved
We have:
- Mass of the body, \( m \)
- Spring constant, \( k \)
- Time period of vibration, \( T \)

### Step 2: Write down the dimensions of the quantities
- The dimension of mass \( m \) is \( [M] \).
- The dimension of the spring constant \( k \) can be derived from Hooke's law, which states that \( F = kx \). The dimension of force \( F \) is \( [M][L][T^{-2}] \) and the dimension of displacement \( x \) is \( [L] \). Therefore, the dimension of \( k \) is:
\[ [k] = \frac{[F]}{[x]} = \frac{[M][L][T^{-2}]}{[L]} = [M][T^{-2}] \]

### Step 3: Set up the dimensional equation
Assuming that the time period \( T \) is related to \( m \) and \( k \) by a power law, we can write:
\[ T \propto m^a k^b \]
where \( a \) and \( b \) are the powers to be determined.

### Step 4: Write the dimensions of \( T \)
The dimension of time \( T \) is \( [T] \).

### Step 5: Substitute the dimensions into the equation
Substituting the dimensions into the equation gives:
\[ [T] = [M]^a [M][T^{-2}]^b \]
This can be rewritten as:
\[ [T] = [M]^a [M]^b [T^{-2b}] \]
Thus, we have:
\[ [T] = [M]^{a+b} [T]^{-2b} \]

### Step 6: Equate the dimensions
For the dimensions to be equal, the powers of \( M \) and \( T \) must match:
1. For mass \( M \):
\[ a + b = 0 \quad (1) \]
2. For time \( T \):
\[ -2b = 1 \quad (2) \]

### Step 7: Solve the equations
From equation (2):
\[ b = -\frac{1}{2} \]
Substituting \( b \) into equation (1):
\[ a - \frac{1}{2} = 0 \implies a = \frac{1}{2} \]

### Step 8: Write the final relationship
Now substituting \( a \) and \( b \) back into the proportionality relation:
\[ T = C m^{\frac{1}{2}} k^{-\frac{1}{2}} = C \sqrt{\frac{m}{k}} \]
where \( C \) is a constant.

### Step 9: Determine the constant \( C \)
The constant \( C \) can be determined to be \( 2\pi \) based on experimental results, leading to the final equation:
\[ T = 2\pi \sqrt{\frac{m}{k}} \]

### Conclusion
Thus, we have proved the equation \( T = 2\pi \sqrt{\frac{m}{k}} \) using the dimensional method.