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 = 2 pi 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 dimensional analysis, we will follow these steps:

### Step 1: Identify the dimensions of the quantities involved
- The period \( T \) has the dimension of time, denoted as \( [T] \).
- The mass \( m \) has the dimension of mass, denoted as \( [M] \).
- The force constant \( k \) (spring constant) has the dimension of force per unit displacement. The dimension of force is \( [M][L][T^{-2}] \) (mass × acceleration), and since it is divided by displacement (length, \( [L] \)), the dimension of \( k \) is:
\[ [k] = \frac{[M][L][T^{-2}]}{[L]} = [M][T^{-2}] \]

### Step 2: Assume a relationship between the quantities
Assume that the period \( T \) is related to \( m \) and \( k \) by a power law:
\[ T = C \cdot m^a \cdot k^b \]
where \( C \) is a dimensionless constant, and \( a \) and \( b \) are the powers we need to determine.

### Step 3: Write the dimensions of both sides of the equation
The left-hand side (LHS) has the dimension of time:
\[ [T] = [T] \]

The right-hand side (RHS) has dimensions:
\[ [m^a] = [M]^a \quad \text{and} \quad [k^b] = [M]^b[T^{-2b}] \]
Thus, the dimensions of the RHS become:
\[ [M]^a \cdot [M]^b \cdot [T^{-2b}] = [M]^{a+b} \cdot [T^{-2b}] \]

### Step 4: Set the dimensions equal
Equating the dimensions from both sides:
\[ [T] = [M]^{a+b} \cdot [T^{-2b}] \]

This gives us two equations:
1. For the dimension of mass:
\[ a + b = 0 \]
2. For the dimension of time:
\[ -2b = 1 \]

### Step 5: Solve the equations
From the second equation, we can solve for \( b \):
\[ b = -\frac{1}{2} \]

Substituting \( b \) into the first equation:
\[ a - \frac{1}{2} = 0 \implies a = \frac{1}{2} \]

### Step 6: Substitute back into the relationship
Now substituting \( a \) and \( b \) back into the assumed relationship:
\[ T = C \cdot m^{1/2} \cdot k^{-1/2} \]
This can be rewritten as:
\[ T = C \cdot \sqrt{\frac{m}{k}} \]

### Step 7: Determine the constant \( C \)
The constant \( C \) is dimensionless, and we can find it through experimental or theoretical means. For the case of a spring, it is known that:
\[ C = 2\pi \]
Thus, we arrive at the final equation:
\[ T = 2\pi \sqrt{\frac{m}{k}} \]

### Conclusion
We have proven that the equation \( T = 2\pi \sqrt{\frac{m}{k}} \) is dimensionally consistent and derived it using the power law assumption.

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