The time period (T) of a spring mass system depends upon mass (m) & spring constant (k) & length of the spring `(l)[k=("Force")/("length")]`. Find the relation among T, m , l & k using dimensional method.

Answer

To find the relation among the time period \( T \), mass \( m \), spring constant \( k \), and length of the spring \( l \) using the dimensional method, we can follow these steps:

### Step 1: Identify the relationship
We start by assuming that the time period \( T \) depends on mass \( m \), spring constant \( k \), and length \( l \). We can express this relationship as:
\[ T \propto m^a k^b l^c \]
where \( a \), \( b \), and \( c \) are the powers we need to determine.

### Step 2: Introduce a proportionality constant
To remove the proportionality, we introduce a constant \( A \):
\[ T = A m^a k^b l^c \]

### Step 3: Write the dimensions
Next, we write the dimensions of each variable:
- The dimension of time \( T \) is \( [T] \).
- The dimension of mass \( m \) is \( [M] \).
- The spring constant \( k \) can be expressed as \( k = \frac{F}{l} \), where \( F \) (force) has the dimension \( [M L T^{-2}] \) and \( l \) has the dimension \( [L] \). Thus, the dimension of \( k \) is:
\[ [k] = \frac{[M L T^{-2}]}{[L]} = [M T^{-2}] \]
- The dimension of length \( l \) is \( [L] \).

### Step 4: Substitute dimensions into the equation
Substituting the dimensions into our equation gives:
\[ [T] = [M^a] [M^{b} T^{-2b}] [L^{c}] \]
This simplifies to:
\[ [T] = [M^{a+b} L^{c} T^{-2b}] \]

### Step 5: Equate dimensions
Now we equate the dimensions on both sides:
- For mass \( M \): \( a + b = 0 \)
- For length \( L \): \( c = 0 \)
- For time \( T \): \( -2b = 1 \)

### Step 6: Solve the equations
From the equation \( -2b = 1 \):
\[ b = -\frac{1}{2} \]
Substituting \( b \) into \( a + b = 0 \):
\[ a - \frac{1}{2} = 0 \implies a = \frac{1}{2} \]
And from \( c = 0 \), we have:
\[ c = 0 \]

### Step 7: Write the final relation
Now substituting the values of \( a \), \( b \), and \( c \) back into the equation for \( T \):
\[ T = A m^{\frac{1}{2}} k^{-\frac{1}{2}} l^{0} \]
This simplifies to:
\[ T = A \sqrt{\frac{m}{k}} \]

### Conclusion
Thus, the relation among the time period \( T \), mass \( m \), and spring constant \( k \) is:
\[ T = A \sqrt{\frac{m}{k}} \]