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 will follow these steps:

### Step 1: Identify the dimensions of each variable
- The dimension of time period \( T \) is \( [T] \).
- The dimension of mass \( m \) is \( [M] \).
- The spring constant \( k \) is defined as \( k = \frac{F}{l} \), where \( F \) is force and \( l \) is length. The dimension of force \( F \) is \( [M][L][T^{-2}] \) and the dimension of length \( l \) is \( [L] \). Therefore, the dimension of spring constant \( k \) is:
\[ [k] = \frac{[F]}{[l]} = \frac{[M][L][T^{-2}]}{[L]} = [M][T^{-2}] \]
- The dimension of length \( l \) is \( [L] \).

### Step 2: Assume a relationship
We assume that the time period \( T \) can be expressed as a function of \( m \), \( k \), and \( l \):
\[ T = C \cdot m^p \cdot k^q \cdot l^r \]
where \( C \) is a dimensionless constant, and \( p \), \( q \), and \( r \) are the powers we need to determine.

### Step 3: Write the dimensions of the equation
Substituting the dimensions into the equation gives:
\[ [T] = [M]^p \cdot [M]^{q} \cdot [T]^{-2q} \cdot [L]^{r} \]
This can be expressed as:
\[ [T] = [M]^{p+q} \cdot [L]^{r} \cdot [T]^{-2q} \]

### Step 4: Equate dimensions
Now we equate the dimensions on both sides:
- For the dimension of mass \( M \):
\[ 0 = p + q \quad \text{(1)} \]
- For the dimension of length \( L \):
\[ 0 = r \quad \text{(2)} \]
- For the dimension of time \( T \):
\[ 1 = -2q \quad \text{(3)} \]

### Step 5: Solve the equations
From equation (2), we have:
\[ r = 0 \]

From equation (3):
\[ q = -\frac{1}{2} \]

Substituting \( q \) into equation (1):
\[ 0 = p - \frac{1}{2} \implies p = \frac{1}{2} \]

### Step 6: Write the final relation
Substituting the values of \( p \), \( q \), and \( r \) back into the assumed relationship gives:
\[ T = C \cdot m^{\frac{1}{2}} \cdot k^{-\frac{1}{2}} \cdot l^{0} \]
Thus, we can simplify this to:
\[ T = C \cdot \sqrt{\frac{m}{k}} \]

### Final Relation
The final relation among \( T \), \( m \), \( l \), and \( k \) is:
\[ T \propto \sqrt{\frac{m}{k}} \]