To find the relation between energy \( E \), mass \( m \), frequency \( n \), and amplitude \( a \) using dimensional analysis, we can follow these steps:
### Step 1: Identify the dimensions of energy
Energy \( E \) can be expressed in terms of force and displacement. The dimensional formula for force is given by:
\[
\text{Force} = \text{mass} \times \text{acceleration} = m \cdot \left( \frac{\text{length}}{\text{time}^2} \right) = m \cdot \frac{L}{T^2}
\]
Thus, the dimensions of force are:
\[
[F] = M L T^{-2}
\]
Energy is defined as force multiplied by displacement:
\[
E = \text{Force} \times \text{Displacement} = F \cdot L
\]
So, the dimensions of energy are:
\[
[E] = [F] \cdot [L] = M L T^{-2} \cdot L = M L^2 T^{-2}
\]
### Step 2: Express energy in terms of \( m \), \( n \), and \( a \)
We assume that energy \( E \) can be expressed as:
\[
E \propto m^a n^b a^c
\]
where \( a \), \( b \), and \( c \) are the powers to which mass, frequency, and amplitude are raised, respectively.
### Step 3: Write the dimensions of \( m \), \( n \), and \( a \)
- The dimension of mass \( m \) is:
\[
[m] = M
\]
- The frequency \( n \) is the reciprocal of time, so its dimension is:
\[
[n] = T^{-1}
\]
- The amplitude \( a \) is a length, so its dimension is:
\[
[a] = L
\]
### Step 4: Substitute the dimensions into the equation
Substituting the dimensions into the expression for energy gives:
\[
M L^2 T^{-2} = (M^a) \cdot (T^{-b}) \cdot (L^c)
\]
This can be rewritten as:
\[
M^a \cdot L^c \cdot T^{-b}
\]
### Step 5: Equate the dimensions
Now, we equate the coefficients of \( M \), \( L \), and \( T \) from both sides:
1. For mass \( M \):
\[
a = 1
\]
2. For length \( L \):
\[
c = 2
\]
3. For time \( T \):
\[
-b = -2 \implies b = 2
\]
### Step 6: Write the final relation
From the values of \( a \), \( b \), and \( c \), we can write the relation for energy:
\[
E \propto m^1 n^2 a^2
\]
or simply:
\[
E = k \cdot m \cdot n^2 \cdot a^2
\]
where \( k \) is a proportionality constant.
### Conclusion
Thus, the relation between energy \( E \), mass \( m \), frequency \( n \), and amplitude \( a \) is:
\[
E = k \cdot m \cdot n^2 \cdot a^2
\]
