Using the method of dimensions , derive an expressions for the energy of a body executing SHM , assuming this energy depends upon its mass m , frequency v and amplitude of vibration r .

Answer

To derive the expression for the energy of a body executing Simple Harmonic Motion (SHM) using the method of dimensions, we will follow these steps:

### Step 1: Identify the Variables
We know that the energy (E) of a body in SHM depends on:
- Mass (m)
- Frequency (v)
- Amplitude (r)

### Step 2: Assign Dimensions
We need to express energy in terms of its dimensions. The formula for kinetic energy is given by:
\[ E = \frac{1}{2} mv^2 \]
The dimensions of energy (E) are:
- Mass (m): \([M^1]\)
- Length (l): \([L^2]\)
- Time (t): \([T^{-2}]\)

Thus, the dimensions of energy are:
\[ [E] = [M^1][L^2][T^{-2}] \]

### Step 3: Express the Variables in Terms of Dimensions
Now, we express the dimensions of the variables:
- Mass (m): \([M^1]\)
- Frequency (v): Frequency is the reciprocal of time period, so its dimension is \([T^{-1}]\)
- Amplitude (r): Amplitude is a measure of length, so its dimension is \([L^1]\)

### Step 4: Formulate the Equation
We assume that the energy can be expressed as:
\[ E = k \cdot m^x \cdot v^y \cdot r^z \]
where \(k\) is a dimensionless constant and \(x\), \(y\), and \(z\) are the powers we need to determine.

### Step 5: Substitute the Dimensions
Substituting the dimensions into the equation gives us:
\[ [E] = [M^1][L^2][T^{-2}] = [M^x][T^{-y}][L^z] \]

### Step 6: Equate the Dimensions
Now we equate the dimensions on both sides:
- For mass: \(1 = x\)
- For length: \(2 = z\)
- For time: \(-2 = -y\) or \(y = 2\)

### Step 7: Solve for x, y, z
From the equations we derived:
- \(x = 1\)
- \(y = 2\)
- \(z = 2\)

### Step 8: Write the Final Expression
Substituting the values of \(x\), \(y\), and \(z\) back into our equation gives us:
\[ E = k \cdot m^1 \cdot v^2 \cdot r^2 \]
Since \(k\) is a dimensionless constant, we can write:
\[ E = C \cdot m \cdot v^2 \cdot r^2 \]
where \(C\) is a constant that can be determined experimentally.

### Conclusion
Thus, the expression for the energy of a body executing SHM, assuming it depends on mass, frequency, and amplitude, can be expressed as:
\[ E = C \cdot m \cdot v^2 \cdot r^2 \]