The total energy of a simple harmonic oscillation is proportional to

Options

A. amplitude
B. square of amplitude
C. frequency
D. velocity

Explanation

To solve the question regarding the total energy of a simple harmonic oscillator (SHM) and its proportionality to various parameters, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Total Energy in SHM**:
The total energy (E) in simple harmonic motion is the sum of kinetic energy (KE) and potential energy (PE).
\[ E = KE + PE \]

2. **Expressions for Kinetic and Potential Energy**:
The kinetic energy of a particle in SHM is given by:
\[ KE = \frac{1}{2} mv^2 \]
The potential energy is given by:
\[ PE = \frac{1}{2} kx^2 \]

3. **Relating Parameters**:
In SHM, the angular frequency (\(\omega\)) is related to the spring constant (k) and mass (m) by:
\[ \omega = \sqrt{\frac{k}{m}} \implies k = m\omega^2 \]

4. **Displacement and Velocity in SHM**:
The displacement of a particle in SHM can be expressed as:
\[ x = A \sin(\omega t) \]
The velocity is:
\[ v = A \omega \cos(\omega t) \]

5. **Substituting into Total Energy**:
Now, substituting the expressions for kinetic and potential energy into the total energy equation:
\[ E = \frac{1}{2} m (A \omega \cos(\omega t))^2 + \frac{1}{2} k (A \sin(\omega t))^2 \]

6. **Simplifying Total Energy**:
Replacing \(k\) with \(m\omega^2\):
\[ E = \frac{1}{2} m A^2 \omega^2 \cos^2(\omega t) + \frac{1}{2} m \omega^2 A^2 \sin^2(\omega t) \]
Factoring out common terms:
\[ E = \frac{1}{2} m A^2 \omega^2 (\cos^2(\omega t) + \sin^2(\omega t)) \]

7. **Using Trigonometric Identity**:
Using the identity \(\cos^2(\theta) + \sin^2(\theta) = 1\):
\[ E = \frac{1}{2} m A^2 \omega^2 \cdot 1 \]
Thus, we find:
\[ E = \frac{1}{2} m A^2 \omega^2 \]

8. **Conclusion**:
From the final expression, we can conclude that the total energy of the simple harmonic oscillator is proportional to the square of the amplitude (\(A^2\)):
\[ E \propto A^2 \]

### Final Answer:
The total energy of a simple harmonic oscillator is proportional to the square of the amplitude.

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