A particle is executing simple harmonic motion. Its total energy is proportional to its

Options

A. displacement from equilibrium position
B. frequency of oscillation
C. square of amplitude of motion
D. velocity at equilibrium position

Answer

To solve the problem, we need to determine how the total energy of a particle executing simple harmonic motion (SHM) is related to its amplitude. Here’s a step-by-step breakdown of the solution:

### Step 1: Understand the Total Energy in SHM
The total energy (E) of a particle in simple harmonic motion is given by the sum of its potential energy (PE) and kinetic energy (KE):
\[ E = PE + KE \]

### Step 2: Write the Expression for Potential Energy
The potential energy of a particle in SHM is given by:
\[ PE = \frac{1}{2} k x^2 \]
where \( k \) is the spring constant and \( x \) is the displacement from the mean position.

### Step 3: Write the Expression for Kinetic Energy
The kinetic energy of the particle is given by:
\[ KE = \frac{1}{2} m v^2 \]
where \( m \) is the mass of the particle and \( v \) is its velocity.

### Step 4: Express Velocity in Terms of Angular Frequency
The velocity \( v \) in SHM can be expressed as:
\[ v = \omega \sqrt{A^2 - x^2} \]
where \( \omega \) is the angular frequency and \( A \) is the amplitude of the motion.

### Step 5: Substitute Velocity into Kinetic Energy
Substituting the expression for \( v \) into the kinetic energy formula gives:
\[ KE = \frac{1}{2} m (\omega \sqrt{A^2 - x^2})^2 \]
This simplifies to:
\[ KE = \frac{1}{2} m \omega^2 (A^2 - x^2) \]

### Step 6: Combine Potential and Kinetic Energy
Now, substituting the expressions for potential and kinetic energy into the total energy equation:
\[ E = \frac{1}{2} k x^2 + \frac{1}{2} m \omega^2 (A^2 - x^2) \]

### Step 7: Simplify the Total Energy Expression
Using the relationship \( k = m \omega^2 \), we can rewrite the total energy:
\[ E = \frac{1}{2} m \omega^2 A^2 \]
This shows that the total energy is constant and depends on the amplitude.

### Step 8: Conclude the Relationship
From the expression \( E = \frac{1}{2} m \omega^2 A^2 \), we can conclude that the total energy is proportional to the square of the amplitude:
\[ E \propto A^2 \]

### Final Answer
The total energy of a particle executing simple harmonic motion is proportional to the square of its amplitude.

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