A particle of mass m executes simple harmonic motion with amplitude a and frequency v. The average kinetic energy during its motion from the position of equilibrium to the ends is

Options

A. `2 pi^(2)ma^(2)v^(2)`
B. `pi^(2)ma^(2)v^(2)`
C. `(1)/(4)pi^(2)ma^(2)v^(2)`
D. `4pi^(2),a^(2)v^(2)`

Explanation

To find the average kinetic energy of a particle executing simple harmonic motion (SHM) from the position of equilibrium to the extreme positions, we can follow these steps:

### Step 1: Understand the Motion
The particle oscillates between two extreme positions (maximum displacement) and passes through the equilibrium position (mean position). At the extreme positions, the velocity is zero, and at the equilibrium position, the velocity is maximum.

### Step 2: Identify Kinetic Energy at Key Points
1. **At the Equilibrium Position (Mean Position)**:
- The velocity is maximum.
- Kinetic Energy (K1) at this position can be expressed as:
\[ K_1 = \frac{1}{2} m v^2 \]
where \(v\) is the maximum velocity.

2. **At the Extreme Position**:
- The velocity is zero.
- Kinetic Energy (K2) at this position is:
\[ K_2 = 0 \]

### Step 3: Calculate Maximum Velocity
The maximum velocity \(v\) in SHM can be related to the amplitude \(a\) and angular frequency \(\omega\):
\[ v = \omega a \]
where \(\omega = 2\pi f\) and \(f\) is the frequency.

### Step 4: Substitute Maximum Velocity into Kinetic Energy
Substituting \(v\) into the expression for kinetic energy at the equilibrium position:
\[ K_1 = \frac{1}{2} m (\omega a)^2 = \frac{1}{2} m \omega^2 a^2 \]

### Step 5: Calculate Average Kinetic Energy
The average kinetic energy (\(K_{avg}\)) from the equilibrium position to the extreme position can be calculated as:
\[ K_{avg} = \frac{K_1 + K_2}{2} = \frac{\frac{1}{2} m \omega^2 a^2 + 0}{2} = \frac{1}{4} m \omega^2 a^2 \]

### Step 6: Substitute \(\omega\) in Terms of Frequency
Substituting \(\omega = 2\pi f\):
\[ K_{avg} = \frac{1}{4} m (2\pi f)^2 a^2 = \frac{1}{4} m \cdot 4\pi^2 f^2 a^2 = m \pi^2 f^2 a^2 \]

### Final Answer
Thus, the average kinetic energy during its motion from the position of equilibrium to the ends is:
\[ K_{avg} = m \pi^2 f^2 a^2 \]

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