In a simple harmonic oscillator, at the mean position

Options

A. (a)kinetic energy is minimum and potential energy is maximum
B. (b)both kinetic and potential energies are maximum
C. (c)kinetic energy is maximum and potential energy is minimum
D. (d)both kinetic and potential energies are minimum

Explanation

To solve the question regarding the kinetic and potential energy of a simple harmonic oscillator at the mean position, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Simple Harmonic Motion (SHM)**:
- In SHM, a particle oscillates about a mean position. The mean position is where the particle experiences no net force and is at equilibrium.

2. **Kinetic Energy (KE) and Potential Energy (PE) in SHM**:
- The kinetic energy (KE) of a particle in SHM is given by the formula:
\[ KE = \frac{1}{2} m v^2 \]
where \( m \) is the mass of the particle and \( v \) is its velocity.
- The potential energy (PE) 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.

3. **At the Mean Position**:
- At the mean position, the displacement \( x = 0 \). Therefore, the potential energy becomes:
\[ PE = \frac{1}{2} k (0)^2 = 0 \]
- The velocity of the particle is at its maximum at the mean position, which means the kinetic energy is at its maximum:
\[ KE = \frac{1}{2} m v_{max}^2 \]
- Since \( v_{max} \) is maximum at the mean position, \( KE \) is also maximum.

4. **Conclusion**:
- At the mean position of a simple harmonic oscillator, the kinetic energy is maximum and the potential energy is minimum (which is zero). Therefore, the correct statement is:
- Kinetic energy is maximum, potential energy is minimum.

### Final Answer:
The correct option is that at the mean position, the kinetic energy is maximum and the potential energy is minimum.

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