A particle of mass m is hanging vertically by an ideal spring of force constant K . If the mass is made to oscillate vertically, its total energy is

Options

A. Maximum at extreme position
B. Maximum at mean position
C. Minimum at mean position
D. Same at all position

Answer

To solve the problem regarding the total energy of a particle of mass \( m \) hanging vertically by an ideal spring of force constant \( K \) and oscillating vertically, we can follow these steps:

### Step 1: Understanding the System
The particle is attached to a spring and is allowed to oscillate vertically. The forces acting on the particle include the gravitational force and the spring force.

### Step 2: Define the Energy Types
In this system, we have two main types of energy:
- **Potential Energy due to Gravity (PE_gravity)**: This is given by \( PE_{gravity} = mgh \), where \( h \) is the height of the mass from a reference point.
- **Potential Energy of the Spring (PE_spring)**: This is given by \( PE_{spring} = \frac{1}{2} k x^2 \), where \( x \) is the displacement from the spring's natural length.

### Step 3: Total Mechanical Energy
The total mechanical energy \( E \) in the system is the sum of the gravitational potential energy and the spring potential energy:
\[ E = PE_{gravity} + PE_{spring} \]

### Step 4: Energy at Different Positions
- **At the Mean Position**: The particle has maximum kinetic energy and minimum potential energy. The total energy is still conserved.
- **At the Extreme Positions**: The particle has maximum potential energy (either gravitational or spring) and zero kinetic energy.

### Step 5: Conservation of Energy
Throughout the oscillation, the total energy remains constant. It transforms between kinetic energy and potential energy but the total energy does not change.

### Conclusion
Thus, the total energy of the system is the same at all positions during the oscillation. Therefore, the answer to the question is that the total energy is the same at all positions.

### Final Answer
The total energy is the same at all positions.

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