For a particle moving in vertical circle, the total energy at different positions along the path

Options

A. is conserved
B. increases
C. decreases
D. may increases or decreases

Explanation

To analyze the total energy of a particle moving in a vertical circle, we can break down the problem step by step.

### Step 1: Understanding the System
The particle is moving in a vertical circle, which means it experiences both kinetic energy (due to its motion) and potential energy (due to its height relative to a reference point).

### Step 2: Identifying the Reference Point
We choose the lowest point of the circle as our reference point for potential energy (h = 0). At this point, the potential energy (PE) is zero, and the total energy will be purely kinetic energy (KE).

### Step 3: Energy at the Lowest Point
At the lowest point:
- Height (h) = 0
- Potential Energy (PE) = mgh = 0
- Kinetic Energy (KE) = (1/2)mv² (where v is the velocity at this point)
- Total Energy (E_total) = KE + PE = (1/2)mv² + 0 = (1/2)mv²

### Step 4: Energy at Other Points
As the particle moves up in the vertical circle:
- At any height h, the potential energy becomes PE = mgh.
- The kinetic energy will change as the particle moves, depending on its speed at that height.

### Step 5: Conservation of Mechanical Energy
Since there are no non-conservative forces acting on the particle (like friction), the mechanical energy is conserved. This means:
- Total Energy at any point in the circle = Total Energy at the lowest point.
- Therefore, (1/2)mv² + mgh = constant.

### Step 6: Analyzing Different Positions
1. **At the Lowest Point (h = 0)**:
- Total Energy = (1/2)mv₀² (where v₀ is the speed at the lowest point).

2. **At the Highest Point (h = 2R, where R is the radius of the circle)**:
- Total Energy = (1/2)mv_h² + mg(2R) = constant.
- Here, the potential energy is maximum, and the kinetic energy is minimum.

3. **At Any Intermediate Point**:
- Total Energy = (1/2)mv² + mgh = constant.

### Conclusion
The total energy remains constant throughout the motion of the particle in the vertical circle, provided no non-conservative forces are acting on it. The energy is continuously converted between kinetic and potential forms as the particle moves along the path.