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To apply the principle of energy conservation to the system, we need to analyze the initial and final energy states in terms of kinetic and potential energies.

### Step-by-Step Solution

Let’s break down the variables involved:

- \( K_i \): Initial kinetic energy
- \( K_f \): Final kinetic energy
- \( U_i \): Initial potential energy
- \( U_f \): Final potential energy

The principle of energy conservation states that the total mechanical energy (sum of kinetic and potential energy) in a closed system remains constant if only conservative forces are acting (e.g., gravity). Therefore:

\[
K_i + U_i = K_f + U_f
\]

Given variables:
- \( \theta \): Initial angle of the object with respect to some reference direction
- \( \phi \): Final angle of the object
- \( L \): Length of the path or pendulum arm (if relevant)
- \( g \): Acceleration due to gravity
- \( m \): Mass of the object
- \( v_i \): Initial velocity of the object
- \( v_f \): Final velocity of the object

### Setting up Initial and Final Energies

1. **Initial Kinetic Energy, \( K_i \):**
   \[
   K_i = \frac{1}{2} m v_i^2
   \]

2. **Final Kinetic Energy, \( K_f \):**
   \[
   K_f = \frac{1}{2} m v_f^2
   \]

3. **Initial Potential Energy, \( U_i \):**
   Assuming the height \( h_i \) depends on the initial angle \( \theta \), we can write:
   \[
   U_i = m g h_i
   \]
   where \( h_i = L (1 - \cos \theta) \). So:
   \[
   U_i = m g L (1 - \cos \theta)
   \]

4. **Final Potential Energy, \( U_f \):**
   Similarly, if the final height \( h_f \) depends on the angle \( \phi \), we have:
   \[
   U_f = m g h_f
   \]
   where \( h_f = L (1 - \cos \phi) \). Thus:
   \[
   U_f = m g L (1 - \cos \phi)
   \]

### Conservation of Energy Equation

Substituting these expressions into the conservation of energy equation:

\[
\frac{1}{2} m v_i^2 + m g L (1 - \cos \theta) = \frac{1}{2} m v_f^2 + m g L (1 - \cos \phi)
\]

This equation represents the conservation of energy in terms of the given variables. Simplifying, we can cancel the mass \( m \) from all terms if it is non-zero:

\[
\frac{1}{2} v_i^2 + g L (1 - \cos \theta) = \frac{1}{2} v_f^2 + g L (1 - \cos \phi)
\]

### Final Answer

So, the principle of energy conservation for this system is given by:

\[
\frac{1}{2} v_i^2 + g L (1 - \cos \theta) = \frac{1}{2} v_f^2 + g L (1 - \cos \phi)
\]

This equation allows us to find relationships between initial and final velocities, angles, and heights based on energy conservation.

Would you like more details on any specific part?

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### Related Questions

1. How can this equation be used to solve for \( v_f \) in terms of the initial conditions?
2. How does changing the length \( L \) affect the kinetic and potential energy terms?
3. If the initial velocity \( v_i \) is zero, what is the final velocity \( v_f \) in terms of the angles and other constants?
4. What would happen if there were non-conservative forces, such as friction, in the system?
5. How would the conservation equation change if the mass \( m \) varied?

### Tip

When working with conservation of energy problems, identify each term in the initial and final states separately before equating them. This will help in setting up equations accurately.

Apply the principle of energy conservation to the system in terms of initial and final kinetic and potential energies, represented as Ki, Kf, Ui, and Uf, in terms of θ, ϕ, L, g, m, vi, and vf.

Explore the principle of energy conservation by analyzing the initial and final kinetic and potential energies in a system. Learn how to use kinetic and potential energy formulas with variables θ, ϕ, length L, mass m, gravitational constant g, and velocities vi and vf. Ideal for physics students in Grades 11-12.

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