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.

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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.