Show that the total energy, kinetic plus potential, is constant.

To show that the total energy (kinetic energy + potential energy) of a system is constant, let’s consider a simple system of a particle with mass \( m \) moving under a conservative force field, where the potential energy \( U(x) \) is a function of position \( x \) only. 

### 1. Define the Kinetic and Potential Energy

1. **Kinetic Energy (K):** For a particle of mass \( m \) moving with velocity \( v = \frac{dx}{dt} \), the kinetic energy is:
   \[
   K = \frac{1}{2}mv^2
   \]

2. **Potential Energy (U):** The potential energy \( U(x) \) is given as a function of position \( x \).

### 2. Total Energy (E)

The total energy \( E \) of the system is the sum of kinetic and potential energy:
\[
E = K + U = \frac{1}{2}mv^2 + U(x)
\]

### 3. Differentiate Total Energy with Respect to Time

To determine if \( E \) is constant, differentiate \( E \) with respect to time \( t \):
\[
\frac{dE}{dt} = \frac{d}{dt} \left( \frac{1}{2}mv^2 + U(x) \right)
\]

Using the chain rule for differentiation:

\[
\frac{dE}{dt} = m v \frac{dv}{dt} + \frac{dU}{dx} \frac{dx}{dt}
\]

Since \( v = \frac{dx}{dt} \), this can be simplified to:
\[
\frac{dE}{dt} = mv \frac{dv}{dt} + \frac{dU}{dx} v
\]

### 4. Substitute \( \frac{dv}{dt} \) Using Newton's Second Law

From Newton's second law, the force \( F \) acting on the particle is:
\[
F = m \frac{dv}{dt} = -\frac{dU}{dx}
\]

Thus, \( \frac{dv}{dt} = -\frac{1}{m} \frac{dU}{dx} \). Substitute this into the energy derivative:

\[
\frac{dE}{dt} = mv \left(-\frac{1}{m} \frac{dU}{dx}\right) + \frac{dU}{dx} v
\]

Simplify:
\[
\frac{dE}{dt} = -v \frac{dU}{dx} + v \frac{dU}{dx} = 0
\]

### Conclusion
Since \( \frac{dE}{dt} = 0 \), the total energy \( E \) is constant over time. This shows that the total energy (kinetic + potential) in a conservative system is conserved.

Would you like further details or have questions about any part of this derivation?

Here are five related questions to deepen your understanding:

1. How does this derivation change if a non-conservative force is present?
2. What role does potential energy play in oscillatory motion?
3. How does energy conservation apply to rotational motion?
4. What are real-world examples where kinetic and potential energy are interchanged?
5. How does this concept apply to quantum mechanics?

**Tip:** Remember that in a closed system with only conservative forces, energy conservation holds true.

Explore a step-by-step derivation of the conservation of total energy in a system. This tutorial covers key concepts, such as kinetic and potential energy, the law of conservation of energy, and Newton’s second law. Ideal for undergraduate physics students.

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