How is energy conserved in a closed physical system?

Conservation of Energy in a Closed Physical System

Definition:
The law of conservation of energy states that within a closed physical system (one that does not exchange energy with its surroundings), the total energy remains constant over time. Energy may transform between different forms (kinetic, potential, thermal, etc.), but the sum of all energies is unchanged.

Mathematically, if $E_{\text{total}}$ is the total energy at any time $t$,
$$ E_{\text{total}}(t) = \text{constant} $$

Worked Example: Pendulum

Consider a simple pendulum of mass $m$ and length $L$, swinging without friction. At its highest point, the pendulum has maximum potential energy and zero kinetic energy. At its lowest point, it has maximum kinetic energy and minimum potential energy.

Let:

• $h$ be the height above the lowest point,
• $v$ be the speed at the lowest point,
• $g$ be the acceleration due to gravity.

Step 1: Energy at the highest point (all potential)
$$ E_{\text{top}} = mgh $$

Step 2: Energy at the lowest point (all kinetic)
$$ E_{\text{bottom}} = \frac{1}{2}mv^2 $$

Step 3: Conservation of energy
Since the system is closed (no friction or air resistance),
$$ E_{\text{top}} = E_{\text{bottom}} $$
$$ mgh = \frac{1}{2}mv^2 $$

Step 4: Solve for $v$
$$ gh = \frac{1}{2}v^2 $$
$$ v = \sqrt{2gh} $$

Takeaways

• In a closed system, energy can change forms but the total amount remains constant.
• Conservation of energy allows us to predict system behavior without knowing all intermediate steps.
• This principle is fundamental across all areas of physics.