7.3 Applications of Energy Conservation

Conservation of Energy in Real-World Applications

The conservation of energy principle says that energy in a closed system is never created or destroyed. It just changes form. This makes it one of the most powerful problem-solving tools in physics: instead of tracking forces and accelerations, you can compare energy at two different points and solve for unknowns directly.

Energy conservation in action

The core idea is straightforward: total energy at one point equals total energy at another point (in a conservative system). Written out:

$KE_i + PE_i = KE_f + PE_f$

This works because energy just shifts between forms. Here's how that plays out in classic examples:

• Roller coaster: At the top of a hill, the car has maximum gravitational potential energy and minimum kinetic energy. As it drops, potential energy converts to kinetic energy, and the car speeds up. At the bottom, nearly all the energy is kinetic. At the next hill or loop, kinetic converts back to potential.
• Pendulum: At the highest point of its swing, the pendulum is momentarily at rest (all gravitational PE, zero KE). At the lowest point of its swing, it's moving fastest (all KE, minimum PE). It oscillates back and forth between these two energy states.
• Spring system: A compressed spring stores elastic potential energy ($PE_s = \frac{1}{2}kx^2$). When released, that energy converts to kinetic energy. During extension and compression, energy shifts continuously between elastic PE and KE.

Energy exchange in conservative systems

A conservative system is one where no energy is lost to friction, air resistance, or other dissipative forces. In these systems, the work done by forces depends only on starting and ending positions, not on the path taken.

The two main energy forms you'll work with:

• Gravitational potential energy: $PE_g = mgh$, where $m$ is mass, $g$ is gravitational acceleration ($9.8 \, m/s^2$), and $h$ is height above a chosen reference point
• Kinetic energy: $KE = \frac{1}{2}mv^2$, where $m$ is mass and $v$ is speed

To solve a conservation of energy problem in a conservative system:

1. Choose your two points (e.g., top of a ramp and bottom of a ramp)
2. Pick a reference height where $h = 0$
3. Write out $KE_i + PE_i = KE_f + PE_f$
4. Substitute known values and solve for the unknown

For example, if a 2 kg ball is dropped from 5 m, you can find its speed at the bottom: set $mgh = \frac{1}{2}mv^2$, cancel $m$, and solve to get $v = \sqrt{2gh} = \sqrt{2(9.8)(5)} \approx 9.9 \, m/s$.

Non-Conservative Systems and Work

Energy transformations in non-conservative systems

Most real situations involve non-conservative forces like friction, air resistance, or applied pushes and pulls. These forces are path-dependent, meaning the work they do depends on the distance traveled, not just the start and end points.

When non-conservative forces act, mechanical energy is not conserved on its own. Some of it gets converted to thermal energy (heat), sound, or deformation. A car skidding to a stop is a clear example: kinetic energy is converted to thermal energy in the brakes and tires. An object sliding down a rough incline arrives at the bottom with less kinetic energy than it would on a frictionless surface, because friction has drained some mechanical energy into heat.

Work done by non-conservative forces

When non-conservative forces are present, the energy equation gets an extra term:

$W_{nc} = \Delta KE + \Delta PE$

This says the work done by non-conservative forces equals the total change in mechanical energy. Since friction removes energy from the system, $W_{nc}$ is typically negative.

To solve problems with non-conservative forces:

1. Identify your initial and final states

2. Calculate the initial mechanical energy: $KE_i + PE_i$

3. Calculate the final mechanical energy: $KE_f + PE_f$

4. Find the difference: $W_{nc} = (KE_f + PE_f) - (KE_i + PE_i)$

5. Relate $W_{nc}$ to the non-conservative force (e.g., $W_{friction} = -f_k \cdot d$, where $f_k$ is the friction force and $d$ is the distance)

Common applications:

• Stopping distance: If you know a car's speed and the friction force from braking, you can solve for how far it travels before stopping ($KE_f = 0$, $W_{nc} = -f_k \cdot d$)
• Energy loss in machines: Comparing mechanical energy input to output tells you how much energy was lost to friction and heat in the system

The key takeaway: in conservative systems, mechanical energy is constant. In non-conservative systems, you have to track where the "missing" energy went.