Conservation of Mechanical Energy and Efficiency

The principle of conservation of mechanical energy states that when only conservative forces such as gravity and spring forces do work, the sum of kinetic and potential energy stays constant: E_k + E_p = constant; when friction or other nonconservative forces act, mechanical energy is reduced and the efficiency is the useful output energy divided by the input energy. In brief: for a frictionless slide, ½mv² = mgh, so v = √(2gh) at the bottom. With friction, the energy lost is W_friction = E_initial − E_final, and it appears as thermal energy.

Roller coasters, pendulums and bouncing balls show the exchange of potential and kinetic energy. In practice, a part of the energy is always dissipated. This guide states the principle, shows how to use it in problems with and without friction, defines efficiency for mechanical systems, compares the cases in a table, works through examples and corrects common mistakes.

What is conserved?

The mechanical energy of a system is the sum of its kinetic and potential energy: E = E_k + E_p. If only conservative forces (gravity, elastic forces) do work, then the energy does not change: E_k,i + E_p,i = E_k,f + E_p,f. A force is conservative when the work it does depends only on the start and end points, not on the path. Normal forces and tension that act perpendicular to the motion do no work, so they do not change the mechanical energy.

Typical applications: the speed of a falling object, a pendulum swinging (energy exchanges at the lowest and highest points), a ball thrown up (kinetic energy turns into potential energy), a mass on a spring (kinetic and elastic energy), and a roller coaster that starts from rest at the top of the first hill.

What happens with friction?

Friction and air resistance are nonconservative: they transfer mechanical energy to thermal energy (and sound). The mechanical energy then decreases, but the total energy is conserved. The work-energy relation becomes

E_initial + W_other = E_final + E_dissipated, with E_dissipated = f × d for a friction force f acting over a path length d.

How is efficiency calculated?

For a mechanical system, efficiency = useful energy output ÷ energy input. For a frictionless system it would be 100 percent. If a block slides down a slope and has less kinetic energy at the bottom than the gravitational energy lost, the efficiency of the conversion is E_k,final ÷ ΔE_p. Machines, such as pulley systems and gears, have efficiencies less than 100 percent because of friction.

Cases compared

Mechanical energy with and without nonconservative forces
Feature	Only conservative forces	With friction
Mechanical energy	Constant	Decreases
Total energy	Constant	Constant (including thermal)
Speed at the bottom of a slide	√(2gh)	Less than √(2gh)
Efficiency	100 percent	Less than 100 percent

Worked example: a slide with and without friction

A 2.0 kg block starts from rest at the top of a slide 1.5 m high. Use g = 9.8 m/s².

Initial energy: E_p = mgh = 2.0 × 9.8 × 1.5 = 29.4 J, E_k = 0.
Frictionless: at the bottom, all is kinetic: ½ × 2.0 × v² = 29.4, so v = √29.4 = 5.4 m/s.
With friction: the measured speed at the bottom is 4.0 m/s. E_k = ½ × 2.0 × 4.0² = 16 J.
Energy dissipated: 29.4 − 16 = 13.4 J, transferred to thermal energy of the block and slide.
Efficiency of the conversion: 16 ÷ 29.4 = 0.54, or 54 percent.
Average friction force: if the slide is 3.0 m long, f = 13.4 ÷ 3.0 = 4.5 N.

A second example: a spring (k = 400 N/m) compressed by 0.10 m launches a 0.10 kg ball upward, with 2.0 J of elastic energy. If the ball rises to 1.5 m only, the gravitational energy gained is 0.10 × 9.8 × 1.5 = 1.47 J, so the efficiency is 1.47 ÷ 2.0 = 74 percent, and 0.53 J is lost to friction and air resistance.

Energy bars for the 2.0 kg block on the 1.5 m slide. The 29.4 J of potential energy at the top becomes 29.4 J of kinetic energy without friction. With friction, only 16 J is kinetic at the bottom and 13.4 J is dissipated as thermal energy, so the total is still 29.4 J.

How was the principle developed?

Galileo observed that a pendulum bob rises to the same height from which it was released, even if the string is caught by a nail. Christiaan Huygens and Leibniz expressed this in terms of vis viva in the 17th century, and Lagrange gave the energy principle a general form in 1788. The recognition of heat as energy in the 19th century made it possible to include friction in the total energy balance.

Which mistakes are common?

Mistake 1: Applying conservation of mechanical energy when there is friction

Include the energy dissipated, or use the work-energy theorem.

Mistake 2: Saying that the energy is destroyed by friction

The mechanical energy is converted into thermal energy. Total energy is conserved.

Mistake 3: Ignoring the shape of the track

Without friction, the speed at a given height depends only on the height, not on the shape of the path.

Mistake 4: Forgetting the mass cancels

In free-fall and frictionless slides, the speed does not depend on the mass.