Conservation of energy

Conservation of energy is an accounting principle: energy can move between objects and change form, but the total energy of a closed system does not disappear. The hard part is usually choosing the system and tracking which forms of energy are relevant.

Energy accounting

ΔE_system = energy transferred into the system − energy transferred out

In mechanics, the most common stores are kinetic energy K, gravitational potential energy U_g, elastic potential energy U_s and thermal energy. Work and heat are transfers of energy, not substances stored inside an object.

Energy can change form without changing the total. For a frictionless descent, gravitational potential energy decreases while kinetic energy increases.

Kinetic and potential energy

For translational motion, K = ½mv². Because speed is squared, doubling speed quadruples kinetic energy.

Near Earth's surface, a convenient gravitational potential-energy model is U_g = mgh. Only changes in U matter, so the zero height can be chosen for convenience. For an ideal spring, U_s = ½kx².

Energy explorer

For an object near Earth's surface, compare gravitational potential and kinetic energy.

Mass m (kg) Height h (m) Speed v (m/s)

Mechanical energy conservation

If only conservative forces such as gravity or an ideal spring do work, the mechanical energy K + U is constant:

K_i + U_i = K_f + U_f

This lets you compare two states without calculating every intermediate acceleration or force.

Conservative forces have a useful path-independence property: their work between two positions depends only on the endpoints. That is why gravity and an ideal spring can be represented with potential energy. Friction is different; the energy it dissipates generally depends on the path length and details of the motion.

Friction changes the mechanical-energy balance

“Lost” mechanical energy is transferred, not destroyed. Friction commonly turns organized mechanical energy into thermal energy.

When friction, drag or an applied force transfers energy, K + U need not stay constant. A useful form is W_nc = Δ(K + U) for work by non-conservative forces on the chosen system. If the system includes the surfaces that warm up, thermal energy can be tracked explicitly and total energy remains conserved.

Choose the system before writing an equation

For a falling ball alone, gravity transfers energy into the ball by work. For the larger ball–Earth system, gravity is internal and the same change is represented as a decrease in gravitational potential energy. Both descriptions can be correct; mixing them in one equation causes double-counting.

Before calculating, state the initial state, final state, system boundary and any energy crossing that boundary.

Energy conservation does not mean every form of energy stays constant. A falling object can gain kinetic energy while gravitational potential energy falls. A sliding block can lose mechanical energy while thermal energy rises. The invariant quantity is the total energy of an appropriately closed system.

Worked examples

1. Falling without air resistance

A 2.0 kg object starts from rest 5.0 m above the ground. What speed does it have just before reaching the ground?

Solution

With the ground as U = 0 and no dissipative work, mgh = ½mv². The mass cancels.

v = √(2gh) = √(2 × 9.81 × 5.0) = 9.90 m s⁻¹.

2. A spring launch

A spring with k = 200 N m⁻¹ is compressed by 0.10 m and launches a 0.50 kg cart on a frictionless track. Find the launch speed after the spring returns to its relaxed length.

Solution

½kx² = ½mv², so v = x√(k/m) = 0.10√(200/0.50) = 2.0 m s⁻¹.

3. Mechanical energy with friction

A 3.0 kg block loses 18 J of mechanical energy to friction while descending. If its gravitational potential energy decreases by 60 J, by how much does its kinetic energy change?

Solution

The 60 J decrease in U supplies energy. Of that, 18 J becomes thermal, leaving 42 J to increase K.