Mechanical Energy: Definition and the Conservation Formula

Work: the condition for transferring energy

According to HyperPhysics, energy is defined as the capacity to do work. Work, in turn, is what a force produces when it causes a displacement along its own direction. If a 20-newton force moves an object 5 metres along its own direction, the work done comes to 100 joules: the International System unit for work, equal to one newton times one metre.

For a constant force, work is calculated by multiplying the force’s magnitude, the displacement, and the cosine of the angle between the two directions (W = F·d·cosθ).

Practical example: a person pushes a lawnmower with a 75-newton force, tilted 35 degrees below horizontal, over 25 metres. According to OpenStax, the work done in this case comes out to about 1,540 joules, less than the same force would do pushing flat, because only part of it points in the direction of motion.

The same principle, applied to electric force, underlies electric current. The description of forces that makes it possible to calculate work in the first place comes from Newton’s three laws of motion.

The work-energy theorem and kinetic energy

Kinetic energy is the energy an object has simply because it’s moving. Giving it to an object requires doing work on it: that’s a specific case of a broader principle, the work-energy theorem, which ties the two quantities together in a single rule.

An object’s kinetic energy is calculated as half the product of its mass and the square of its velocity, KE = ½mv². It also measures the work that moving object could do on whatever it hits, purely because of its motion: a car travelling at 100 km/h doesn’t have twice the kinetic energy of one at 50 km/h, it has four times as much, because velocity is squared in the formula.

A simple pendulum one metre long, released from 30 degrees with no friction, reaches a speed of about 1.62 m/s at the lowest point of its swing. That calculation links work and velocity in a case of oscillating motion.

Potential energy and the conservation formula

An object’s mechanical energy is defined as the sum of its kinetic energy and its potential energy. In its gravitational form, potential energy is calculated by multiplying an object’s mass, the acceleration of gravity, and its height above a chosen reference level, PE = mgh.

When an object with gravitational potential energy falls, that energy converts into kinetic energy. A 10-kilogram rock falling from a 20-metre cliff turns potential energy into kinetic energy as it drops. In a pendulum, the same exchange runs both ways: the kinetic energy at the lowest point of the swing equals the potential energy at the highest point, in a continuous back-and-forth between the two forms, with the total staying the same.

An object’s mechanical energy stays constant as long as any non-conservative forces, such as friction, do no work on it: in that case, the change in mechanical energy between two instants is zero. It’s the same broader principle behind the fact that, when a system doesn’t interact with its surroundings, certain quantities, energy among them, stay constant, the so-called conserved quantities: in an isolated system, total energy is neither created nor destroyed. The formula that captures the conservation of mechanical energy is Em = KE

• PE = constant.

When mechanical energy isn’t conserved

When non-conservative forces such as friction act, mechanical energy is no longer exactly conserved: its change equals the work those forces do. With dissipative forces such as air resistance, the difference between initial and final mechanical energy is exactly the energy that got dissipated, 130 kJ in one worked example from OpenStax. On a roller coaster, as the car goes down the first drop, potential energy converts into kinetic energy, and by the lowest point it has already picked up most of the potential energy it started with. Treating a system as closed, in this context, means assuming no energy is lost to friction or air resistance along the way; real systems rarely match that assumption exactly.

Energy conservation remains one of physics’s foundational principles, alongside the conservation of mass and of momentum. The same transformation between forms of energy, made possible by the work a force does, also underlies how a nuclear power plant works, where steam does work on the turbines.