In conservative systems, we can define another form of energy, based on the configuration of the parts of the system, which we call potential energy. This quantity is related to work, and thus kinetic energy, through a simple equation. Using this relation we can finally quantify all mechanical energy, and prove the conservation of mechanical energy in conservative systems.
Since mechanical energy must be conserved under conservative forces, but the kinetic energy can fluctuate based on the speed of the particles in the system, there must be an additional quantity of energy that is a property of the structure of the system. This quantity, potential energy, is denoted by the symbol U and can be easily derived from our knowledge of conservative systems.
Consider a system under the action of a conservative force. When work is done on the system it must in some way change the velocity of its constituent parts (by the Work Energy Theorem), and thus change the configuration of the system. We define potential energy as the energy of configuration of a conservative system, and relate it to work in the following way:
|ΔU = - W|
In other words, work applied by a conservative force reduces the energy of configuration of a system (potential energy), converting it to kinetic energy.
To see exactly how this conservation works, let's derive the expression for the potential energy of a system acted upon by gravity. Consider a ball of mass m dropped from a height h. The only force acting upon the ball is gravity, so we know the system is conservative, since we proved it last section. How much work is done during the fall? A constant gravitational force of mg acts over a distance of h, so W = mgh. Thus, over the course of the fall, the potential energy is reduced by a factor of - mgh. We may define the potential energy to be zero when the ball hits the ground and calculate the potential energy at height h: ΔU = Uf - Uo = - mgh. Thus:
|UG = mgh|
Since our choice of height h was arbitrary, this equation holds for all h relatively close to the center of the earth, and the equation is a universal definition of the gravitational potential energy.
An important property of energy is that it is a relative quantity. Just as observers moving with different velocities observe different values for the kinetic energy of a given particle, observers at a different height observe different values for gravitational potential energy, for example. When working problems we are free to choose whatever origin we like, to correspond to a convenient value for our potential energy.
Having defined potential energy, we can now see how it relates to kinetic energy, and generate our principle of conservation of mechanical energy.