If all the forces acting on a system, such as gravity or spring forces, are conservative, the Principle of Work and Energy can be written as

This equation is referred to as the Principle of Conservation of Energy. This relationship is true since the work of a conservative force is independent of its path.

Frictional forces are not conservative forces since energy is lost to frictional heat.

Work on a rigid body is the same as work on a particle, with the addition of rotational energy.

Potential and Kinetic
Energy Relationship

Gravity is a conservative force and thus the amount of work needed to move an object through the gravitational field is the same regardless of the path.

Gravitational potential energy is directly related to how high the object is above a given reference plane (called a datum line),

     V_g = mgy

Objects below the datum line require additional work to raise them up to the datum line. Thus those objects have negative potential energy.

A compressed or stretched spring has stored energy that will produce work if released. This energy is called elastic potential energy and is modeled as

     V_s = 1/2 k s²

Recall from the Rotational Kinetic Energy and Work section section, kinetic energy for a rigid body is

     T = 1/2 m v_G² + 1/2 I_G ω²

Here, V_G is the linear velocity and ω is the angular velocity about the body's center of gravity G.