plane of height h, its speed at the bottom is gh irrespective of the angle of inclination. Thus, at the bottom of the inclined plane it acquires a kinetic energy, mgh . If the work done or the kinetic energy did depend on other factors such as the velocity or the particular path taken by the object, the force would be called non- conservative. The dimensions of potential energy are [ML T – ] and the unit is joule (J), the same as kinetic energy or work.
To reiterate, the change in potential energy, for a conservative force, ∆ V is equal to the negative of the work done by the force ∆ V = − F ( x ) ∆ x ( . ) In the example of the falling ball considered in this section we saw how potential energy was converted to kinetic energy. This hints at an important principle of conservation in mechanics, which we now proceed to examine. .
THE CONSERVATION OF MECHANICAL ENERGY For simplicity we demonstrate this important principle for one-dimensional motion. Suppose that a body undergoes displacement ∆ x under the action of a conservative force F . Then from the WE theorem we have, ∆ K = F ( x ) ∆ x If the force is conservative, the potential energy function V ( x ) can be defined such that − ∆ V = F ( x ) ∆ x The above equations imply that ∆ K + ∆ V = ∆ ( K + V ) = ( . ) which means that K + V, the sum of the kinetic and potential energies of the body is a constant.
Over the whole path, x i to x f , this means that K i + V ( x i ) = K f + V ( x f ) ( . ) The quantity K + V ( x ), is called the total mechanical energy of the system. Individually the kinetic energy K and the potential energy V ( x ) may vary from point to point, but the sum is