a constant. The aptness of the term ‘conservative force’ is now clear. Let us consider some of the definitions of a conservative force. l A force F ( x ) is conservative if it can be derived from a scalar quantity V ( x ) by the relation given by Eq.
( . ). The three-dimensional generalisation requires the use of a vector derivative, which is outside the scope of this book. l The work done by the conservative force depends only on the end points.
This can be seen from the relation, W = K f – K i = V ( x i ) – V ( x f ) which depends on the end points. l A third definition states that the work done by this force in a closed path is zero. This is once again apparent from Eq. ( .
) since x i = x f . Thus, the principle of conservation of total mechanical energy can be stated as The total mechanical energy of a system is conserved if the forces, doing work on it, are conservative . The above discussion can be made more concrete by considering the example of the gravitational force once again and that of the spring force in the next section. Fig.
. depicts a ball of mass m being dropped from a cliff of height H . Fig. .
The conversion of potential energy to kinetic energy for a ball of mass m dropped from a height H. The total mechanical energies E , E h , and E H of the ball at the indicated heights zero (ground level), h and H , are E H = mgH ( . a) h h E mgh ( . b) E = ( / ) mv f ( .
c) The constant force is a special case of a spatially dependent force F ( x ). Hence, the mechanical energy is conserved. Thus E H = E or, mgH gH a result that was obtained in section . for a freely falling body.
Further, E H = E h which implies,