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The Conservation of Mechanical Energy

For simplicity we demonstrate this important principle for one-dimensional motion. Suppose that a body undergoes displacement Δx under the action of a conservative force F. 

Then from the Work-Energy theorem we have,
ΔK = F(x) Δx

If the force is conservative, the potential energy function V(x) can be defined such that
-ΔV = F(x)Δx

The above equations imply that
ΔK +ΔV = 0
Δ(K +V) = 0

which means that K + V, the sum of the kinetic and potential energies of the body is a constant. Over the whole path, xi - xf this means that
Ki + V(xi) = Kf + V(xf)

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 a constant. The aptness of the term 'conservative force' is now clear.

Let us recapitulate the various definitions of a conservative force.

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.

The work done by the conservative force depends only on the end points. This can be seen from the relation,
W = Kf - Ki = V(xi) - V(xf)

Which depends on the end points.

A third definition states that the work done by this force in a closed path is zero. 
Since xi = xf.

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.

The ball's total mechanical energies Eo, Eh and EH at the indicated heights zero (ground level), h and H are
EH = mgH
Eh = mgh +
Eo = (1/2) m

The constant force is a special case of a spatially dependent force F(x). Hence, the mechanical energy is conserved. Thus
EH = Eo
Or, mgH =
vf =
EH = Eh

Which implies,
= 2g(H - h) and is a familiar result from kinematics.

At the height H, the energy is purely potential. It is partially converted to kinetic at height h and is fully kinetic at ground level. This illustrates the conservation of mechanical energy.

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