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The Work-Energy Theorem for a Variable Force

We confine ourselves to one dimension. The time rate of change of kinetic energy is
= m v
= F v (from Newton's second Law)
= F
dK = F dx

Integrating from the initial position (xi) to final position (xf), we have

where Ki and Kf are the initial and final kinetic energies corresponding to Ki and Kf .
or Kf - Ki =
Kf - Ki = W

In simpler terms it is a scalar version of Newton's Second Law.

The Conservation of Mechanical Energy

Suppose that a body undergoes displacement Δx under the action of a conservative force F. Then from the WE 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.

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.

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 given figure depicts a ball of mass m being dropped from a cliff of height H.

The conversion of potential energy to kinetic energy for a ball of mass m 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

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

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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