The Chemistry Maths Book, Second Edition

156 Chapter 5Integration

We note that, whilst the kinetic energy has a well defined absolute value, this is not

true of the potential energy, for which only relative values are defined by equations

(5.56) and (5.57). Examples 5.18 and 5.19 below show how the zero of potential

energy is chosen in two different physical situations.

A system in which all the forces are conservative is called a conservative system.

In such a system, the work done in moving a body round a closed loop A 1 → 1 B 1 → 1 Ais

zero:

W

ABA

1 = 1 W

AB

1 + 1 W

BA

1 = 1 (V

A

1 − 1 V

B

) 1 + 1 (V

B

1 − 1 V

A

) 1 = 10 (5.61)

Dissipative forces such as those due to friction are not conservative forces because the

work done against friction is always positive.

Combining the expressions (5.54) and (5.56) we have the result

T

A

1 + 1 V

A

1 = 1 T

B

1 + 1 V

B

(5.62)

and it follows that, in a conservative system, the quantity T 1 + 1 Vis constant. This

quantity is called the total energyof the system,E 1 = 1 T 1 + 1 V, and (5.62) is an expression

of the principle of the conservation of energy: if the forces acting on a body are

conservative, then the total energy of the body,T 1 + 1 V, is conserved.

EXAMPLE 5.18A body moving under the influence of gravity

Consider a body of mass mat height habove a horizontal

surface, as in Figure 5.22. The force of gravity acting on

the body isF 1 = 1 −mg(negative because the force acts in the

negative x-direction) and the work done on the body as it

falls freely from heightx 1 = 1 honto the surface atx 1 = 10 is

This work is the change of potential energy,

W 1 = 1 mgh 1 = 1 V(h) 1 − 1 V(0)

whereV(x)is the potential energy of the body at height x. The natural choice of zero

of potential energy in this example isV(0) 1 = 10 , zero at the surface. ThenV(x) 1 = 1 mgx

is the potential energy of the body at height x, and the force is related to it by

F 1 = 1 −dV 2 dx 1 = 1 −mg.

Let the body fall from rest atx 1 = 1 hand let the kinetic energy at heightxbeT(x).

Then T(h) 1 = 10 and, by equation (5.54), the kinetic energy of the body when it reaches

the surface is T(0) 1 = 1 mgh. In addition, because the force (a constant) is conservative,

the total energy of the body is conserved and is equal to E 1 = 1 mgh, which is the potential

energy at x 1 = 1 h(where the kinetic energy is zero) and is the kinetic energy atx 1 = 10

W F dx mg dx mgh

hh

==− =ZZ

00

Figure 5.22