The Chemistry Maths Book, Second Edition

5.7 Dynamics 153

EXAMPLE 5.16A body falling under the influence of gravity

A body of mass mfalling freely under the influence of gravity experiences the constant

force F 1 = 1 mgdirected downwards. This is the total force acting on the body in the

absence of air resistance and other frictional forces. The work done by gravity on the

body as it falls through a height his therefore

This is also the work that must be done againstthe force of gravity to raise the body

through the distance h.

EXAMPLE 5.17Electrostatic work

The force acting between two electric charges q

1

and q

2

separated by distance xin a

vacuum is given by Coulomb’s inverse-square law

where ε

0

is the permittivity of a vacuum.*Like charges (charges with the same

sign, such as two nuclei or two electrons) repel, so that, in Figure 5.20, the force

acting on q

2

due to the presence of q

1

acts in the positive x-direction, away

from q

1

. Unlike charges (of opposite signs, such as the proton and electron in the

hydrogen atom) attract, and the force on q

2

is directed towards q

1

(Fis negative

in Figure 5.20).

Consider two like charges, initially infinitely far apart. Because the charges repel,

work must be done on the system to bring q

2

from infinity to the distance xfrom q

1

.

The force−Fmust be applied to overcome the repulsion, and the work done is

WFxdx

qq

dx

x

qq

xx

x

=−ZZ=− =− −

∞∞∞

()

12

0

2

12

0

44

1

ππεεxx

qq

x

















=

12

0

4 πε

Fx

qq

x

()=

12

0

2

4 πε

W F dx mg dx mgh

hh

== =ZZ

00

*The presence of the factor 4 πε

0

ensures that the force acting between two charges of one coulomb separated by

one metre is one newton;N 1 = 1 C

2

2 (4πε

0

m

2

)in terms of SI units.

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F

• •

0 x

q

1

q

2

x

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Figure 5.20