Environmental Engineering Reference
In-Depth Information
provides us with the opportunity to solve for the motion of two gravitating bodies
without too much hard work. The general motion for more than two bodies is rather
more complicated and we won't address any problems of that nature, although of
course no new physics is involved.
In Section 3.2, we showed that a uniform gravitational field of force is 'con-
servative' and hence that it can be described using a potential energy function.
Specifically, we showed that the work done against gravity in moving a body
around in a uniform gravitational field does not depend upon the details of the
body's journey, rather it just depends on the difference in height between its start-
ing and finishing points. Consequently, we can define a potential energy function
such that for a particle moving around under the action of a conservative force the
sum of the kinetic and potential energies of the particle is a constant. The fact that
the law of conservation of energy can be expressed so simply is often very helpful
when it comes to solving problems.
According to Newton's Law, Eq. (9.1), the gravitational field in the vicinity of
a point on the Earth's surface is not exactly uniform: it decreases slightly as the
distance from the centre of the Earth increases and it always points towards the
centre of the Earth. Of course it is often a good approximation to assume the field is
uniform but we should keep in mind that really it varies in strength and direction
from point to point. Let us now show that the gravitational force described by
Newton's Law is also conservative and hence that we can go ahead and define a
potential energy function.
To be specific let us consider the gravitational force on the mass m due to the
mass M . Let us compute the work done by this force as the particle of mass m
moves from a point A to a point B in the field of the other mass M which we
consider as being at some fixed point in space. Let us start by assuming that the
gravitational force acting on m is conservative. It means that the work done on the
particle can be written
B
F
ยท
d x
=โˆ’
U( x B )
+
U( x A ),
(9.2)
A
where U( x ) is the potential energy of the particle when it is at position x . Notice
that the sign of U( x ) is purely a matter of convention and that for any U( x ) we
can also add or subtract an overall constant without changing Eq. (9.2). It is our
job to introduce a potential energy function and we must be careful to interpret it
correctly. The minus sign in Eq. (9.2) was chosen so that the increase in the kinetic
energy of the mass m as it moves in the gravitational field (no other forces are
present) from A to B is given by
T( x B )
โˆ’
T( x A )
=โˆ’
U( x B )
+
U( x B )
(9.3)
i.e.
T( x A )
+
U( x A )
=
T( x B )
+
U( x B )
(9.4)
and so the sum of the kinetic and potential energies is a constant, which we usually
call the total energy. Applied locally, to infinitesimal displacements, Eq. (9.2) can
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