1. 11.4 Conservative Forces
Work done by a force when it undergoes a
differential displacement has been defined as
dU = F cosθ ds
If the force is displaced over a path that has
finite length s, the work is determined by
integrating over the path
U = ∫ F cos θds
s
To evaluate the integral, obtain a
relationship between F and the component of
displacement ds cosθ

2. 11.4 Conservative Forces
In some instances, however, the work done
by a force will be independent of its path and
instead, will depend only on the initial and
final locations of the force along the path
As force with such a property is called a
conservative force

3. 11.4 Conservative Forces
Weight
Consider body initially at P’
If the body is moved down along arbitrary path a
to second position, then for a given displacement
ds along the path, the displacement
component in the direction
of W has a magnitude
of dy = ds cos θ

4. 11.4 Conservative Forces
Weight
Since both the force and displacement are in
the same direction, the work is positive
y
U = ∫ W cosθds = ∫ Wdy
or s 0
U = Wy
Similarly, for work done by the weight when
the body moves up a distance y back to P’,
along arbitrary path A’,
U = −Wy

5. 11.4 Conservative Forces
Weight
Weight of a body is therefore a
conservative force since the work done
by the weight depends only on the
body’s vertical displacement and is
independent of the path along which
the body moves

6. 11.4 Conservative Forces
Elastic Spring
Force developed by an elastic spring (Fs = ks) is
also a conservative force
If the spring attached to a body and the body is
displaced along any path, such that it causes the
spring to elongate or
compress from position s1
to s2, the work will be negative
since the spring exerts a force
Fs on the body that is opposite
to the body’s displacement

7. 11.4 Conservative Forces
Elastic Spring
For either extension or compression, work is independent
of the path and is simply
s2 s2
U = ∫ Fs ds = ∫ (−ks)ds
s1 s1
1 2 1 
= − ks2 − ks12 
Friction 2 2 
Work done by a frictional force depends on the path;
longer the path, the greater the work
Frictional forces are non-conservative and work done is
dissipated in the form of heat