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First of all, it is misleading to suggest that gravity might
"cause" a tension or normal force and, somehow, confer its
conservative nature to the tension or normal force.
to be a *candidate* for being "conservative," a force
must be directly calculable from a "configuration."
For instance, ... the spring force is determined from
the extension of the spring.
On the other hand it's easy to relax the "inextensible" and
"perfectly rigid" assumptions. After all, an extensible rope is a
"spring" and a nonperfectly rigid surface is a "trampoline"--a
more complicated "spring." Then, one may be able to calculate the
force from the strain and consider the force to be conservative.
Note carefully, however, that the conservative nature is connected
with an ability of the spring or the trampoline *itself* to store
potential energy; that energy should not be considered to be
stored in the supported body or even in the interaction of the
supported body with the spring or trampoline.
Finally, note that real ropes and real surfaces are generally at
least somewhat "dissipative." They exert forces that are not
completely calculable from their configurations and may exhibit
hysteresis and/or velocity-dependence. For instance, a rope may
(and usually does) exert a stronger pull at each specific length
while being extended than when it is subsequently allowed to
relax. As a result, the work done *by* an external agent to
extend the rope is smaller than the work done *on* the external
agent as it is allowed to relax.