141
relaxed
spring
force F
being
applied
xo
x
Back in Newton’s time, experiments like this were considered cutting-
edge research, and his contemporary Hooke is remembered today for doing
them and for coming up with a simple mathematical generalization called
Hooke’s law:
F
˜
k(x-x
o
) [force required to stretch a spring; valid for small
forces only] .
Here k is a constant, called the spring constant, that depends on how stiff
the object is. If too much force is applied, the spring exhibits more compli-
cated behavior, so the equation is only a good approximation if the force is
sufficiently small. Usually when the force is so large that Hooke’s law is a
bad approximation, the force ends up permanently bending or breaking the
spring.
Although Hooke’s law may seem like a piece of trivia about springs, it is
actually far more important than that, because all solid objects exert
Hooke’s-law behavior over some range of sufficiently small forces. For
example, if you push down on the hood of a car, it dips by an amount that
is directly proportional to the force. (But the car’s behavior would not be as
mathematically simple if you dropped a boulder on the hood!)
Discussion questions
A car is connected to its axles through big, stiff springs called shock absorbers,
or “shocks.” Although we’ve discussed Hooke’s law above only in the case of
stretching a spring, a car’s shocks are continually going through both stretch-
ing and compression. In this situation, how would you interpret the positive and
negative signs in Hooke’s law.
Section 5.5Objects Under Strain