Resource Lesson
Springs: Hooke's Law
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In 1678, Robert Hooke announced the invention of the spring scale and the relationship for elastic materials that is now known as
Hooke's Law
. When an object is acted upon by a force, it can be compressed, stretched or bent. If when the force is removed, the object returns to its original shape, it is said to be elastic. Solids that do not return to their original configuration once they have been distorted are categorized as plastics.
Hooke discovered that not only are certain materials (steel bars, rods, wire, springs, diving boards, and rubber bands) elastic, but the stretch they experience is directly proportional to the load that they support.
Elastic media will stretch until the reach their elastic limit, or yield point. After that point, they exhibit plastic deformation and will never return to their original shape. Ductile materials stretch thinner and thinner, while brittle materials break without any plastic deformation. Eventually all will rupture at their breaking point.
To simplify our discussion, we are going to use springs as our example of an elastic medium. The formulas used to calculate the force required to stretch or compress an elastic medium with respect to its equilibrium position and its elasticity constant, k, are:
F
_{internal}
= - kx
force supplied by the spring to restore itself to equilibrium (Hooke's Law)
F
_{external}
= kx
force supplied by an external agent on the spring distorting it from equilibrium
This formula is only applicable to force acting on an ideal spring that has not surpassed its elastic limit. Note that the amount of force required by an external agent to stretch the spring depends on how far it has been displaced from its equilibrium position. That is, the force is not constant, it is
variable
.
When two or more springs are combined in
parallel
(side by side) so that any applied force must stretch both springs simultaneously, the spring constant for the combination will be
k
_{parallel}
= k
_{1}
+ k
_{2}
When two or springs are combined in
series
(one after another), an applied force may stretch one more than another. Recall a saying that a chain is only as "strong as its weakest link." The spring constant for the combination will be
k
_{series}
= (1/k
_{1}
+ 1/k
_{2}
)
^{-1}
Work done and energy stored
The formula used to calculate the work required to stretch the spring OR the amount of elastic potential energy subsequently stored in a spring is:
PE
_{e}
= ½kx
^{2}
This energy is calculated graphically as the area under a F vs x graph.
PE
_{e}
= Work done on the spring
= average force times distance
= area under the graph
= ½ bh
F is measured in newtons
= ½(x)(F)
x is measured in meters
= ½(x)(kx)
k is measured in nt/m
= ½ kx
^{2}
PE
_{e}
is measured in joules
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