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Definition: The spring constant, k , is a measure of the stiffness of a spring (large k stiff spring, small k soft spring).
To compress a spring by a distance x we must apply a force
F ext = kx . By Newton's 3rd law, if we hold a spring in a
position, the spring exerts a force
Fs = - kx . This is called a
linear restoring force because the force is always in the opposite
direction from the displacement.
In Figure (5.2a)
x = xf - xi = - 5 which gives
Fs = - k(- 5) = 5k . This
positive and therefore directed to the right. This means that the spring
compression. In Figure (5.2b)
x = xf - xi = 3 which gives Fs = - 3k .
The negative sign indicates that the force is to the left and that the spring
resists the stretching.
To find the potential energy stored in a compressed (or stretched) spring,
we calculate the work to compress (or stretch) the spring: the force to
spring varies from
F ext = F0 = 0 (at xi = 0 ), to
F ext = Fx = kx
(at xf = x ).
Since force increases linearly with x , the average force that must be
|= (F0 + Fx) = kx|
|PEs = kx 2.||(4)|