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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
compressed
position, the spring exerts a force
F_{s} = - kx . This is called a
linear restoring force because the force is always in the opposite
direction from the displacement.
Note:
For Example:
In Figure (5.2a)
x = x_{f} - x_{i} = - 5 which gives
F_{s} = - k(- 5) = 5k . This
force is
positive and therefore directed to the right. This means that the spring
resists the
compression. In Figure (5.2b)
x = x_{f} - x_{i} = 3 which gives F_{s} = - 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
compress a
spring varies from
F_{ ext} = F_{0} = 0 (at x_{i} = 0 ), to
F_{ ext} = F_{x} = kx
(at x_{f} = x ).
Since force increases linearly with x , the average force that must be
applied is
= (F_{0} + F_{x}) = kx |
PE_{s} = kx^{ 2}. | (4) |
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