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l
rest
denote the rest length, then the magnitude of the restoring force f
r
is
calculated by Equation (12.7).
Hooke's Law for Spring Forces
f
r
= k(l
rest
− l).
(12.7)
The constant k is known as the spring constant and essentially describes
how “stiff” the spring is. The constant is not dimensionless. In order for
Equation (12.7) to make sense, we must have
[ML/T
2
] = k[L],
[ML/T
2
]/[L] = k,
[M/T
2
] = k,
or you can just think of k as having units of “unit force per unit length.”
The really interesting thing about springs is how they behave over time.
To see this, let's restate Hooke's law in a way that focuses on the kinemat-
ics of a particle that is being acted on by restorative forces. Specifically,
we're interested in functions for the position, velocity, and acceleration of
a particle.
Things get easier if we adopt a reference frame where the position x = 0
designates the “rest” position, where there are no restorative forces. Fur-
thermore, since we are interested in the acceleration of the particle rather
than the forces acting on it, we will introduce a constant K = k/m, and
since K contains both the spring constant k and the mass of the particle m,
it measures the spring's ability to accelerate the specific particle of interest
to us. With those notational changes, we can rewrite Equation (12.7) as
Acceleration due to
Hooke's law
a(t) = −Kx(t).
(12.8)
You should convince yourself that this is equivalent to Equation (12.7)
before continuing.
Equation (12.8) makes a statement about the relationship between the
position function and the acceleration function; but what we really want is
the function x(t) itself. Equations like this are called differential equations;
they describe the relationship between some unknown function (in this case,
x(t)) and one or more of its derivatives (remember that acceleration is the
second derivative of position). To “solve” a differential equation is to find
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