Potential Energy of Springs
The force of a spring is
determined by Hooke's Law, which is
Fsp=-kΔx,
where Fsp
is the force of a spring, k is the spring
constant (measured in newtons/meter), and Δx
is the displacement of the spring. Moreover, the force of a spring is
equivalent to
Fsp=-kx=-dUsp/dx.
In other words, the
force of a spring is equivalent to the change in potential
energy of the spring, with respect
to distance [3]. If we were
to solve this differential, we would get
Usp=1/2kΔx2
which means the potential energy of a spring is proportional to the displacement of the spring squared [3]. It was through the work of Calculus made people easily understand the relationship between potential energy and springs.
Simple Harmonic Motion
Earlier we showed that
the force of a spring is equivalent to the spring constant
times the displacement. In terms of Newton's Second Law,
this means the force
of a spring equals mass times acceleration, or rather,
ma=-kx,
where
m is the mass of the
spring and a is the
acceleration of the spring (Δx=x,
assuming the spring starts from
equilibrium).
This is assuming there are no other external
forces acting on the spring (friction, for
example, would make the spring move slower)! This
happens to imply
a+(k/m)x=0
[2].
So what's
the big deal? One might think that
calculating the acceleration of a spring is very hard, so
this doesn't really help. This may be true, except we happen to know
the equation of the position of a spring with respect to
time, which is
x(t) = A sin(ωt+ φ) ,
where A is the amplitude of the spring, ω is
the angular velocity of the spring, t is the time,
and φ is the phase
shift of the spring (the phase shift isn't that
important right now). From
this equation, we may find that the acceleration
of a spring will be
a(t)=d2x/dt2=-ω2A
sin(ωt+ φ).
Plugging this into a+(k/m)x=0,
we get that 0=(k/m)A sin(ωt+
φ) -ω2A
sin(ωt+
φ),
which implies (k/m)1/2=ω
[3].
In
other words, the
angular velocity
of a spring is
equivalent to
the square root
of the spring
constant divided
by the mass of
the spring.
Perhaps
this is not the
most
exciting
discovery in
physics, but
through some
simple
manipulation of
physics
equations, we
can get a better
understanding of
how springs work.
(The graph on the left is an example
of an oscillating graph. As you can see
the shape of the graph looks like a spring that is stretched
out, but it's really a sine function. From this graph we
should be able to determine the period, amplitude, and
position of a spring based on its spring constant and mass.
The animation on the right shows a spring oscillating. You
may notice that the position of the spring varies with
respect to time. In the sine function, the graph reaches a
maximum and minimum height, just like the spring in the
animation.)
We
will now discuss the conservation of momentum to continue
our understanding of calculus in physics.