Professor Dave here, let's discuss simple
harmonic motion.
Sometimes when we examine the motion of
an object, it will involve a
single action, like a rock falling from a
cliff down to the ground.
This will probably just happen one time
after which the rock will remain at rest.
But some motion is periodic, meaning
repeated, like the motion of the pendulum
on a grandfather clock, or the vibration
of a spring. We will refer to this kind
of motion as simple harmonic motion. Say
we have a spring that is attached to
some stationary surface, and on the other
end of the spring there is a block of a
particular mass. If we pull this block so
as to expand the spring and then we
release the block, it will vibrate back
and forth between more compressed and
less compressed states. If we assume that
the surface of motion is completely
frictionless, then the elastic potential
energy reaches a maximum when the spring
is most or least compressed and the
block is changing direction, while
kinetic energy is at a maximum when the
block is right in the middle and moving
the fastest. If we were to graph the
position of the block against time it
would display sinusoidal behavior, just
like a trigonometric sine function, where
the block continues to occupy the same
positions over a particular period of
time, which is why we call this periodic
motion. x equals 0 at the equilibrium
position of the block, where it was at
before we pulled it, and assuming zero
friction the block will oscillate
between positive x and negative x
indefinitely. In reality, frictional
forces will dampen the spring's activity
and it will eventually come to a stop,
but for many systems an ideal
mass-spring system will be a decent
approximation for a real one. Of course
every spring is different, some are loose
like a slinky and some are stiff like
the spring that launches a pinball into
play. This factor is represented by the
spring constant k, which will be unique
to a particular spring.
The force that a spring can exert is equal
to negative k times the displacement of
the object it acts upon, and this
relationship is called Hooke's law.
The negative sign indicates that the force
of the spring is always opposite the
direction of the movement of the object.
When the object compresses the spring it
will push out, and when the object
stretches the spring it will pull in.
In both cases, the spring is attempting
to move the object back to its
equilibrium position, which is why the
force applied by the spring can be
called a restoring force. The units on a
spring constant will be Newtons per
meter, so that when you multiply by some
distance x, you get the force that must
be applied to compress that spring that far.
Let's also note that the elastic
potential energy of a spring will be
equal to one-half kx squared, where k is
the spring constant and x is the
distance it is stretched or compressed.
If at equilibrium the distance is zero, so it
has zero elastic potential energy, and
the maximum elastic potential energy
will occur at the maximum distance from
this point on either side of the
oscillation. This expression looks
similar to the expression for kinetic
energy, which is convenient since the two
will interchange as the mass oscillates.
A pendulum also displays periodic motion
since it swings back and forth between
the same two positions,
although this involves gravitational
potential energy rather than elastic
potential energy, and we will look at
this behavior next. Let's check comprehension.
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