


Springs
Questions about springs on SAT II Physics are usually
simple matters of a mass on a spring oscillating back and forth.
However, spring motion is the most interesting of the four topics
we will cover here because of its generality. The harmonic
motion that springs exhibit applies equally to objects moving
in a circular path and to the various wave phenomena that we’ll
study later in this book. So before we dig in to the nittygritty
of your typical SAT II Physics spring questions, let’s look at some
general features of harmonic motion.
Oscillation and Harmonic Motion
Consider the following physical phenomena:
 When you drop a rock into a still pond, the rock makes a big splash, which causes ripples to spread out to the edges of the pond.
 When you pluck a guitar string, the string vibrates back and forth.
 When you rock a small boat, it wobbles to and fro in the water before coming to rest again.
 When you stretch out a spring and release it, the spring goes back and forth between being compressed and being stretched out.
There are just a few examples of the widespread phenomenon
of oscillation. Oscillation is the natural world’s
way of returning a system to its equilibrium position,
the stable position of the system where the net force acting on
it is zero. If you throw a system offbalance, it doesn’t simply
return to the way it was; it oscillates back and forth about the
equilibrium position.
A system oscillates as a way of giving off energy. A system
that is thrown offkilter has more energy than a system in its equilibrium
position. To take the simple example of a spring, a stretchedout
spring will start to move as soon as you let go of it: that motion
is evidence of kinetic energy that the spring lacks in its equilibrium
position. Because of the law of conservation of energy, a stretchedout
spring cannot simply return to its equilibrium position; it must
release some energy in order to do so. Usually, this energy is released as
thermal energy caused by friction, but there are plenty of interesting
exceptions. For instance, a plucked guitar string releases sound
energy: the music we hear is the result of the string returning
to its equilibrium position.
The movement of an oscillating body is called harmonic
motion. If you were to graph the position, velocity, or acceleration
of an oscillating body against time, the result would be a sinusoidal
wave; that is, some variation of a y = a sin bx or
a y = a cos bx graph.
This generalized form of harmonic motion applies not only to springs
and guitar strings, but to anything that moves in a cycle.
Imagine placing a pebble on the edge of a turntable, and watching
the turntable rotate while looking at it from the side. You will
see the pebble moving back and forth in one dimension. The pebble
will appear to oscillate just like a spring: it will appear to move
fastest at the middle of its trajectory and slow to a halt and reverse
direction as it reaches the edge of its trajectory.
This example serves two purposes. First, it shows you
that the oscillation of springs is just one of a wide range of phenomena
exhibiting harmonic motion. Anything that moves in a cyclic pattern
exhibits harmonic motion. This includes the light and sound waves
without which we would have a lot of trouble moving about in the
world. Second, we bring it up because SAT II Physics has been known
to test students on the nature of the horizontal or vertical component
of the motion of an object in circular motion. As you can see, circular motion
viewed in one dimension is harmonic motion.
Though harmonic motion is one of the most widespread and
important of physical phenomena, your understanding of it will not
be taxed to any great extent on SAT II Physics. In fact, beyond
the motion of springs and pendulums, everything you will need to know
will be covered in this book in the chapter on Waves. The above
discussion is mostly meant to fit your understanding of the oscillation
of springs into a wider context.
The Oscillation of a Spring
Now let’s focus on the harmonic motion exhibited by a
spring. To start with, we’ll imagine a mass, m,
placed on a frictionless surface, and attached to a wall by a spring.
In its equilibrium position, where no forces act upon it, the mass
is at rest. Let’s label this equilibrium position x = 0.
Intuitively, you know that if you compress or stretch out the spring
it will begin to oscillate.
Suppose you push the mass toward the wall, compressing
the spring, until the mass is in position .
When you release the mass, the spring will exert a force,
pushing the mass back until it reaches position , which is called the amplitude of
the spring’s motion, or the maximum displacement of the oscillator.
Note that .
By that point, the spring will be stretched out, and will
be exerting a force to pull the mass back in toward the wall. Because
we are dealing with an idealized frictionless surface, the mass
will not be slowed by the force of friction, and will oscillate
back and forth repeatedly between and .
Hooke’s Law
This is all well and good, but we can’t get very far in
sorting out the amplitude, the velocity, the energy, or anything
else about the mass’s motion if we don’t understand the manner in which
the spring exerts a force on the mass attached to it. The force, F,
that the spring exerts on the mass is defined by Hooke’s Law:
where x is
the spring’s displacement from its equilibrium position and k is
a constant of proportionality called the spring constant.
The spring constant is a measure of “springiness”: a greater value
for k signifies a “tighter” spring,
one that is more resistant to being stretched.
Hooke’s Law tells us that the further the spring is displaced
from its equilibrium position (x) the
greater the force the spring will exert in the direction of its
equilibrium position (F).
We call F a restoring
force: it is always directed toward equilibrium. Because F and x are directly
proportional, a graph of F vs. x is
a line with slope –k.
Simple Harmonic Oscillation
A mass oscillating on a spring is one example of a simple
harmonic oscillator. Specifically, a simple harmonic oscillator
is any object that moves about a stable equilibrium point and experiences
a restoring force proportional to the oscillator’s displacement.
For an oscillating spring, the restoring force, and consequently
the acceleration, are greatest and positive at . These quantities decrease as x approaches
the equilibrium position and are zero at x = 0.
The restoring force and acceleration—which are now negative—increase
in magnitude as x approaches and are maximally negative at .
Important Properties of a Mass on a Spring
There are a number of important properties related to
the motion of a mass on a spring, all of which are fair game for
SAT II Physics. Remember, though: the test makers have no interest
in testing your ability to recall complex formulas and perform difficult
mathematical operations. You may be called upon to know the simpler
of these formulas, but not the complex ones. As we mentioned at
the end of the section on pulleys, it’s less important that you
memorize the formulas and more important that you understand what
they mean. If you understand the principle, there probably won’t
be any questions that will stump you.
Period of Oscillation
The period of oscillation, T,
of a spring is the amount of time it takes for a spring to complete
a roundtrip or cycle. Mathematically, the period of oscillation
of a simple harmonic oscillator described by Hooke’s Law is:
This equation tells us that as the mass of the block, m,
increases and the spring constant, k, decreases,
the period increases. In other words, a heavy mass attached to an
easily stretched spring will oscillate back and forth very slowly,
while a light mass attached to a resistant spring will oscillate
back and forth very quickly.
Frequency
The frequency of the spring’s motion tells us how quickly
the object is oscillating, or how many cycles it completes in a
given timeframe. Frequency is inversely proportional to period:
Frequency is given in units of cycles per second, or hertz
(Hz).
Potential Energy
The potential energy of a spring () is sometimes called elastic energy, because
it results from the spring being stretched or compressed. Mathematically, is defined by:
The potential energy of a spring is greatest when the
coil is maximally compressed or stretched, and is zero at the equilibrium
position.
Kinetic Energy
SAT II Physics will not test you on the motion of springs
involving friction, so for the purposes of the test, the mechanical
energy of a spring is a conserved quantity. As we recall, mechanical
energy is the sum of the kinetic energy and potential energy.
At the points of maximum compression and extension, the
velocity, and hence the kinetic energy, is zero and the mechanical
energy is equal to the potential energy, U_{s}=
^{1}/_{2} .
At the equilibrium position, the potential energy is zero,
and the velocity and kinetic energy are maximized. The kinetic energy
at the equilibrium position is equal to the mechanical energy:
From this equation, we can derive the maximum velocity:
You won’t need to know this equation, but it might be
valuable to note that the velocity increases with a large displacement,
a resistant spring, and a small mass.
Summary
It is highly unlikely that the formulas discussed above
will appear on SAT II Physics. More likely, you will be asked conceptual
questions such as: at what point in a spring’s oscillation is the
kinetic or potential energy maximized or minimized, for instance.
The figure below summarizes and clarifies some qualitative aspects
of simple harmonic oscillation. Your qualitative understanding of
the relationship between force, velocity, and kinetic and potential
energy in a spring system is far more likely to be tested than your
knowledge of the formulas discussed above.
In this figure, v represents
velocity, F represents
force, KE represents kinetic energy,
and represents potential energy.
Vertical Oscillation of Springs
Now let’s consider a mass attached to a spring that is
suspended from the ceiling. Questions of this sort have a nasty
habit of coming up on SAT II Physics. The oscillation of the spring
when compressed or extended won’t be any different, but we now have
to take gravity into account.
Equilibrium Position
Because the mass will exert a gravitational force to stretch
the spring downward a bit, the equilibrium position will no longer
be at x = 0, but at x
= –h, where h is the
vertical displacement of the spring due to the gravitational pull
exerted on the mass. The equilibrium position is the point where
the net force acting on the mass is zero; in other words, the point
where the upward restoring force of the spring is equal to the downward
gravitational force of the mass.
Combining the restoring force, F =
–kh, and the gravitational force, F = mg,
we can solve for h:
Since m is in the numerator
and k in the denominator of the fraction,
the mass displaces itself more if it has a large weight and is suspended
from a lax spring, as intuition suggests.
A Vertical Spring in Motion
If the spring is then stretched a distance d,
where d < h,
it will oscillate between and .
Throughout the motion of the mass, the force of gravity
is constant and downward. The restoring force of the spring is always
upward, because even at the mass is below the spring’s
initial equilibrium position of x = 0.
Note that if d were greater than h, would be above x = 0,
and the restoring force would act in the downward direction until
the mass descended once more below x = 0.
According to Hooke’s Law, the restoring force decreases
in magnitude as the spring is compressed. Consequently, the net
force downward is greatest at and the net force upward
is greatest at .
Energy
The mechanical energy of the vertically oscillating spring
is:
where is gravitational potential
energy and is the spring’s (elastic) potential energy.
Note that the velocity of the block is zero at and , and maximized at the
equilibrium position, x = –h. Consequently,
the kinetic energy of the spring is zero for and and is greatest at x
= –h. The gravitational potential energy of the
system increases with the height of the mass. The elastic potential
energy of the spring is greatest when the spring is maximally extended
at and decreases with the extension of the
spring.
How This Knowledge Will Be Tested
Most of the questions on SAT II Physics that deal with
spring motion will ask qualitatively about the energy or velocity
of a vertically oscillating spring. For instance, you may be shown
a diagram capturing one moment in a spring’s trajectory and asked
about the relative magnitudes of the gravitational and elastic potential
energies and kinetic energy. Or you may be asked at what point in
a spring’s trajectory the velocity is maximized. The answer, of
course, is that it is maximized at the equilibrium position. It
is far less likely that you will be asked a question that involves
any sort of calculation.
