The Potential Energy Function of a Spring
As we study the idea
that energy is neither created nor destroyed, we see that energy can be stored
in various physical ways, and it becomes important to understand how this
happens. Today’s lab investigates energy storage in a simple coiled spring. (We
choose this because surprisingly such a spring is a good model for a very wide
variety of physical happenings, including such unlikely items as the absorption
and emission of infra-red light by molecules.)
The basic idea is to
store energy in a spring by gradually adding weights to a bucket on the end of
the spring. Each additional weight stretches the spring a bit more, lowering
the weight together with all pre-existing weights by a small amount. The
lowering of the weights represents a loss of the potential energy of gravity.
If energy is conserved, we can expect that that lost gravitational energy has
gone into the stretching of the spring, and by adding up these small inputs of
energy, we should be able to say how much energy is stored in the spring at any
given degree of stretching.
Equipment: § spring
§
washers as standard weight
§
bucket holding enough washers to achieve good extension of the spring (20–60 cm
of stretch)
§
centimeter scale (meter stick or other)
Procedure: Make a data table which records the
amount of weight held by the bucket and the length of the spring at that
weight, and also records the loss in gravitational potential energy in the step
that led to the given length, and the cumulative loss in gravitational
potential energy from the very first step in the process.
The basic procedure is
pretty straightforward. Record the starting length of the spring, then add a
small amount of weight (1-3 washers), and record the new length of the spring.
Continue adding weights and recording the resulting spring length until you
have several data points (weights vs. spring lengths) to use in your
calculations. Use the basic formula for
gravitational P.E. to calculate the gravitational energy put into the spring at
this step
Note: This input of
gravitational energy is just the mass at the beginning of the step, times the
small distance it moves downward before it comes to equilibrium with the
spring. It is not the mass times the
whole distance from the start (because the total mass you have at a given step
only stretched the spring the small additional amount, not the whole distance).
Analysis: (a) Find a
power law function (possibly shifted) which fits your table of cumulative
gravitational P.E. loss as a function of the spring’s length. You can interpret
this as the potential energy function of the spring (because we are assuming
that the lost gravitational potential energy is stored in the spring as
potential energy of a different kind). Is the function approximately one of the
familiar ones (e.g. linear or quadratic)?
(b) Find a function
which fits your table of step by step
potential energy changes.
(c) Find the function
which gives the slope of the step by
step change function. You can interpret this as the force function of the
spring. Is the function approximately one of the familiar ones (e.g. linear or
quadratic)?
Dessert Course (Optional )
(a) Total Energy
Predicts Kinetic Energy
If you think about
today’s lab so far, you can see that it measures a loss in gravitational
potential energy and interprets it as
a gain in the spring’s potential energy. It doesn’t actually show that the
potential energy is in the spring and can be got out again.
One way to check is to
let the weight on the end of the spring drop from a height somewhat above its
equilibrium height, and see if the kinetic energy at any given point is what
you would expect if total energy (the sum of gravitational P.E., spring P.E.
and kinetic energy) stays constant.
Equipment: § photogate
/ interface / computer rig (as in the “super-collider” lab)
§
straw or equivalent (for light-blocking obstacle of known width)
Procedure: Experiment
with the spring / weight / lab stand set up to see how far the weight will
travel downward if it is released from above its equilibrium level. Be careful
not get it moving to fast that it stretches the spring. Now set up the
photogate to measure the speed of the weight as it passes a given level. If
possible, set things up so you can measure at several different levels without
difficulty. Take speed and height data.
Analysis: Write down
the formula for total energy of this system as the sum of gravitational
potential energy (relative to your chosen reference level), spring potential
energy (relative to the same level), and kinetic energy. Use the standard
formula for gravity and kinetic energy, and your function derived from the main
lab for the spring. Calculate the value of each term, and the sum.
Interpretation: Answer
the question of what the sum of energy terms should come to (numerically) if
total energy is unchanging during the falling and rising of the weight, discuss
your observed/caculated results, and then discuss in what ways this experiment
sheds light on whether energy is freely exchanged between these three different
forms.
Write-up: In normal format. Include graphs of spring P.E. and slope of PE
as functions of stretch.