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Coefficient of Restitution
Name: Jim
Status: Student
Grade: 6-8
Location: AP
Country: United States
Date: March 2009
Question:
We are working on a module on Sports Materials. We have
done an activity where we measure the rebound of sports balls. We
calculate the coefficient of restitution (COR) using:
COR = (rebound height/drop height)^0.5
A COR of 1 would be a perfectly elastic collision, and a COR of 0
would be a perfectly inelastic collision. We have dropped ping
pong, tennis, basketball, soccer ball, baseball, golf ball, and others.
The class decided to do an extension of this activity. We tried
different metal balls. We have dropped metal balls (varying mass)
on metal surfaces, different balls on hard surfaces, and done both
of these on soft surfaces. We got results that seemed to make
sense. We then dropped (60 cm height)steel balls (1", 1/2", 1/8"
diameters) on a drum membrane. Why is it that the steel on steel
trials the impact (and presumably deformation/heat production) the
lower the COR, but for steel on a drum membrane the greater the
impact (and presumably deformation/heat production) the higher the COR?
I accept, for the steel on steel, the collisions have nonlinear
effects, as shown by the square root relationship shown above. The
greater the collision, the greater the losses (so balls dropped
from greater heights or are thrown at greater speeds will have
lower CORs). But for the drum membrane, the greater the impact
speed, the higher the COR and the more elastic the collision. Are
small impacts being absorbed by the membrane without stretching it?
I have done this with constant volume, but different density balls
(lead, steel, and glass), and again I find opposite trends. For the
steel plate, the marble, then steel, then lead ball bounce highest,
but it is the opposite for the drum. Perhaps the radius of
curvature of the impact has something to do with the outcome.
Perhaps the membrane is a polymer and reacts significantly
different than a non-polymer. I am not sure. Again, Why is the
membrane more elastic the more it is deformed? What would you guess
would happen if I continued the experiments for balls of brass,
rubber, cork, and aluminum?
Replies:
Jim,
Polymers have a vexing property called glass transition that is the point at
which the polymer moves from a solid glassy state to a solid rubbery state. The
vexing thing about it is that it is not set to a particular temperature like
melting point. It is not a first order transition.
On top of this, is that most polymers will exhibit viscoelastic properties giving it
both viscous and elastic properties at any given temperature. This means that the
polymer can both absorb energy and dissipate it as heat, and also return the energy
in an elastic way.
The combined situation makes polymer collisions very non-linear.
For example, if you take silly putty and stretch it slowly, or allow its own weight
to make it stretch, the putty will flow like a liquid (be viscous). However, if you
pull on it really fast, it snaps and breaks like a solid rubber (be elastic with a
low energy to failure). This points out that the flow characteristics of silly putty
is a function of the frequency and amplitude of the force acting on it. I've even seen
videos where silly putty was formed into a hollow ball with thin walls and with a very
high frequency, high amplitude impact - as with a very fast hammer strike, make the
silly putty break like glass - with a sound very much like breaking glass.
Another example is in the case of dropping steel balls onto solid glassy polystyrene.
Above and below the glass transition, the ball will bounce. Below the glass transition
the polymer is acting like a solid glass and not much energy dissipation takes place.
Above the glass transition the polymer is acting like a rubber which returns the
energy to the ball (as the polymer returns to its low energy state) and not much energy
dissipation takes place. But, at the glass transition, the ball does not bounce at
all. At the glass transition all the energy is dissipated as the polymer flows - and
flows to an equally low energy state so no rebound occurs.
I am guessing because the drum skin is made of some polymeric material that is
stretched tight, it tends to exhibit elastic, viscous and energy dissipation -
depending on impact strength and frequency (like I said- very non-linear). You might
even find that depending on drum-head material and manufacturer, that you would get
varying results.
Greg (Roberto Gregorius)
Hi Jim,
These are great questions, and as you have noticed there are tons of variables
involved. I also want to congratulate you on performing some great
experiments with your students. Your challenge now is how to better organize
and visualize your data -- that will help you to come up with explanations
for the data (and generate new experiments to test them).
Looking at the big picture, there are lots of factors at play here beyond
just the material of the ball and the height it is dropped. The biggest
factors are the deformation of the ball and the deformation of the surface
when they impact, the time frames in which they deform, and how much of the
energy of that deformation is returned to the ball. Depending on the
materials (ball and surface), the deformation could be large or small, and
depending on the velocity/energy involved in the impact, the return could be
(comparatively) fast or slow. All of these factors affect the apparent COR.
As you can see, things get complicated very fast. If you want to cut to the
chase, I did find a very good article helping explain some of the details
from the American Journal of Physics.
http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=AJPIAS000067000003000222000001&idtype=cvips&gifs=yes
This article does not discuss
the surface as much as the ball, but it does a good job of breaking down the
factors involved. The article is probably at too high a level for the vast
majority of your students, but you might find it useful, and perhaps you can
'translate' in terms that are appropriate for them.
With respect to your drum experiments, I would guess that yes, the membrane is
able to absorb some amount of energy before it returns it elastically.
Polymers have a 'time constant' where they deform when moving slowing but
like a solid when moving quickly (think of silly putty stretching slowing,
but breaking/snapping when pulled quickly). The more energy in the
collision, the more solid the polymer will act, and thus the more elastic
the collision. So the heavier/higher the ball, the closer to 'full' COR you
will see. Did you plot the CORs against drop height? Do they reach an
asymptote?
I would recommend you try to get quantitative and use graphs to examine your
data. Start plotting your data of COR versus height and COR versus diameter
(for the same ball). Perhaps visualization will help you develop better
explanations for your data. I would also be interested in hearing what
experiments you plan next to test your hypothesis.
Hope this helps,
Burr Zimmerman
This may not completely answer your question but in some cases increased force
increases elastic behavior. An example of this is playing pool. A a light strike
of cue ball and object ball will leave both moving after the collision - not a very
good elastic collision. But a very firm hit will stop the cue ball dead in its
tracks and send the object ball streaking away in a highly elastic collision.
Robert Avakian
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