2007 AAPT Summer Meeting
Greensburo, NC
Scaling With Bolts
Brett
Carroll
Green River Community College
12401 SE 320^{th} St.
Auburn, WA 98092
2538339111 x4322
bcarroll@greenriver.edu
Abstract
When
an object is Òscaled upÓ by a factor X, its surface area increases by X^{2
}and its volume increases
by X^{3}. Students often struggle with this
concept, particularly regarding objects with irregular shapes. Stacking cubes into larger cubes is
helpful, but does little to eliminate questions about Òreal worldÓ scaling.
After
much searching, industrial bolts have proven an excellent example of scaling of
irregular objects. They are
available in many sizes in ratios of 1:2:3:4, and lend themselves easily to
investigations of surface area and volume ratios.
Construction of Apparatus:
In its simplest form, scaling with bolts can be
explored with nothing more than a set of bolts, some paper, and a scale or
graduated cylinder to measure volume.
Standard bolts are specified using two sets of numbers
– the thread size, a
pair of numbers that indicate both the diameter and number of threads per inch
of the bolt, and the length of the threaded portion. A typical designation is ¼Ó20 by ½Ó,
indicating a bolt with a ¼ inch diameter and 20 threads per inch that is
½ inch long.
The following is a very useful set of bolts for
scaling investigations, and can be found at McMasterCarr (mcmaster.com):
¼Ó28 by ½Ó
(McMasterCarr #92620A562, 27 or more)
½Ó13 by 1Ó (McMasterCarr
#92620A712, 8 or more)
¾Ó10 by 1 ½Ó (McMaster Carr #92620A839, 3 or more)
1Ó8 by 2Ó
(McMasterCarr #92620A953, 1 or more)
These are in linear ratios of 1:2:3:4,
simplifying the mathematics required to compare measurements.
Area comparisons can be done with paper and
pencil, or simply by stacking the bolts to see how many heads of a smaller bolt
it takes to cover the head of a larger bolt etc.
Volume comparisons can be done most simply be
weighing the bolts to see how many of a smaller bolt it takes to equal the
weight of a larger bolt. Since
weight is proportional to the amount of steel in a bolt and thus to its volume,
this gives a very accurate comparison of volumes. Another way to measure volumes is through water displacement
in a graduated cylinder or other tall thin transparent container.
Use of Apparatus:
Although scaling of surface area and volume a
objects get larger are important concepts for the understanding of many
physical systems, good realworld examples that can be analyzed easily are
difficult to find. To work well,
the objects must be available in a variety of sizes in simple ratios, have a
somewhat complex (but not too complex!) geometry, be faithful ÒscaledupÓ
versions in all dimensions, and be made from a material with a constant density
so that volume can be determined simply by measuring weight.
Systems of objects that fill all these needs are
hard to find. Industrial bolts
have proven to be an excellent option that satisfies all the requirements, and
they are available at your local hardware store or online through merchants
such as McMasterCarr (mcmaster.com).
Be sure that the bolts are all made of the same material (such as Grade
8 steel bolts) and have threads that run the entire length of the bolt. The simplest simple set that will allow
scaling comparisons is two sizes of bolts in a 1:2 ratio, with 8 of the smaller
bolts and one of the larger bolt.
Compare linear dimensions first by laying two of
the smaller bolts endtoend, and noting that the length of the two is equal to
the length of the larger bolt. The
same comparison can be made for the heads of the bolts by placing them
sidebyside. The width of the
threaded portions of the bolts are a little harder to compare, but putting two
small bolts sidebyside in opposite directions allows you to put the threaded
parts together and compare them to the corresponding diameter on the larger
bolt. Or simply note that one of
the smaller bolts extends across half the diameter of a larger bolt.
Surface areas for the two bolts can be compared
by placing four of the smaller bolt heads together, and noting that they cover
approximately the same area as the head on a larger bolt. Because the bolt heads will not ÒpackÓ
together in the same shape as the larger head, the students will have to do a
bit of estimating. But it should
be clear that two of the smaller bolt heads will not cover the area of one of
the larger bolt heads, and that four are a very close match. The area of the head of the bolt that
is twice as large is four times as great.
The surface areas of the threaded portions can be
compared by cutting a piece of paper or tape that will just wrap around and
completely cover the surface on the larger bolt. That same piece of paper or tape will wrap twice around the
diameter of two small bolts placed endto end, showing that it takes the
threaded area of four small bolts to equal the threaded area of one larger
bolt. Or the piece of paper can be
divided into four smaller parts (by cutting or with a pencil), each of which
will cover the threaded area of a smaller bolt. Similar methods can be used to compare the areas of
all corresponding parts on the two bolts and show its dependence on the square
of the linear dimension.
To measure volume, simply weigh the larger bolt,
and note that is does not take two small bolts to equal the weight of one bolt
that is twice as large. Eight of
the smaller bolts should be very close in weight to that of the larger bolt,
showing the dependence of the volume on the cube of the linear dimension. Since the bolts are made of steel with
a constant density, the measured weight of steel is directly proportional to
the volume of the bolt.
A more direct way to compare volume is by
displacement of water in a graduated cylinder or other tall, transparent
container. It is a simple matter
to show that eight of the smaller bolts displace the same amount of water as
one larger bolt. While a more
direct measure of volume, this method is less accurate, more time consuming,
and much messier than volume comparisons by weight.
All of the above can be extended to a larger
range of bolts so that sizes in a 1:2:3:4 ratio can be compared – see the
list in the construction section above.
The smallest of the listed bolts (1/4Ó28 by ½Ó) has some small
deviations from scale, due largely to the fact that the failure strength of a
bolt varies only with the square of its linear dimension (the Òshear strengthÓ
is directly proportional to the area of the threads that must be sheared off to make the bolt fail, or
the area of the shank that must be snapped to break the bolt), so threads can
be proportionally smaller on larger bolts and still be much stronger. Thread counts also do not scale in
exact proportions because of standardized thread counts used by industry. But for all practical purposes,
the accuracy of the scaling factors is more than adequate to show the X^{2
}and X^{3} relationships.
The simplest and closest comparisons are found
using eight ½Ó13 by 1Ó bolts with one 1Ó8 by 2Ó bolt. But the additional ¼Ó28 and
¾Ó10 bolts add additional data to the comparisons if the time is
available for the extra measurements and comparisons. In addition, the variations from ÒidealÓ scale can be used
to introduce the concept of adjusting factors such as thread size and thread
count to make the bolts work properly at different scales.
Those same factors come into play when creating
Ògiant antsÓ and other creatures of science fiction legend, except that in
these cases the scaling ratios are often 1:1000 or higher. The adjustments that would be necessary
for such an overgrown insect (such as proportionally much thicker legs to have enough area to support the
huge volume) would make it unrecognizable as an ant. The much smaller scaling adjustments that are made to bolts
of increasing size can provide a good introduction to concepts like these when
they (almost inevitably) arise in class.
Equipment and costs required
to construct apparatus:
Item 
Source 
Part
number 
Cost 
¼Ó28
by ½Ó bolts (27) 
McMasterCarr 
2.62 

½Ó–13
by 1Ó bolts (8) 
McMasterCarr 
3.15 

¾Ó10 by 1 ½Ó bolts (3) 
McMasterCarr 
4.63 

1Ó8 by 2Ó bolts (1) 
McMasterCarr 
3.25 

Metal rods 
Home
Depot 

5.79 
Tape, paper 
Grocery
store etc. 

0.50 
Total Cost 
19.94 