What it shows:
A small crude spindly model of a cow is able to support five times its own weight. Another model, scaled up exactly six times in all dimensions, collapses under its own weight! Assuming that strength is proportional to cross-sectional area (∝ dimension 2) and weight is proportional to volume (∝ dimension 3), simply scaling the model up geometrically leads to the situation where the weight is too great for its strength.
How it works:
This demonstration was inspired by R.H. Stinson's apparatus note in the AJP (see References below). The small cow's body is made of 1-in. plywood and measures about 2 3/4 " × 8" (7 × 20.5 cm). The head is of 1/2-in. plywood, joined to the body by a neck of 1/4-in. diameter wooden dowel. The legs are cotton swab sticks of 1/16-in. diameter hardwood. They protrude from the body a distance of 5 in. Viewed from the side, the legs are vertical; viewed from the front they are angled 20° from the vertical. The model weighs 188 gm. It will support an additional 1 kg weight placed on its back.
The large cow is scaled up a factor of six in all dimensions. The same materials are used throughout. The body is laminated with six 1-in. plywood sheets; the head is also laminated and has a 1 1/2" dia. wooden dowel neck; the legs are 3/8" dia. × 29" long hardwood dowels. A rope serves as the tail and an inflated rubber glove represents the udder. It weighs in at 38 kg (84 lbs.) which is just 6% off from the theoretical weight. During the lecture the cow is supported from underneath by a tall stool. When lifted off the stool and allowed to stand freely on the floor it momentarily sways precariously before collapsing in a most satisfactory hail of broken legs.
Mother Nature is quite aware of this problem and scales animals differently: lengths and radii of body segments scale with mass as
length ∝ m1/4 and radius ∝ m3/8.
Since the large cow is 6 times larger, its weight is 63 = 216 times the weight of the little cow. Thus Mother Nature would make the legs 2161/4 = 3.8 times as long (instead of 6) and 2163/8 = 7.5 times as thick (instead of 6). To demonstrate this empirical scaling law (Kleiber's Law), we have a second set of legs scaled accordingly; these are 1/2" dia dowels measuring 19" long. Indeed, She's right - the large cow is stable and stands!
Setting it up:
Both the lecture bench top and the cement floor are too smooth - the legs of the models slip outwards under the weight of the cow. Lay a piece of carpet on the floor and a small piece on the bench top to prevent this. There are two sets of holes to accommodate the two sizes of legs. Sometimes the legs break off flush with the body making it difficult to remove the interior bits. A drill will remove them but take care not to enlarge the collet holes.
Definitely a fun demonstration. It should convince the audience why elephants need fat legs and that Hollywood's giant flies are impossible.
1. R.H. Stinson, Am J Phys 45, 498 (1977).
2. Kane & Sternheim, Physics (Wiley & Sons, N.Y., 1978) p 163.
3. McMahon & Bonner, On Size and Life (Scientific American, N.Y., 1983) pp 128-131.