# Springs

## Discussion

### introduction

Elasticity is the property of materials to return to their original size and shape after being deformed (that is, after the deforming force has been released). Since it is really a property of materials, a more complete discussion will have to wait until later. For now we'll stick to a simple elastic system (the coil spring) and a simple law (Hooke's law).

Hooke's law isn't about hooks. It's about springs — coil springs — the kind of spring found in a car's suspension or a retractable pen, the kind that look like a pig's tail or a lock of curly hair. Coil springs are also known as helical springs since the mathematical name for this kind of shape is a helix. The law is named in honor of its discover, the English scientist, mathematician, and architect Robert Hooke (1635–1703). Although Hooke's name is now usually associated with elasticity and springs, he was interested in many aspects of science and technology. His most famous written work is probably the *Micrographia*, a compendium of drawings he made of objects viewed under a magnifying glass. In this book, he was the first to use the word "cell" to described the walled-in regions he saw when looking at a magnified slice of plant tissue (in Hooke's case, a slice of cork). The standard story is that he compared these walled-in regions to the cells in a prison or monastery, but I could find no mention of this in the *Micrographia*. He also compared biological cells to pores, pumice, and honeycombs, but cell was the word that stuck.

As I said, Hooke was interested in many areas of science. He not only looked through magnifying glasses and microscopes, but also through telescopes. Like Galileo he pointed his telescope at the sun, and like Galileo he did not look at the sun with his eye. That would have been stupid. Instead, like every sensible person since Galileo, he placed a sheet of white paper several inches in front of the eyepiece and looked at the projected image of the sun. Such a device is called a helioscope. (In Greek, ήλιος "elios" is the sun and σκοπεῖν "skopein" is to observe.) In 1675, Hooke wrote a book on the helioscope and added this little bit of text to fill up the white space leftover at the bottom of the last page…

To fill the vacancy of the enſuing page, I have added a decimate centeſme [a thousandth] of the Inventions I intend to publiſh, though poſſibly not in the ſame order, but as I can get opportunity and leaſure; moſt of which, I hope, will be as uſeful to Mankind, as they are yet unknown and new.

He then went on to list ten inventions and discoveries he had made. (This was not followed by any later list with the remaining 990 inventions he promised, by the way.) These included a way to regulate pendulum clocks, a method for constructing arches, and other inventions in optics, hydraulics, and mechanical engineering. The third item on his list is of importance to us right now.

3. The true Theory of Elaſticity or Springineſs, and a particular Explication thereof in ſeveral Subjects in which it is to be found: And the way of computing the velocity of Bodies moved by them. ceiiinosssttuu

That weird bit that looks like someone fell asleep on their computer keyboard is not a mistake. It's an anagram. In the time before patents and other intellectual property rights, publishing an anagram was a way to announce a discovery, establish priority, and still keep the details secret long enough to develop it fully. Hooke was hoping to apply his new theory to the design of timekeeping devices and didn't want the competition profiting off his discovery. He was successful in this regard and in 1678 Hooke made the solution to the anagram, and the true theory of springiness that now bears his name, public knowledge.

About two years ſince I printed this Theory in an Anagram at the end of my Book of the Deſcriptions of Helioſcopes, viz. ceiinosssttuu, that is

Ut tenſio sic vis; That is, The Power of any Spring is in the ſame proportion with the Tenſion thereof: That is, if one power ſtretch of bend it one ſpace, two will bend it two, and three will bend it three, and ſo forward. Now as the Theory is very ſhort, ſo the way of trying it is very eaſie.

The Latin…

Ut tensio sic vis

literally translated into English would read something like…

As extension, so is force

but in contemporary English, we would probably say something more like…

Extension is directly proportional to force.

The remainder of the quoted passage that follows his Latin phrase is a description of what is means for two things to be directly proportional. Try not to get confused with his apparent misuse of words, however. Scientific terminology in the English language is much more precise now than it was in the Seventeenth Century. By "tension" he means extension and by "power" he means force. The directly proportional relationship is between extension and force, not tension and power.

We can write Hooke's law as a proportionality statement in mathematical shorthand like this…

Δ*x* ∝ *F*

where…

F = |
force, spring force, elastic force, applied force, deforming force, …. You get the idea. Versions with subscripts are also common (F, _{s}F, etc.)._{e} |

Δx = |
extension or compression of the spring; that is, the change in length from the spring's relaxed, natural, or original length (x_{0}). Use of ∆ [delta] is optional as the idea of "change" is implied. |

Since equations are so popular nowadays (meaning the last 150 years or so) we should probably finish by writing Hooke's law as an equation…

*F* = − *k*Δ*x*

The constant of proportionality (*k*), which is needed to make the units work out right, is called the spring constant — an apt name since it is a constant that goes with a particular spring. It is not a constant that goes with a particular material. Materials don't have constants in elasticity, they have moduli (plural of modulus). Hooke's law is now recognized as being approximately true for a variety of elastic applications, not just springs, but as I said earlier, a more complete discussion of this will have to wait until later in this book.

The SI unit of the spring constant is the newton per meter, which has no special name.

k = |
F |
⇒ | ⎡ ⎣ |
N | = | N | ⎤ ⎦ |

x |
m | m |

Since most springs would never stretch anything close to a meter, other units like the newton per centimeter [N/cm] or newton per millimeter [N/mm] are also common.

You may have noticed a negative sign in the equation above. This gives the spring force its direction. If the spring is stretched in the positive direction (+*x*) the spring force pulls back in the negative direction (−*F*). If the spring is compressed in the negative direction (−*x*), the spring force pushes back in the positive direction (+*F*).

Newton and Hooke. Hooke and Newton. Reputed to be the ugliest scientist of all times, no portrait of Hooke is known to exist (no undisputed, original portrait). Hooke was also short and Newton mocked him with his famous "shoulders of giants" line.