Springs – The Physics Hypertextbook (2024)




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).

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.

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



F=force, spring force, elastic force, applied force, deforming force, …. You get the idea. Versions with subscripts are also common (Fs, Fe, etc.).
x=extension or compression of the spring; that is, the change in length from the spring's relaxed, natural, or original length (x0). Use of [delta] is optional as the idea of "change" is implied.

Hooke's law as an equation is written…


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.




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).

elastic potential energy

Tell a story that ends with…



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.

Hooke 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.

Robert Hooke

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

Robert Hooke

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.

Robert Hooke

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 17th 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.


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.

Springs – The Physics Hypertextbook (2024)


What is the physics theory of springs? ›

Hooke's Law: The Physics of Springs

Hooke's Law states that the more you deform a spring, the more force it will take to deform it further. Using the example of a common compression spring, the more you compress the spring, the more force it will take to compress it further.

What is the spring rule in physics? ›

Hooke's law: The extension of a spring is directly proportional to the force applied, provided that the limit of proportionality is not exceeded.

What is the formula for springs in physics? ›

F = -kx. The proportional constant k is called the spring constant. It is a measure of the spring's stiffness.

What is the science behind springs? ›

Elasticity and Restoring Force

Hooke's law is considered to be the earliest explanation of this concept. Restoring force enables the spring to return to its original shape after undergoing manipulation. In the context of Hooke's Law, the restoring force is usually proportional to the amount of stretch experienced.

What is the principle of springs? ›

The Fundamental Principle Behind Springs

Hooke's Law is a fundamental principle in the world of springs, dictating how they behave under force. Simply, this law posits that the force needed to either stretch or compress a spring is directly proportional to the extent of its elongation or compression.

What is the Newton's formula for springs? ›

Step 2: Identify or calculate the distance the spring has been stretched or compressed from its equilibrium length. Step 3: Calculate the force exerted by the spring using Hooke's law: F s = − k Δ x . The spring exerts a force of 7.5 Newtons in the opposite direction of the compression.

What law of motion is a spring? ›

Newton's Third Law, for every action there is an equal and opposite reaction, is demonstrated with the help of Hooke's Law, where the force on a spring is equal to the spring constant multiplied by the displacement from the equilibrium point of the spring.

What is the Hooke's Law? ›

Hooke's law states that the strain of the material is proportional to the applied stress within the elastic limit of that material. When the elastic materials are stretched, the atoms and molecules deform until stress is applied, and when the stress is removed, they return to their initial state.

Do springs get stiffer over time? ›

Over time, the elastic modulus of a spring can degrade due to thermal, mechanical, and environmental factors. The most significant factor is cyclic loading, which causes damage to the material, leading to a decrease in its stiffness or an increase in its overall deflection.

Why is spring force negative? ›

The spring force is called a restoring force because the force exerted by the spring is always in the opposite direction to the displacement. This is why there is a negative sign in the Hooke's law equation. Pulling down on a spring stretches the spring downward, which results in the spring exerting an upward force.

What is the K value in physics springs? ›

The spring constant represents the stiffness of a spring or an elastic material. It is symbolized by k and has an SI unit of newton per meter (N/m). The larger the value of k, the stiffer the spring is and the larger the force needed to compress or stretch it.

What is the property of springs in physics? ›

Properties of Springs in Physics

In physics, springs have the following basic properties: They are elastic, meaning they return to their original shape after being deformed by a force. They store potential energy. The force the spring exerts increases as the distance it is stretched/compressed increases.

What is the law of springs? ›

Hooke's Law is a principle of physics that states that the force needed to extend or compress a spring by some distance is proportional to that distance.

What is the theory of spring? ›

In spring, material is arranged in such a way that it can undergo a considerable change of shape, Without getting permanently distorted. A spring is used to absorb energy in the form of resilience whicn may De. restored wnen required. The quality of a spring is judged from the energy it can absorb and the natural.

Who invented springs? ›

In fact, spring technology goes back to the “bow and arrow” ages. A non-coiled spring was used in a bow and arrow dating back 64,000 thousand years ago. The first coiled spring was invented in 1763 by R. Tradewell, but the first steel coil spring wasn't developed until 1857.

What is the physics behind spring balance? ›

Spring balance is used to measure the weight of the object. When a load is hung on the spring, it stretches and the spring's extension is proportional to the weight of the object. The working principle of the spring balance is Hooke's law which is F=−kx.

What is the spring theory of the universe? ›

String theory turns the page on the standard description of the universe by replacing all matter and force particles with just one element: tiny vibrating strings that twist and turn in complicated ways that, from our perspective, look like particles.


Top Articles
Latest Posts
Article information

Author: Msgr. Refugio Daniel

Last Updated:

Views: 6212

Rating: 4.3 / 5 (54 voted)

Reviews: 85% of readers found this page helpful

Author information

Name: Msgr. Refugio Daniel

Birthday: 1999-09-15

Address: 8416 Beatty Center, Derekfort, VA 72092-0500

Phone: +6838967160603

Job: Mining Executive

Hobby: Woodworking, Knitting, Fishing, Coffee roasting, Kayaking, Horseback riding, Kite flying

Introduction: My name is Msgr. Refugio Daniel, I am a fine, precious, encouraging, calm, glamorous, vivacious, friendly person who loves writing and wants to share my knowledge and understanding with you.