The **spring constant** is a mathematical parameter present in *Hooke’s law*, the mathematical law that describes the stored potential energy of a coiled or stretched spring. Hooke’s Law, named after the English natural philosopher Robert Hooke who originally formulated the principle, states that the distance a spring is stretched or compressed is directly proportional to the force applied. Hooke’s law can be expressed as follows:

**F _{s} = −kx**

Where F_{s }is the force required and x is the distance. The spring constant, written as *k* in the equation, can be viewed as a measure of how difficult it is to stretch a spring. The spring constant tells you the force exerted by the spring per unit distance from the equilibrium state of the spring (the state where it is neither compressed nor stretched). Therefore, the spring constant can be thought of as a measure of the stiffness of the spring: how much force must be exerted to stretch or compress the spring and move it out of balance. The unit of the spring constant *k* is the **newton per meter (N / m).**

Let’s say that a force of 1000N extends a spring at rest by 3 meters. What is the spring constant in this situation? The spring constant can be determined by a simple algebraic analysis:

1000N = -k (3m)

1000N / 3m = -k

-k = 333.33 N / m

k = -333.33 N / m

That is, a spring which is stretched 3 meters by the application of a force of 1000 N has a spring constant value of **-333.33 N / m. **This value essentially means that it takes 333.33 Newtons to move such a spring over a distance of 1 meter. The value is negative because the force exerted by the spring is in the opposite direction to the external force stretching the spring.

The exact value of the spring constant depends on the spring itself. Highly flexible springs would have a small spring constant and would be easy to stretch or compress, whereas heavy, thick springs would have a much higher spring constant and would be more difficult to stretch or compress. In addition to real springs, Hooke’s law is applicable (to a certain extent) in most cases where an elastic body is deformed under the application of a certain force: pinching of a guitar string, the wind which blows and bends tall buildings and fills an elastic balloon part.

## Simple examples of the spring constant in action

Let’s say we have a block attached to a horizontally oriented spring and the spring is at rest. In this position, the applied force is 0 and the displacement is 0. Now suppose we apply a force F to compress the spring by a distance of Δx, then bring the spring back to equilibrium. Now let’s say we apply the same force F to stretch the spring the same distance of Δx. Hooke’s law tells us that the relationship between the displacement of the spring Δx and the application of the force F during compression and stretching can be expressed by:

| F | = k | Δx |

where k is the spring constant. In other words, the spring constant is the mathematical entity that mediates the relationship between applied force and the stretch and compression of a spring.

Let’s put some real numbers on the scenario described previously: let’s say we have a 40cm long spring which is compressed to 35cm under this application of a force of 3 N. What is the spring constant of such a spring?

In this case, our Δx is 5 cm (0.05 m)) and F is 3N. Plugging these values into our equation gives us:

| 3 | = k | 0.05 |

| 3 | / | 0.05 | = k

**k = 60 N / m.**

That is, the spring has a spring constant k of 60 N / m. It would take 60 newtons of force to compress or stretch the spring one meter.

Likewise, working with a given spring constant can help you predict how much a spring will stretch or compress under a given force. Say we have a spring with a spring constant of 87 N / m, and we tried to stretch it with a force of 212 N. How far would the spring stretch?

Again, plugging these values into our equation gives us:

| 212 | = 87N / m | Δx |

| 212 | / 87N / m = x

**x = 2.43 meters**

So applying a force of 212 Newtons to a spring with a spring constant of 87 N / m would stretch the spring 2.43 meters.

Finally, if we know the spring constant and the desired displacement, we can determine the force that we would need to apply to the spring to move it that distance. Let’s say we have a spring that we need to compress 13cm (0.13m) and that has a known spring constant of 57 N / m. How much strength would we need to accomplish this task? The branching of the values gives us:

| F | = 57N / m | 0.13 |

F = 7.41 N

It would take 7.41 newtons of force to move a spring with a spring constant of 57 N / m over a length of 13 cm.

## Hooke’s law in a two-spring system

Say we have an object attached to *of them* springs parallel to each other, each with a different spring constant. How are we supposed to model the combined properties of the two springs and how they affect the stretch and compression of the total system?

Fortunately, there is a simple rule for describing such a system. In a system with two parallel springs each with a different spring constant, the spring constant of the total system is just the sum of the individual spring constants. That is to say in a system with 2 springs and 2 spring constants k_{1} and k_{2}, the total spring constant of this configuration is just

**k _{total} = k_{1} + k_{2}**

Parallel spring systems can be represented mathematically as a single spring whose spring constant is the sum of the individual spring constants.

So let’s say we have a block attached to two springs parallel to each other, the first with a spring constant k_{1}= 100 N / m and the second with a constant k_{2}= 200N / m. How much force would be required to move the entire spring system 5 cm?

Since we know the individual spring constants, we can simply add the two values to get the constant for the entire system. Plugging these values into our equation gives:

| F | = (k_{1}+ k_{2}) (Δx)

| F | = (100 N / m + 200 N / m) (0.05 m)

| F | = (300 N / m) (0.05 m) = **15 N**

It would therefore take 15 newtons of force to move the entire spring system 5 cm. The linear nature of the relationship between force and compression / stretch allows us to combine the properties of the two springs to model the behavior of the entire 2-spring system.

## Limits of Hooke’s Law

Hooke’s law is a first order approximation of the behavior of elastic materials and is not applicable in all fields. Hooke’s Law only works for certain stretch / compression ranges, as there is a maximum distance an elastic material can be stretched or compressed before permanently deforming. Hooke’s law is therefore applicable on a relatively small scale of forces. Despite this limitation, Hooke’s Law is extremely useful for engineers and forms the basis of disciplines that study spring-like phenomena, such as seismology and acoustics.

In the more general case of compressing / stretching elastic bodies, the mathematical form of Hooke’s law is accurate, but its specific parameters may not be. In more complex systems, stresses and strains have independent components, so it would not be fair to represent them with a single real number, but a *tensor* which maps multiple vectors to a single point in the material.

## Physical importance of the spring constant

These equations are all great, but what exactly is the spring constant for? *mean *in the context of physics? The other parameters of Hooke’s Law, like force and displacement, clearly refer to a physical thing (force) or property (displacement), so what object or property does the spring constant refer to? ?

We can think of it this way: the spring constant refers to a physical property of the spring, in particular the force of the spring. *himself* exerts in response to compression or stretching. According to Newton’s third law of dynamics, every action has an equal and opposite reaction. So when you pull on the spring, it moves back with equal force. When you press the spring, it pushes back with equal force. The spring constant is then only a measure of the relation between this force exerted and the distance traveled by the spring; the amount of force exerted by the spring per unit of displacement.

Alternatively, one can think of the spring constant as a measure of how much *potential energy* a compressed or stretched spring has stored in its turns. Compressing or stretching a spring beyond its point of equilibrium places stress on the materials of the spring, a stress which is mathematically represented as a force in the opposite direction of pushing or pulling. When the spring is released, this stored energy is converted into kinetic energy and the spring returns to its equilibrium state. Thus, Hooke’s law can be considered as a special case of the more general principles governing the relationship between kinetic and potential energy.

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