Spring Constant Formula Using Hooke's Law

Springs are something that all of us are familiar with. We usually see them in pens, toys and other household objects. One of the distinct things that make a spring special is its ability to maintain it’s length even after stretching. It just comes back into its original shape and is fascinating.  In this section, the student will get a chance to explore this concept further.

Whenever the spring is stretched, a force is applied to elongate the spring in the direction that is away from the center of the spring. This happens whenever someone or something pulls the spring and this creates a tension in the spring that causes it to snap back toward the center of the spring when the force is released, i.e., when the person or thing holding it lets go. This is the force that the student will calculate using Hooke’s law.

Before deriving the formula for the spring constant, here is a simple explanation of why the spring behaves this way. Spring constant is the numerical representation of the internal property of the spring that enables it to retain its shape and length even after stretching.

Spring constant formula is an integral part of simple harmonic motion. In order to understand the formula for the spring constant, firstly we will look at what SHM or what we call Simple Harmonic Motion is. Once we are thorough with the concept of SHM, we will look at how springs are related to the simple harmonic motion and then finally derive the spring constant formula. The detailed explanation provided here also attempts to elaborate the spring constant formula using Hooke’s Law.

Simple Harmonic Motion

Simple harmonic motion is basically the repeated back and forth movement through a central position, such that the maximum displacement on one side of that position is equal to the maximum displacement on the other side. Each complete vibration's time interval is the same. The force responsible for the motion is always directed towards the direction of equilibrium and is directly proportional to the distance from it.

Some common examples of SHM are pendulums. If the resting position of the pendulum is assumed to be the center, then when the pendulum moves, it is observed that the distance it covers on either side of the center is equal, and the pendulum always moves toward the center after covering it’s maximum distance. We shall refer to the center as equilibrium.

Springs usually have SHM. Springs have their own natural “spring constants” which define how stiff they are. Hooke's law is a famous law that explains the SHM and gives a formula for the force applied using the spring constant.

When applied to a spring, we can observe that when the spring is stretched in both directions then both the ends of the spring recoil toward equilibrium. This is very similar to how a pendulum behaves even though it does not move in a circular fashion. 

Hooke’s Law

Hooke's law defines the relation between the force applied and the distance stretched in the spring. The force required to compress or extend a spring is directly proportional to the distance it is extended.

This is based on Newton's third law of motion which states that for every action, there is an equal and opposite reaction. Think of stretching the spring as the action, and the natural recoiling of the spring to be the reaction.

According to Newton's Third Law of Motion, it pulls back with a restoring force when spring is pulled. This restoring force follows the Law of Hooke, which relates the force of the spring to the spring constant. This relation when visualised mathematically, is called the spring constant formula.

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Force of the Spring =  -(Spring Constant) x (Displacement)

F=−K*X

F=−KX

The negative sign indicates the opposite direction of the reaction force.

Where,

F: The spring's restoring force directed towards equilibrium.

K: The constant of spring in N.m-1.

X: The displacement of the spring from its position of equilibrium.

Spring Constant (K)

Now, the Spring constant is defined as the force required to restore the spring to its original shape per unit of extension of the spring. This also means that after knowing the spring constant we can easily find how much force is needed to deform the spring. Because if we apply more force than what the spring can handle, then it will lose the property to exhibit SHM any more.

K=−F * x

K=−Fx

Its unit is N/m (Newton per metre).

How Does the Spring Constant Depend on the Length?

Suppose we have a spring of 6 cm with a spring constant k. What happens if we split the spring into two bits of equal size? There will be a new spring constant for one of these shorter springs, which will be 2k. More generally, a spring's spring constant is inversely proportional to the spring's length, assuming we are talking about a specific material spring and thickness.

This means that more the length of the spring, the lesser the force it will require to reach its equilibrium state and similarly vice versa.

So suppose we cut the spring exactly in two in the above example, making two shorter springs each 3 cm in length. A spring constant, which is twice the original, will be applicable for the smaller springs. That is because it is inversely proportional to the spring constant and the length of the spring. That means that on a shorter spring, the original mass of 30 gm would only yield a 1 mm stretch. The larger the constant of the spring, the smaller the extension that a given force generates. This is why smaller springs are more difficult to stretch!

Hooke’s Law Graph

The graph for Hooke’s law has been provided below. Here the material exhibits elastic behaviour up to the yield strength point, after which the material loses elasticity and exhibits plasticity.

This means that once we successfully apply a force larger than the threshold, the spring loses its elasticity and therefore further application of force will make the spring change its shape and retain it. When a material changes its shape and retains that shape when any force is applied to it, then this property is called plasticity.

Think of the items you have seen that are plastic. Once you apply force, it either bends or breaks. It makes no attempt to stretch back or go back into its original shape like a well functioning spring.

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The straight line implies that the material follows Hooke's law from the origin to the proportional limit approaching yield power. The material loses its elastic existence beyond the elastic limit between the proportional limit and yield strength and begins to exhibit plasticity. The area from the origin to the proportional limit under the curve falls below the elastic range. Under the plastic range, the area under the curve is from a proportional limit to the rupture/fracture point.

To help the students with some of the quick and easy tips to study Physics, we have listed a few points that shall be considered to succeed in the preparation.

• Focus more on the basics of the topic, as you strengthen the basic concepts, the later journey becomes easier.

• Take a note of the small details, missing out on any of the concepts can cost you a lot.

• Always try to use some visual learning techniques such as flowcharts, drawings, diagrams etc to make learning fun and easy.

• Try to simplify the situations and focus on understanding what the questions are asking. Without that, you won't be able to answer them accurately.

• Make use of different study resources, such as online study material, quizzes, your study notes, classroom videos etc. 

• Physics has a lot of practical learning and it involves the use of equations. Students shall make sure to learn and understand the equations and the related formulas.

• Another important skill to ace in this subject is strengthening your Maths skills. Since it comes with a lot of numerical based questions, you shall be quick and accurate with the calculations.