Spring Constant

Simple Harmonious motion i.e. SHM is a veritably intriguing type of stir. It's constantly applied in the oscillatory motion of the objects. Springs generally have SHM. Springs have their own native “ spring constants'' which define how stiff they are. Hooke's law is a notorious law that explains the SHM and gives a formula for the force applied using spring constant.

For example, ask if a group of car designers will knock on your door and help develop a suspension system. "Of course," you say. They say the car weighs 1,000 kilograms and requires the use of four shock absorbers, each 0.5 meters long. How strong should the feathers be? Assuming these shock absorbers use springs, each must support a weight of at least 250 kilograms: 

 F = mg = (250 kg) (9.8 m / s²) = 2,450 N 

Where F is the force, m is the mass of the object, g is the gravitational acceleration, 9.8 meters / sec². The spring in the shock absorber will, at a minimum, have to give you 2,450 newtons of force at the maximum compression of 0.5 meters. What does this mean the spring constant should be? In order to figure out how to calculate the spring constant, we must remember what Hooke`s law says:

F = –kx

Now, we need to rework the equation so that we are calculating for the missing metric, which is the spring constant, or k.

The springs used in the shock absorbers must have spring constants of at least 4,900 newtons per meter.

Spring Constant Units

Spring constant definition is related to simple harmonic motions and Hooke's law. So, before we try to define the spring constant and understand the workings of the spring constant, we need to look at Hooke's law. According to the theory of elasticity, when a load is applied to spring it will naturally extend proportionally, as long as the load applied is less than the elastic limit. Now we know that when force is applied to an object, it tends to deform in some way. 

Consider a spring, when we apply force on one side of the spring, it will get compressed, as they are elastic. At this time the spring exerts its force in the direction opposite to the applied force, to expand to its original size. Therefore, to define the spring constant, we first define Hooke's law. Hooke's law is defined as the force required by the spring to revert to its size is directly proportional to the distance of the compression of the spring.

(Image to be added soon)

The image shows the movement of the spring when force is applied to one side.

Spring Constant Definition

To understand the spring constant definition, we will look at Hooke’s law formula. Hooke’s law formula is also known as the spring constant formula. The formula is given below.

Where F represents the restoring force of the spring, x is the displacement of the spring, and k is known as the spring constant. The spring constant units are given as Newton per meter. 

Now that we know that k is the spring constant, we will look at the spring constant definition. We define spring constant as the stiffness of the spring. In other words, when the displacement of the spring is one unit, we can define the spring constant as the force applied to cause that said displacement. Therefore, it is clear to say that, the stiffer the spring is, the higher will be its spring constant.

Spring Constant Dimensional Formula

According to Hooke’s law, we know that,

F= -kx

Therefore, 

k= -F/x

Now we know that the unit of force is given as Newton (N), or as kg m/s2.

Therefore, we can write the dimensional unit as [MLT-2].

We also know the dimensional unit of x is given as L

Applying spring constant formula, we get, 

\[[\frac{MLT^{-2}}{L}]\]

k= - [MT-2]

The spring constant unit is in terms of Newton per meter (N/m).

Solved Problems

Question 1) A spring is stretched by 40cm when a load of 5kg is added to it. Find the spring constant.

Answer 1) Given,

Mass m = 5kg,

Displacement x = 40cm = 0.4m

To find the spring constant, we first need to find the force that is acting on the spring. 

We know that F = m * x

Therefore, F = 5 * 0.4

F = 2N

The load applies a force of 2N on the spring. Hence, the spring will apply an equal and opposite force of – 2N.

Now, by substituting the values in the spring constant formula we get,

k = -F/x

k = \[\frac{-2}{0.4}\]

k = 5 N/m

Therefore, the spring constant of the spring is 5N/m

Question 2) Consider a spring with a spring constant of 14000N/m. A force of 3500N is applied to the spring. What will be the displacement of the spring?

Answer 2) Given,

Force F = 3500N,

Spring constant k = 14000N/m,

We can calculate the displacement of the spring by using the spring constant formula.

x= \[\frac{-F}{k}\]

The load applies a force of 3500N on the spring. Hence the spring will apply an equal and opposite force of – 3500N.

Thus,

x = \[-(\frac{-3500}{14000})\]

x = 0.25 m

x = 25 cm

Therefore, the spring is displaced by a distance of 25cm.

Conclusion

This is how the spring constant is determined and calculated. The formula is defined using the terms to determine the various physical quantities. Understand how this constant is utilized in solving different problems from the given examples.