# Spring Constant: Definition, Formula, Unit, Calculation

In this article, you will learn a complete overview of the spring constant such as its definition, formula, unit, dimensional formula, affecting factor, calculation, and many more.

So, without wasting time let's get started.

## What is a Spring Constant?

Spring Constant is the restoring force exerted by spring to restore the position against a unit displacement caused by an external force.

The higher the value of the spring constant higher will be the stiffness of the spring which means a higher load will be required to compress or deflect a spring.

It is also called spring rate and stiffness constant.

It is always a positive or a non-zero value.

## Spring Constant Formula

According to Hooke's Law,

Force is directly proportional to the deflection or displacement of the spring.

Mathematically,

F ∝ x

F = K.x

Where K is a spring constant.

The restoring force (F) developed in spring works opposite to the force applied.

So,

Fr = - F

Fr = - K.x

Which can also be written as,

K = - (Fr/x)

or

K = F/x

Where,

Fr = Restoring force in spring (N)

x = Deformation or displacement in spring (m)

F = Force applied to spring

K = Spring Constant

### Spring Rate Calculation

Spring rate refers to the amount of force required to compress or extend or twist a spring by unit displacement or deflection.

Spring rate does not primarily depend on the material grade of spring.

Putting the values in the formula derived from Hooke's Law,

K = F/x = (W2 - W1)/(h2 - h1)

### Theoretical Method of Calculation

Formula for calculating spring constant

K = (G × d⁴)/(8 × Dm³ × Nac)

Where,

G - Modulus of Rigidity

d - Wire Diameter

Dm- Mean Diameter of Spring

Nac- No. of Active Coils

## Spring Constant Unit

### Spring Constant Unit in SI System

As we know the unit of force and distance in the SI system are Newton (N) and Metre (M) respectively.

So,

Spring Constant,

K = F/x

K = N/m

Hence in the  SI system, the unit of the spring constant will be N/m.

### Spring Constant Unit in FPS System

As we know the unit of force and distance in the SI system are Pound (Ib) and Feet (ft) respectively.

So,

Spring Constant,

K = F/x

K = Ib/ft

Hence in the FPS system, the unit of the spring constant will be Ib/ft.

## Spring Constant Dimensional Formula

As we know,

K = F/x

Since,

Force = mass (m) × acceleration (a)

Hence,

K = (m × a)/x

As we know the unit of mass and acceleration are kg and m/s² respectively.

So,

K = kg × m s ⁻²/m

K = M T⁻²

Hence the dimensional formula for spring constant K will be M T⁻².

## Combination of Spring

Show in the figure combination of spring in series and parallel which are described below in detail.

### Spring in Series

In series combinations, tension is the same in all the springs & extensions will be different.
( If k is the same then deformation is also the same ).

In a series combination, an extension of springs will be reciprocal of its spring constant.

The total displacement when the spring is in series,

x = x₁ + x₂

As we know spring Constant

F = K × x

So,

F/K = x

Hence,

F/Keq = F/K₁ + F/K₂

So, the equivalent constant in the series combination Keq is given by :

1/Keq = 1 / K₁ + 1 / K₂

### Spring in Parallel

In parallel combination, extension is the same for both springs but force acting will be different.

If the force acting in the spring is F then the total force acting on the spring,

F = F₁ + F₂

As we know,

F = K × x

So,

F = ( K₁x + K₂ x )

F = ( K₁ + K₂ )x

K × x  =  ( K₁ + K₂ )x

So, the equivalent constant in parallel combination Keq is given by :

Keq = K₁ + K₂

## Factors Affecting Spring Constant

The spring constant of a spring depends on the following factors
• Wire Diameter of Spring
• Modulus of Rigidity
• Coil Diameter
• No. of Active or Effective Coils

### Wire Diameter of Spring

If the diameter of the spring wire is increased, the spring constant will increase.

### Modulus of Rigidity

If the modulus of rigidity of the spring material is increased, the spring constant will increase.

### Coil Diameter

If the diameter of the spring coil is increased, the spring constant will decrease.

### No. of Active or Effective Coils

If the number of coils of the spring is increased, the spring constant will decrease.

## Spring Constant Calculation

### Question .1

What is the spring constant if it takes 30 N to stretch a spring 0.2 m?

Solution

Given Data,

F = 30 N

x = 0.2 m

K =?

As we know,

F = K × x

So,

K = F/x

K = 30/0.2

K = 150 N/m

### Question .2

A spring of force constant k is cut into lengths of ratio 1: 2 : 3 . They are connected in series and the new force constant is Ks. Then they are connected in parallel and the force constant is Kp. Then what is the ratio of Ks:Kp?

Solution

As we know,

K ∝ (1/l)

So,

K₁l₁ = Constant

Here K is the spring constant and l is the spring length So in Series combination,

1/Ks = 1 / K₁ + 1 / K₂ + 1/K₃

1/Ks = 1 + 2 + 3

1/Ks = 6

Ks = 1/6

Now in parallel combination,

Kp = K₁ + K₂ + K₃

Kp = 1 + 1/2 + 1/3

Kp = 11/6

The ratio of the Series and Parallel combination of the spring will be,

Ks:Kp = 1:11

## FAQ Related to Spring Constant

### What is the spring constant k?

The spring constant, denoted by the symbol k, is a measure of the stiffness of a spring. It is defined as the force required to stretch or compress the spring by a certain distance. The spring constant is a measure of how much force is required to move the spring a certain distance.

### What is spring's constant formula?

The formula for the spring constant is:

k = F / x

where F is the force applied to the spring, and x is the distance the spring is stretched or compressed from its rest position. The spring constant is typically measured in units of newtons per meter (N/m).

### Can the spring constant be zero?

Technically, it is possible for a spring to have a spring constant of zero. This would mean that no matter how much force is applied to the spring, it would not stretch or compress at all. In other words, the spring would be completely rigid and not able to deform in any way.

However, it is not possible for a real-world spring to have a spring constant of exactly zero. This is because all materials have some degree of elasticity, or the ability to stretch or compress when a force is applied to them. Even if a material is very stiff and does not deform much under a given force, it will still have a non-zero spring constant.

So, while it is theoretically possible for a spring to have a spring constant of zero, it is not possible for a real-world spring to have a spring constant of exactly zero.

### Why is spring constant important?

The spring constant is an important property of a spring because it determines how much force is required to stretch or compress the spring by a certain distance. It is used in a variety of calculations involving springs, such as determining the period of oscillation of a mass on a spring or the force required to compress a spring a certain distance.

For example, if you are designing a device that uses a spring to apply a force, the spring constant will determine how much force the spring can exert over a given distance. If you need the spring to exert a large force over a small distance, you will need to use a spring with a high spring constant. On the other hand, if you need the spring to exert a small force over a large distance, you will need to use a spring with a low spring constant.

The spring constant is also important in understanding the behavior of a spring. For example, if you know the spring constant of a spring, you can calculate how much the spring will stretch or compress under a given load. This can be useful in designing structures or machines that rely on springs, such as cars or trampolines.

Overall, the spring constant is an important property of a spring that determines how the spring will behave under different loads and is used in a variety of calculations involving springs.

So here you have to know all aspects related to the spring constant. If you have any doubts then you are free to ask me by mail or on the contact us page.

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