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.

spring constant

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 constant formula in series and parallel

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

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

Thank You.

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