**Spring Constant**

**Contents : What is spring constant and what applications does it have ?**

**Spring constant Definition****Spring constant formula****Spring constant unit****How to find spring constant****?****Constant force spring****Spring constant values****Spring constant graph****Effective Spring constant****Torsional spring constant****Factors affecting spring constant****Constant torque spring****Spring constant range****Ideal spring constant****Constant force spring assembly****FAQs****Spring constant problems**

**What is spring constant ?**

**Spring constant definition:**

Spring constant is the measure of the stiffness of the spring. Springs having higher stiffness are more likely difficult to stretch. springs are elastic materials. when applied by external forces spring deform and after removal of the force, regains its original position. The deformation of the spring is a linear elastic deformation. Linear is the relationship curve between the force and the displacement.

**Spring constant formula: **

F= -Kx

Where,

F= applied force,

K= spring constant

x = displacement due to applied load from normal position.

**Spring constant units: **

the spring-constant represented as K, and it’s unit is N/m.

**How to find spring constant?**

**Spring constant equation: **

The Spring-constant is determined according to the Hooke’s law stated as below:

The applied force on the springs is directly proportional to the displacement of the spring from the equilibrium.

The proportionality constant is the spring’s constant. The spring force is in opposite direction of force. So, there is a negative sign between the relation of the force and the displacement.

F= -Kx

Therefore,

K= -F/x(N/m)

**Dimension of spring constant:**

K=-[MT^-2]

**Constant force spring:**

Constant force spring is the spring that does not obey Hooke’s law. The spring has the force it exerts over its motion range is constant and does not vary by any means. Generally, these springs are constructed as springs rolled up such that the spring is relaxed when fully rolled up and after unrolling the restoring force takes place as the geometry remains constant as the spring unrolls. The constant force spring exerts the constant force for unrolling due to the change in radius of curvature is constant.

**Applications of constant spring force:**

**Brush springs for motors****Constant force motor springs****Counterbalance springs for window****Carriage returns springs of typewriters****Timers****Cable retractors****Movie cameras****Extension springs**

The constant spring force does not give constant force at all the time. Initially, it has a finite value and after the spring is deflected 1.25 times its diameter it reaches full load and maintains the constant force in the spring despite the deformation. These springs are made with metal strips and not with wires The springs are made up of materials like stainless steel, High carbon steel, etc. springs give tension in the linear direction.

The performance, corrosion elements, temperature affects the fatigue of such springs. They are more likely to have a lifespan of 2500 cycles to more than one million depending on the size and load applied.

**Spring constant examples**

**Spring constant of a rubber band:**

Rubber band acts like spring within certain limitations. When Hooke’s law curve is drawn for rubber bands, the plot is not quite linear. But if we stretch the band slowly it might follow Hooke’s law and have spring-constant value. Rubber band can stretch only its elastic limit that

also depends on the size, length, and quality.

**Spring constant values:**

Spring constant value is determined using the Hooke’s law. As per the Hooke’s law, when spring is stretched, the force applied is directly proportional to the increase in length from the original position.

**How to determine spring constant?**

F=-Kx

K=-F/x

**Spring constants of materials :**

## Spring constant for **Steel** **=21000 kg/m**^{3}

^{3}

## Spring constant for **Copper = 12000 kg/m**^{3}

^{3}

**How to find spring constant from graph ?**

**Spring constant graph:**

**Can the spring constant be negative?**

This can not be negative.

**Spring constant formula with mass:**

T=

where,

T= period of spring

m=mass

k=spring constant

**Effective spring constant:**

Parallel: When two massless springs which obey Hooke’s law and connected through the thin vertical rods at the ends of the springs, connecting two ends of springs are said to be parallel connection.

The constant force direction is perpendicular to the force direction.

Spring constant K written as,

K=K1+K2

Series:

When springs are connected to each other in a series manner such that the total extension combination is the sum of total extension and spring’s constant combination all the springs.

The Force is applied at the end of the end spring. The force direction is in the reverse direction as the springs compressed.

Hooke’s law,

F1=k1x1

F2=k2x2

x 1+ x 2 =

**Equivalent spring constant:**

K =

**Torsional spring constant:**

A torsion spring is twisted along the axis of the spring.When it is twisted it exerts torque in opposite direction and is proportional to the angle of the twist.

A torsional bar is a straight bar that is subjected to twisting gives shear stress along the axis torque applied at its end.

**Examples: **

Helical torsional spring, torsion bar, torsion fiber

**Applications:**

clocks-clocks has spring coiled up together in a spiral, It is a form of helical torsional spring.

**torsional spring constant formula | Torsion coefficient**

Within elastic limit torsional springs obey Hook’s law as it twisted within elastic limit,

Torque represented as,

τ = -Kθ

τ = − κ θ

K is displacement called the torsional spring coefficient.

The -ve sign specifies that torque is acting in reverse to twist direction.

The energy U, in Joules

U= ½*Kθ^2

**Torsional balance:**

Torsion balance is torsional pendulum. It works as a simple pendulum.

To measure the force, first, need to find out the spring’s constant. If the force is low, it’s difficult to measure the sparing constant. One needs to Measure the resonant vibration period of the balance.

The frequency depends on the Moment of Inertia and the elasticity of the material. So, the frequency is chosen accordingly.

Once the Inertia is calculated, springs constant is determined,

F=Kδ/L

**Harmonic Oscillator:**

Harmonic oscillator is a simple harmonic oscillator when undergoes deformation from the original equilibrium position experiences restoring force F is directly proportional to the displacement x.

Mathematically written as follows,

F= -Kx

**Torsional Spring rate:**

Torsional spring rate is the force of spring travelled around 360 degrees. This can be further calculated by the amount of force is divided by 360 degrees.

**Factors affecting spring constant:**

**Wire diameter: The diameter of the wire of the spring****Coil diameter: The diameters of the coils, depending on the stiffness of the spring.****Free length: Length of the spring from equilibrium at rest****The number of active coils: The number of coils that compress or stretch.****Material: Material of the spring used to manufacture.**

**Constant torque spring:**

Constant torque spring is a type of spring that is a stressed constant force spring traveling between 2 spools. After the release of the compressed spring torque is calculated from the output spool as the spring returns back to its original equilibrium position in the storage spool

**Spring constant range:**

*k* = *k*’ *δ*’/*δ*,

K Varies from

Minimum= 0.9N/m

Maximum=4.8N/m

Spring’s constant depends on the number of turns n.

**Ideal spring constant:**

The spring constant is the measure of the stiffness of the springs. The larger the value of k, the stiffer is the spring and it is difficult to stretch the spring. Any spring that obeys Hooke’s law equation is said to be an **ideal spring**.

**Constant force spring assembly:**

A Constant Force Spring is mounted on a drum by wrapping it around the drum. The spring has to be tightly wrapped. Then the free end of the spring is attached to the loading force such as in a counterbalance uses or vice-versa.

**The drum diameter should be larger than the inside diameter.****Range: 10-20% drum diameter> Inside diameter.****One and a half-wrapped spring should be on the drum at extreme extension.****The strip will be unstable at the larger extensions so it is advisable to keep it smaller.****Pulley diameter must be greater than the original diameter.**

**FAQs:**

**Why is spring constant important?**

The spring-constant is important as it shows the basic material property. This gives exactly how much force is required to deform any spring of any material. The higher spring’s constant shows the material is stiffer and the lower spring’s constant shows the material is less stiff.

**Can spring constant change?**

Yes. spring-constant can change as per the force applied and the extension of the material.

**Can spring constant be 0 ?**

No. The spring-constant cannot be zero. If it is zero, the stiffness is zero.

**Can spring constant has negative value?**

No. the Spring-constant always has a positive value.

**When are Young’s modulus and Hooke’s spring constant equal?**

When the ratio of the length to that area of the spring is unity, then the young’s modulus and the spring’s constant value will be equal.

Spring constant is represented as, K=-F/x,

The above mentioned equation shows the relationship between springs constant and the extension of the spring for the same applied force

**Why a spring is cut in half, its spring constant changes?**

This is inversely proportional to the extension of the spring. when the spring is cut into half, the length of the spring reduces hence the spring’s constant will be doubled.

**Does Newton’s third law fails with a spring ?**

**Answer : No**

**Spring constant problems:**

**Q1) A spring is stretched by 20cm and a 5kg load is added to it. Find the spring constant.**

Given:

Mass m = 5kg.

Displacement x=20cm.

Solution:

## 1.Find out the force applied on the spring

F= m*x

= 5*20*10^-2

= 1N.

The load applied on the spring is 1N. So, the spring will apply an equal and opposite load of -1N.

## 2. Find out the spring constant

K= -F/x

=-(-1/20*10^-2)

= 5N/m

The constant of the spring is 5N/m.

**Q2)A force of 25 KN is applied on the spring of spring constant of 15KN/m.Find out the displacement of the spring.**

Given:

Applied force= 2.5KN

Spring-constant=15KN/m

Solution:

1.Find out the displacement of the spring

The spring will apply equal and opposite force of -2.5KN

F=-Kx

X=-F/K

= – 2.5/15

= 0.167m

Hence the spring is displaced by 16.67cm.

**Q3)A spring with a force constant of 5.2 N/m has a relaxed length of 2.45m and spring’s perpendicular length 3.57m. When a mass is attached to the end of the spring and permitted to rest. What is elastic potential energy stored in the spring?**

Solution:

Given:

Force constant= 2.45m

x = 2.45m

L= 3.57m

Force constant spring:

F= -Kx

The work was done due to stretching of the spring= Elastic potential energy of the spring.

W=Kx^2/2

Extension x = 3.57-2.45

=1.12

W=5.2*1.12^2/2

=3.2614 J.

**Q4) A massless spring with force constant k 400** **N/m hangs vertically from the ceiling. A 0.2 kg block is attached to the end of the spring and released. The highest elastic strain energy kept in the spring is (g= 10m/s^2).**

Given:

Force constant= 400N/m

m = 0.2kg

g= 10m/s^2

Solution:

Maximum elastic strain energy=1/2*K*x^2

=

=0.02J

**Spring constant with multiple springs **

**A spring is cut into 4 equal parts and 2 are parallel What is the new effective spring constant of these parts?**

The spring’s constants of the four springs is k1, k2, k3, k4

respectively,

Parallel:

Equivalent spring’s constant (k5) = k1 + k2

Series;

**Total equivalent springs constant of the system:**

K=

**If a spring constant of 20N /m and it is stretched by 5cm what is the force acting on the spring:**

Given:

K=2 N/m.

x = 5cm.

According to Hooke’s law,

F= -Kx

= – 20*5*10^-2

=-1N

Spring force is in opposite direction

Hence spring force = 1N.

An object with a weight of 5.13 kg placed on top of a spring compresses it by 25m What is the force constant of the spring How high will this object go when the spring releases its energy.

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