# How to Calculate Spring Constant

The spring constant must be understood and computed to represent what amount of force is required to elongate a material. Before moving ahead, it’s very important to Understand the Hooke’s law Statement; which states that the extension of the Spring force is directly Proportional to the force used to stretch the spring. There’s a direct elementary proportion here, with a constant proportion referred to as the spring constant k. Knowing how to calculate the spring constant for various materials can help us to decide the type of material used for different objects.

Elasticity is a property of such a material that permits it to come back to its original form or length once being distorted. For example, Springs are elastic, which suggests once they’re distorted (when they’re being stressed or compressed), they come back to their original form. Springs are found in several objects that we use in our daily life. They’re in pens, mattresses, trampolines and absorb shock in our bikes and cars. Therefore, determining the spring constant is an important parameter. The materials are stretchable because they contain long-chain molecules bound up in a bundle and might straighten out once stretched.

This article will enable you to understand the constant spring formula, how to calculate the spring constant step by step, and give practical examples of where it can be implemented.

## What is Spring Constant

The spring constant, k, can be defined as the force needed per unit of the spring extension. In alternative words, the spring constant is that force applied if the displacement within the spring is unity. It tells us about the stiffness of the spring. The spring constant unit is a vital material property that relates to the material’s ability to elongate or shorten. It’s different for various springs and materials. Springs with larger spring constants tend to have smaller displacements than springs with lesser spring constants for identical mass added.

Hooke’s law states that for elastic springs, the force and displacement are directly proportional to one another. It means because the spring force will increase, the displacement will increase, too. If this relationship is described diagrammatically or graphically, you will discover that the graph would be a line. Its inclination depends on the constant of proportion, referred to as the spring constant. It cannot be a negative value. Some materials don’t seem to be elastic as they’re brittle and can snap before they bend or stretch. Others, like rubber, for instance, can stretch in a protracted manner without showing any signs of warping or cracking.

## Spring Constant Formula

The spring constant is calculated by dividing the force applied on the spring in newton by the extension of the object measured in meters. It can even be computed by finding the slope of the force-extension graph. The spring constant formula is given as:

Where,

F = the normal force applied on the spring in Newton’s (N)

k = spring constant, in Newton’s per meter (N/m)

x = displacement of the spring from its Original position.

The negative sign represents that the restoring force is acting in the opposite direction of displacement. Its units are Newtons per meter (N/m).

## How to Calculate Spring Constant

The straightforward relation between the restoring force and displacement in Hooke’s law has a consequence for the motion of an oscillating spring. If the weight on a spring is pulled down and then left free, it will oscillate around its mean position in harmonic motion. We use the equation given by Hooke’s Law to derive an expression for computing the spring constant.

### Calculation Step by Step

1. Measure the force applied on the spring in Newton (N). If the spring’s load is in kg, convert it into N by multiplying it with gravitational acceleration 9.81 m/s2
2. Determine the displacement in the spring, the distance by which it is compressed or stretched. The change in length must be noted.
3. Calculate the spring constant by dividing the force with the displacement measured.

### Example 1

A man weighing 20 lbs stretches a spring by fifty centimeters. Calculate the spring constant.

Mass conversion from lbs to kg, (=A3/2.2)

Force calculation, F= 9.09*9.8 (A4*9.8)

Displacement Unit conversion, cm to m (D3/100)

Calculate Spring Constant, k = -F/x = 89.09/0.5 (=C5/D5)

Ans, K = 178.18 (N/m)

## Tips for Calculating Spring Constant

• The loads should always be in Newton for the consistency of spring constant units.
• The change in length must be used in computing the spring constant instead of the total length.
• Several measurements can be taken for displacements against different loads and plotted to obtain a straight line on the force-extension graph. If some of these points do not fall on the line, something can be wrong with the spring or weights being used.

## Graphical Explanation of Spring Constant

The most common  method to get values for a graph representing Hooke’s law is to suspend the spring from a hook and connect a series of weights whose values are weighted accurately. To plot a line, take a minimum of 2 measurements; however, additional measures are preferred.

To plot the points on graph, suspend the spring vertically from a hook and record its extension with the help of a ruler. Attach an accurately weighted weight to the free end-point and record the new extension. The difference between the two is x. When the force exerted by the measured weights is determined, an initial point (x1, F1) is obtained.

Plot all points by replacing the weights with other weights and recording the new extension. Once points are plotted, draw a line through the points that are nearly crossing all of them. Find the slope of the Force-Extension Graph. If the initial point is (x1, F1), and the 2nd point is (x2, F2), the slope of that line is:

This gives us the value needed of the spring constant, k. Despite the sign in the Hooke’s law equation, the spring constant is always greater than zero because the slope in the Hooke’s law graph is always positive.

### Example 2

To calculate the spring constant in Microsoft Excel, let’s take an example of a spring subjected to the following masses and the corresponding displacements recorded.

All the masses of objects are noted in kg, so they will be converted into newtons by using the following formula in cell number C3 on the excel sheet:

= A3 * 9.81

Use the same formula for all masses in column C.

Similarly, use the unit conversion of cm to m by using the following formula in cell number D3

=B3/100

Use the same formula for all masses in column D.

Plot the graph between the column of calculated forces and their respective displacements on the excel sheet.

Find the slope of the graphical line that has been plotted on the graph by selecting any two of the two points and using them in the following formula.

Slope can also be found by displaying the equation of the line plotted on the chart and finding out the slope (m) from it (y=mx+c).

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