# Hooke's Law Calculator

Created by Davide Borchia
Last updated: Aug 06, 2022

Spring into action with our Hooke's law calculator: handy and foolproof, our tool will tell you the answer to any of your problems involving springs and elastic materials. Well, almost any.

• What is the spring constant in an elastic material;
• What is Hooke's law formula;
• How to find the spring constant by repeating measurements;
• Examples of applications of the equation for a spring force and deformation.

To learn about another force related to the movement of an object, check how to find the coefficient of friction in our another tool.

## A primer on springs: what is the spring constant

Springs are simple mechanical devices capable of storing mechanical energy in the elastic potential energy. Their secret lies in their elasticity, the property of an object to return to a defined shape after a force has been applied.

By experience, we know springs have different stiffness: this depends on the material they are made of, their fabrication properties, and even on the environment.

Microscopically speaking, elasticity is a characteristic of many materials: lattices (for metals) and chains (for polymers) are able to "accept" a deformation, changing their spatial structure (but not their chemical one) due to the tension force. Lattices bend while chains stretch.

## Hooke's law equation: the spring constant formula

The microscopic behavior of a spring can surely be modeled in a comprehensive way, considering each atom, their degree of freedom, angles, and so on: it would be thoroughly impractical. Luckily, when studied from a macroscopic point of view, a spring has a rather simple behavior, described by the Hooke's law equation:

$F =- k\cdot x$

Three letters: let's check what they mean:

• $F$ is the force (in this case the restoration force);
• $k$ is the spring constant: in the calculations, this is the only part we assume to be constant;
• $x$ is the displacement, or deformation; it equals the change in length of the spring.

Hooke's law formula is surprisingly straightforward: a simple proportionality relationship between length and force. The negative sign models the fact that the restoration force and the deformation act in the opposite direction. If you are using the applied force, the negative sign is not necessary.

In the formula, the spring constant is measured in $\text{N}/\text{m}$, which — intuitively — suggests that for each $k$ newtons, our spring would deform by $1$ meter.

We "spoiled" you that we measure the force in newtons and the deformation of the spring in meters, but, of course, you can change the units freely as long as you maintain the correct dimensions and are consistent across your Hooke's law calculations.

## How to calculate the spring constant from the Hooke's law formula

Here you will learn how to find the spring constant. There are no methods to measure it directly; however, you can use Hooke's law to calculate the spring constant by taking samples, pair of measurements of both force and displacement. If you take enough of them, you can build a statistic and calculate the value of $k$ to predict the behavior of a spring in an experimental setting.

How do you take force measurements for the calculation of the spring constant? Easy peasy! Hang weights to the spring, held vertically. If you know the mass, you can calculate the force simply by multiplying that value by $g$, the gravitational acceleration.

Assume we measured the following set of pair of measurements. Apply then the reversed formula for the

$F$

$x$

$k$

$0.98\ \text{N}$

$-1.28\ \text{cm}$

$76.64\ \text{N}/\text{m}$

$1.96\ \text{N}$

$-2.7\ \text{cm}$

$72.59\ \text{N}/\text{m}$

$4.90\ \text{N}$

$-6.5\ \text{cm}$

$75.38\ \text{N}/\text{m}$

$9.81\ \text{N}$

$-12.9\ \text{cm}$

$76.05\ \text{N}/\text{m}$

We can find the average value of the spring constant: $75.17$. Now that we know the value of the spring constant, the formula of Hooke's law will allow us to calculate the value of the spring deformation.

## Applications of the spring force equation

Use our Hooke's law calculator to find the value of the deformation of a spring, and not only.

We are almost sure you know slinkies, those springy toys that can walk down the stairs. If you have ever touched one of them, you would know that the spring is extremely "soft": its spring constant is a mere $0.84\ \text{N}/\text{m}$. If you have one at home, you know how to calculate its spring constant: try it!

Imagine hanging a small banana from an average slinky: with a weight of $110\ \text{g}$, our spring would elongate by:

\begin{align*} x &= \frac{F}{k}= \frac{0.110\ \text{kg}\cdot 9.81 \frac{\text{m}}{\text{s}^2}}{0.84\ \frac{\text{N}}{\text{m}}}\\ & = 1.28\ \text{m} \end{align*}

We expected it! On the contrary, take a really stiff spring, like the suspension spring of a car. The value of the spring constant is $49\ \text{kN}/\text{m}$. Assume that the weight of the car is $1,400\ \text{kg}$. How much compression would be experienced by each spring?

\begin{align*}x &=\frac{\frac{1,\!400}{4}\ \text{kg}\cdot 9.81 \frac{\text{m}}{\text{s}^2}}{49\cdot 10^3\ \frac{\text{N}}{\text{m}}}\\ & = 0.07\ \text{m} \end{align*}

We divided by $4$ assuming that the car mounts the same springs on each wheel, hence dividing the total weight on all of them. The compression of $7\ \text{cm}$ is in the safe range.

Davide Borchia
Spring displacement (Δx)
ft
Spring force constant (k)
N/m
Force (F)
N
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