# Hooke's Law Calculator

Created by Bogna Szyk
Reviewed by Steven Wooding
Last updated: Dec 21, 2022

We created the Hooke's law calculator (spring force calculator) to help you determine the force in any spring that is stretched or compressed. You can also use it as a spring constant calculator if you already know the force. Read on to get a better understanding of the relationship between these values and to learn the spring force equation.

## Hooke's law and spring constant

Hooke's law deals with springs (meet them at our spring calculator!) and their main property - the elasticity. Each spring can be deformed (stretched or compressed) to some extent. When the force that causes the deformation disappears, the spring comes back to its initial shape, provided the elastic limit was not exceeded.

Hooke's law states that for an elastic spring, the force and displacement are proportional to each other. It means that as the spring force increases, the displacement increases, too. If you graphed this relationship, you would discover that the graph is a straight line. Its inclination depends on the constant of proportionality, called the spring constant. It always has a positive value.

## Spring force equation

Knowing Hooke's law, we can write it down it the form of a formula:

$F = -k Δx$

where:

• $F$ — The spring force (in $\mathrm{N}$);
• $k$ — The spring constant (in $\mathrm{N/m}$); and
• $Δx$ is the displacement (positive for elongation and negative for compression, in $\mathrm{m}$).

Where did the minus come from? Imagine that you pull a string to your right, making it stretch. A force arises in the spring, but where does it want the spring to go? To the right? If it were so, the spring would elongate to infinity. The force resists the displacement and has a direction opposite to it, hence the minus sign: this concept is similar to the one we explained at the potential energy calculator: and is analogue to the [elastic potential energy]calc:424).

🙋 Did you know? the rotational analog of spring constant is known as rotational stiffness: meet this concept at our rotational stiffness calculator.

## How to use the Hooke's law calculator

1. Choose a value of spring constant - for example, $80\ \mathrm{N/m}$.
2. Determine the displacement of the spring - let's say, $0.15\ \mathrm{m}$.
3. Substitute them into the formula: $F = -kΔx = -80\cdot 0.15 = 12\ \mathrm{N}$.
4. Check the units! $\mathrm{N/m \cdot m} = \mathrm{N}$.
5. You can also use the Hooke's law calculator in advanced mode, inserting the initial and final length of the spring instead of the displacement.
6. You can now calculate the acceleration that the spring has when coming back to its original shape using our Newton's second law calculator.

You can use Hooke's law calculator to find the spring constant, too. Try this simple exercise - if the force is equal to $60\ \mathrm{N}$, and the length of the spring decreased from $15\ \mathrm{cm}$ to $10\ \mathrm{cm}$, what is the spring constant?

## FAQ

### Does Hooke's law apply to rubber bands?

Yes, rubber bands obey Hooke's law, but only for small applied forces. This limit depends on its physical properties. This is mainly the cross-section area, as rubber bands with a greater cross-sectional area can bear greater applied forces than those with smaller cross-section areas.
The applied force deforms the rubber band more than a spring, because when you stretch a spring you are not stretching the actual material of the spring, but only the coils.

### Why is there a minus in the equation of Hooke's law?

The negative sign in the equation F = -kΔx indicates the action of the restoring force in the string.

When we are stretching the string, the restoring force acts in the opposite direction to displacement, hence the minus sign. It wants the string to come back to its initial position, and so restore it.

### What is the applied force if spring displacement is 0.7 m?

Let's consider the spring constant to be -40 N/m. Then the applied force is 28N for a 0.7 m displacement.

The formula to calculate the applied force in Hooke's law is:
F = -kΔx

where:
F is the spring force (in N);
k is the spring constant (in N/m); and
Δx is the displacement (positive for elongation and negative for compression, in m).

### What happens if a string reaches its elastic limit?

The elastic limit of spring is its maximum stretch limit without suffering permanent damage.
When force is applied to stretch a spring, it can return to its original state once you stop applying the force, just before the elastic limit. But, if you continue to apply the force beyond the elastic limit, the spring with not return to its original pre-stretched state and will be permanently damaged.

Bogna Szyk
Spring displacement (Δx)
ft
Spring force constant (k)
N/m
Force (F)
N
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