Springs are found in many everyday devices. They play an important role in click pens, door closers, vehicle suspensions, and more.

Springs are useful due to the way they interact with other objects. But, what determines the magnitude and direction of the force a spring exerts on an object? Let's analyze the results of a sample experiment to find out.

Collecting spring force data

A group of students has a spring. One end of the spring is attached to a wall, and the other end is attached to a block. The block is on a horizontal surface with negligible friction.

If the block is at rest and the students do not apply force to the block, the block remains at rest with the spring at a particular length. This is the spring's equilibrium length, \[L_0.\] The students designate a coordinate system with the origin at the spring's equilibrium length, as modeled below:

The spring exerts zero force on the block when at its equilibrium length.

Now, the students want to determine how much force the spring exerts on the block when the spring is not at its equilibrium length. The students stretch the spring by pulling the block to the right. They measure the force \[\vec{F}_a\] they must apply to the block to keep it at rest when the spring is stretched by an amount \[\Delta x\] from equilibrium:

Current

The block modeled above remains at rest while the students apply a horizontal force \[\vec{F}_a\] to the block.

Compare the force the students apply to the block, \[\vec{F}_a,\] to the force the stretched spring exerts on the block, \[\vec{F}_s.\]

The magnitude of \[\vec{F}_a\] is

the magnitude of \[\vec{F}_s.\]

The direction of \[\vec{F}_a\] is

the direction of \[\vec{F}_s.\]

CheckExplain

The students measure how much force they must apply to the block to gradually stretch the spring by different amounts \[\Delta x.\] They use their applied force to determine the force the spring exerts on the block, \[\vec{F}_s,\] for each stretched (positive) displacement, \[\Delta x.\]

The students repeat this process in the other direction too. They gradually compress the spring by pushing the block to the left. They record the force the spring exerts on the block, \[\vec{F}_s,\] for each compressed (negative) displacement, \[\Delta x.\]

The students plot their \[\vec{F}_s\] vs. \[\Delta \vec{x}\] data from stretching the spring in the positive direction and compressing the spring in the negative direction. They get the following graph:

Current

Based on the shape of the graph, how are the magnitudes \[F_s\] and \[\Delta x\] related?

Choose 1 answer:Choose 1 answer:

(Choice A)   

\[F_s\] is proportional to \[\dfrac{1}{\Delta x}\]

A

\[F_s\] is proportional to \[\dfrac{1}{\Delta x}\]

(Choice B)   

\[F_s\] is proportional to \[\Delta x\]

B

\[F_s\] is proportional to \[\Delta x\]

(Choice C)   

\[F_s\] is proportional to \[\sqrt{\Delta x}\]

C

\[F_s\] is proportional to \[\sqrt{\Delta x}\]

(Choice D)   

\[F_s\] is proportional to \[(\Delta x)^2\]

D

\[F_s\] is proportional to \[(\Delta x)^2\]

How do the directions of \[\vec{F}_s\] and \[\Delta \vec{x}\] compare?

Choose 1 answer:Choose 1 answer:

(Choice A)   

\[\vec{F}_s\] and \[\Delta \vec{x}\] point in the same direction

A

\[\vec{F}_s\] and \[\Delta \vec{x}\] point in the same direction

(Choice B)   

\[\vec{F}_s\] and \[\Delta \vec{x}\] point in opposite directions

B

\[\vec{F}_s\] and \[\Delta \vec{x}\] point in opposite directions

CheckExplain

The students repeat their experiment with two different springs \[(\text{B}\] and \[\text{C}).\] They plot their results on the same graph as the original spring \[(\text{A}).\]

Current

Based on the graph above, how do springs \[\text{B}\] and \[\text{C}\] compare physically to spring \[\text{A}?\]

Spring \[\text{B}\] is

than spring \[\text{A}.\]

Spring \[\text{C}\] is

than spring \[\text{A}.\]

CheckExplain

As suggested by the three straight lines on the graph above, all springs demonstrate the same proportional relationship between \[\vec{F}_s\] and \[\Delta \vec{x}.\] So, we're ready to construct a general mathematical model that applies across all springs.

Hooke's law

The sample experiment above revealed these features of the spring force:

The magnitude of the force a spring exerts is directly related to how much its length is changed from its equilibrium length. When the spring is at equilibrium length, it exerts no force. The farther the spring is stretched or compressed from this length, the stronger the force it exerts.

All springs demonstrate this same proportional relationship between spring force and change in length. However, the constant of proportionality between \[\vec{F}_s\] and \[\Delta \vec{x}\] depends on the spring. Some springs are stiffer or looser than others.

The direction of the force a spring exerts on an object is always directed towards the equilibrium position. If the spring is stretched, it pulls the object back towards equilibrium. If the spring is compressed, it pushes the object back towards equilibrium.

Current

Which mathematical model for the force exerted by a spring, \[\vec{F}_s,\] and the change in the spring's length, \[\Delta \vec{x},\] matches the properties above?
Assume \[k\] is a positive constant.

Choose 1 answer:Choose 1 answer:

(Choice A)   

\[\vec{F}_s=-k\Delta\vec{x}\]

A

\[\vec{F}_s=-k\Delta\vec{x}\]

(Choice B)   

\[\vec{F}_s=k\Delta\vec{x}\]

B

\[\vec{F}_s=k\Delta\vec{x}\]

(Choice C)   

\[\vec{F}_s=\Delta\vec{x}+k\]

C

\[\vec{F}_s=\Delta\vec{x}+k\]

(Choice D)   

\[\vec{F}_s=\Delta\vec{x}-k\]

D

\[\vec{F}_s=\Delta\vec{x}-k\]

Check

This model is called Hooke's law. It relates the force a spring exerts on an object \[(\vec{F}_s)\] to the spring's change in length from equilibrium \[(\Delta\vec{x}):\]

\[\boxed{\vec{F}_s=-k\Delta\vec{x}}\]

The positive constant of proportionality \[(k)\] is called the spring constant. Each spring has its own \[k\] value which represents how stiff or loose the spring is. The spring constant has units of \[\text{N/m}.\]

Finally, the negative sign in Hooke's law represents that the spring force is a restoring force. It works to restore equilibrium, always pulling or pushing an object back towards the position where the spring is at equilibrium length. If the spring is stretched in the positive direction so \[\Delta\vec{x}\] is positive, then \[\vec{F}_s\] points in the negative direction (and vice versa).

Assumptions and notes

In the sample experiment above, we assumed each spring was an ideal spring. An ideal spring has negligible mass, therefore it does not contribute inertia or weight to the system. Additionally, an ideal spring perfectly follows Hooke's law.

However, an ideal spring is an approximation. Real springs do have mass. And in reality, a spring only follows Hooke's law within a limited range of displacements. If a real spring is stretched or compressed too far, the force it exerts may no longer match the predicted value from Hooke's law. Beyond some limit, the spring becomes permanently deformed.

We also represented all springs in this article as coils. But it's worth noting that springs exist in a variety of other shapes as well.