KAHIBARO
Table of Contents
Restoring force of a spring
A spring force appears when an elastic object, such as a spring, is stretched or compressed away from its natural length. The key idea is that the spring resists the deformation and tries to return to its original shape. Because of this, the spring force is called a restoring force.
If a spring is left alone at its natural length, it exerts no force. If it is stretched, it pulls inward. If it is compressed, it pushes outward. In both cases, the force points toward the equilibrium position, which is the position where the spring is neither stretched nor compressed.
Hooke's law
For many ordinary springs, as long as the deformation is not too large, the spring force is proportional to the displacement from equilibrium. This is Hooke's law:
$$F_s = -kx$$
Here, $F_s$ is the spring force, $k$ is the spring constant, and $x$ is the displacement from the spring's natural length. The minus sign shows that the force is opposite to the displacement.
If $x > 0$, the spring is stretched and the force points back toward equilibrium. If $x < 0$, the spring is compressed and the force again points back toward equilibrium.
Hooke's law for an ideal spring is
$$F_s = -kx$$
The spring force always acts opposite to the displacement from equilibrium.
Meaning of the spring constant
The spring constant $k$ measures how stiff the spring is. A large value of $k$ means the spring is hard to stretch or compress. A small value of $k$ means the spring is easier to deform.
Since force is measured in newtons and displacement in meters, the unit of $k$ is
$$[k] = \text{N/m}$$
A spring with $k = 200\ \text{N/m}$ is stiffer than a spring with $k = 50\ \text{N/m}$.
Direction of the force
The direction of spring force depends on whether the spring is stretched or compressed.
| Situation | Displacement $x$ | Spring force direction |
|---|---|---|
| Natural length | $0$ | No spring force |
| Stretched to the right | Positive | To the left |
| Compressed to the left | Negative | To the right |
It is very important to choose a coordinate direction first. Once that is done, the sign of $x$ determines the sign of the spring force.
Natural length and deformation
The natural length of a spring is the length at which it is not deformed. If the current length is $L$ and the natural length is $L_0$, then the deformation is
$$x = L - L_0$$
If $L > L_0$, the spring is stretched. If $L < L_0$, the spring is compressed.
This deformation, not the total length itself, is what determines the force.
Example of spring force
Suppose a spring has spring constant $k = 100\ \text{N/m}$ and is stretched by $0.20\ \text{m}$. Then
$$F_s = -kx = -(100)(0.20) = -20\ \text{N}$$
The negative sign means the force points opposite to the positive displacement.
If the same spring is compressed by $0.10\ \text{m}$, then $x = -0.10\ \text{m}$ and
$$F_s = -(100)(-0.10) = 10\ \text{N}$$
Now the force is positive, so it points in the positive direction.
Spring force on attached objects
When an object is attached to a spring, the spring exerts a force on the object. At the same time, the object exerts an equal and opposite force on the spring. This follows Newton's third law, but the detailed discussion belongs elsewhere. Here, the important point is that the spring force acting on the object depends on how much the spring is deformed.
For example, if a block is attached to a horizontal spring and pulled to the right, the spring pulls the block to the left. If the block is pushed to the left and compresses the spring, the spring pushes the block to the right.
Validity of Hooke's law
Hooke's law is an approximation that works well only within the elastic limit of the spring. If the spring is stretched or compressed too much, the force may no longer be proportional to displacement. In extreme cases, the spring can be permanently deformed or even break.
Hooke's law is valid only when the spring remains in its elastic range.
If the deformation is too large, $F_s = -kx$ may no longer describe the spring correctly.
Springs in everyday physics
Spring forces are used to model many real systems, not only metal springs. Rubber bands, bows, mattresses, vehicle suspensions, and some atomic bonds can show spring-like behavior over a limited range. In physics, the spring model is important because it gives a simple and useful way to describe restoring forces.
Key idea to remember
A spring force is a restoring force that depends on how far the spring is stretched or compressed from its natural length. For an ideal spring, the relation is linear:
For an ideal spring,
$$F_s = -kx$$
where $x$ is measured from equilibrium, not from an arbitrary point.
Understanding this force is essential because springs appear in many mechanical systems and are the foundation for later study of oscillations.