What do slinkies, longbows, car suspensions, rubber bands, and hair clips have in common? They are all springs. Anything we might say is, well, "springy" can be classified as a spring. This article will focus mainly on the spring force and how to calculate it since we are discussing types of contact forces, but we will also briefly explain what springs are, properties of springs, and types of springs. Since springs have many different properties and applications, other articles will focus on other aspects of springs, such as the potential energy of springs, work done by springs, the motion of springs, and spring-mass systems.
Fig. 1-A slinky is a prime example of a spring.
Springs - Definition and Meaning
Springs are elastic and can store energy.
Springs are objects that produce a restoring force when stretched or compressed by an applied force and will return to their natural shape after the applied force is taken away.
Properties of Springs in Physics
In physics, springs have the following basic properties:
- They are elastic, meaning they return to their original shape after being deformed by a force.
- They store potential energy.
- The force the spring exerts increases as the distance it is stretched/compressed increases.
Types of Springs
The most common spring, especially in physics problems, is a simple metal coil spring. This is what you probably think of when you think of the word spring, like slinkies or the springs in pens. However, there are many different types of springs. Anything that has the properties we mentioned above technically counts as a spring. For example, if you hold a plastic ruler on the end of your desk, bend it downwards, and release it, it will spring back and forth until it becomes straight again. Therefore, the ruler acts as a spring.
Even within coil springs, there are different types for different applications. The two most common types of coil springs are compression springs and extension springs. Compression springs are designed such that they have the most potential energy when they are compressed, and they have space to be compressed, such as the spring in a pogo stick. Extension springs have the most potential energy when extended and typically don't have much space for compression, such as the springs on a trampoline. There are also torsion springs that are used in rotation, such as the spring in a mousetrap.
Even though springs come in many shapes and sizes, all springs have similar characteristics and exert the same type of force.
Spring Force
Let's define and discuss spring forces.
The spring force is the push or pull of a spring when it is compressed or extended.
If you squish a spring between your fingers, you can feel it fighting back against the force you apply; this is the spring force. The spring force acts in the direction opposite of the compression or extension of the spring.
If you squish a spring, the spring force pushes outward, and if you release your force suddenly, the spring force might catapult the spring across the room. If you stretch a spring between your hands, the resulting spring force is directed inward, and if you release the spring with one hand the spring force might cause the spring to snap back against your other hand.
The spring force is sometimes referred to as a restoring force because it wants to restore itself to its relaxed position.
Spring Force Equation - Hooke's Law
The equation for spring force is known as Hooke's Law:
$$F_\mathrm{s}=-kx$$
The variable \( k \) represents the spring constant, which depends on the stiffness of the spring. The stiffer the spring, the larger the spring constant, the larger the spring force. The spring constant depends on the spring's material properties, the thickness of the spring, and--for coil springs--how many coils it has. It has units of \( \mathrm{\frac{N}{m}} \).
The variable \( x \) represents the distance the spring has been compressed or extended. The larger the distance, the larger the spring force. Since the spring constant is always the same for a single spring, springs will extend or compress linearly with the amount of force applied to them.
The negative in the equation tells us that the spring force acts in the opposite direction of the displacement of the spring. We might also see the spring force equation as follows:
$$\left| F_\mathrm{s} \right |=-k\left | x \right |$$
The absolute values just indicate that the magnitude of the force increases as the magnitude of the distance increases. Hooke's law applies to all springs as long as they aren't stretched farther than their elastic zones (if you stretch a spring so far that it doesn't return back to its correct shape, the equation doesn't work as exactly).
The Spring Force is Caused by Interatomic Electric Forces
The spring force is a contact force (because something has to come into contact with the spring for there to be a force), and all contact forces are caused by interatomic electric forces. Microscopically, springs are made up of many atoms and molecules that are bonded together. The atoms of a spring prefer to stay in their natural state. When the spring is compressed, the interatomic electric forces push out, and when the spring is stretched, the electric forces pull back to keep the atoms and their bonds at their proper distance.
Spring Force Examples - Mass Without Motion
To look at examples using the spring force, we will use a mass attached to a common coil spring.
Mass on a Spring Attached to a Wall
First, we consider an example of a block on a spring hoving horizontally.
A spring is fixed to a wall on one side and attached to a block on the other along a frictionless surface, as in the image on the left below. The block moves \( 0.12\,\mathrm{m} \) when pushed with a force of \( 5.0\,\mathrm{N} \) to the left. What is the spring constant? How far would it move with a force of \( 10.0\,\mathrm{N} \)?
Fig. 2 - Spring attached between a wall and a block with an associated free body diagram
Looking at the free-body diagram in the figure on the right above, we can see that in the x-direction the only forces are the applied force and the spring force. Since the block is not accelerating, these forces equal each other, so the spring force is also \( 5.0\,\mathrm{N} \), but to the right. We can then use Hooke's Law:
$$F_\mathrm{s}=-kx$$
We can substitute our values. We use a negative distance since the block moved to the left, which we have selected as the negative direction:
$$5.0\,\mathrm{N}=-k(-0.12\,\mathrm{m})$$
Solve for
k:
$$k=42\,\mathrm{\frac{N}{m}}$$
This is our answer for the spring constant. Then we can use our \( k \) in our new equation with the \( 10\,\mathrm{N} \) force:
$$10\,\mathrm{N}=(-42\,\mathrm{\frac{N}{m}})x$$
Solve for \(x\):
$$x=-0.24\,\mathrm{m}$$
Again the distance is negative because the block moved in the negative direction. You might notice that this distance is exactly double the first distance, just as the force was double the initial force. This is because the spring force and distance have a linear relationship. If we weren't asked to find the spring constant, we could have skipped that step and set up a ratio to solve for the second distance instead. You can choose whichever method you are comfortable with.
Mass Hanging from a Spring
What happens when the mass is instead hanging vertically?
A mass hangs from a vertical spring as shown in the figure on the left below. How far has the spring extended from its natural state?
Fig. 3 - Mass hanging from a vertical spring with associated free-body diagram.
We drew a free-body diagram above on the right to show the forces acting on the block--the spring force and the gravitational force. Since the block isn't accelerating, these forces equal each other.
$$F_\mathrm{s}=mg$$
We can write our Hooke's Law equation:
$$F_\mathrm{s}=-kx$$
Substituting the spring force and solving for x:
$$x=\frac{mg}{-k}$$
The distance the spring stretched from its natural state is equal to the weight of the block divided by the spring constant. This number will also be negative since it stretched in the downward direction, which we defined as negative.
We can also analyze the motion of a mass on a spring, which follows the rules for simple harmonic motion. The spring, when extended with a force and then released, will compress and extend back and forth--overcorrecting itself-- until it reaches its natural state. For more in-depth information on this topic, see our article on spring-mass systems.
Springs - Key takeaways
- Springs are objects that are elastic and store energy. They will return to their natural shape after being compressed or extended by a force.
- The spring force is a restoring force that tries to push or pull the spring back to its natural shape when an outside force is applied.
- The spring force increases linearly with the distance the spring is compressed or extended, represented by Hooke's Law: \( F_\mathrm{s}=-kx \).
- The interatomic electric forces within the spring don't like to be extended or compressed from their natural positions, so will fight against anything that causes them to. These tiny forces make up the spring force.
References
- Fig. 1 - (https://www.pexels.com/photo/wave-construction-industry-pattern-7123048/) by Tara Winstead (https://www.pexels.com/@tara-winstead/) public domain
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