Vibration - physics explained

It is time for some fun science again! I have been thinking about what this post will be for the last two days. The first idea was to write about sound and then I realized how that topic is really huge and a post about it would be just too long. I realized that before sound I should explain waves first and in my next lesson move on to sound. That was all fine but there is no understanding of waves without vibration and periodic movement first. So, the vibration it is. Before we start, here is a list of previously published posts in my series Physics explained:

To follow along this post, no previous knowledge of physics is necessary, however, this post will be needed to understand those that will follow it so think about this as lesson 1, waves as lesson 2 and then finally, sound as the lesson 3. Enjoy the post and feel free to contact me if you have any questions or want to know more. Are you ready to learn about vibration? Let's go!

When things vibrate, they move back and forth from their equilibrium position. That is the same position in which an object would be if it was not moving at all, the position where the net force on that object is zero. So, if we have an object that can vibrate and we apply an external force on it to disturb that state of equilibrium, it will start moving away from the equilibrium, stop at some point and come back to equilibrium, pass it and move to the other side, then stop and come back and so on and so on...

We see vibration everywhere in our daily lives. From bobblehead dolls in our cars, pendulum clocks, balls bouncing off walls or floors, children on swings, trees swinging in the wind or any other motion of shaking and swinging. There is usually elastic thing involved or a one that is hanging from something like a pendulum. These are all examples that we can see but if we go deeper we will find that in fact, everything vibrates, even solid objects that seem like they are still. In solids, particles are vibrating too and only at the temperature of absolute zero (-273 °C) all movement stops. Since I have covered electrons in my previous posts, we will stick to macrophysics this time and discuss vibration and periodic motion that we can clearly see.

When vibrations are regular and repeated, we categorize them as periodic. An object moves over the same path over the course of time and the motion repeats itself until the damping stops it. Mass on a spring and pendulums are the best examples to learn about periodic motion so let's see the mass on a spring. It is in the gif right from this text.

Look at that animation and pick a starting position the mass is in. To be on the same page with this, let's start with mass being at the bottom. From there, it will go up to equilibrium, pass it and continue going up where it will stop and change direction and start coming back. On its way back it will again pass the equilibrium and come back to the bottom. We call this a cycle and it is always made of 4 parts. If we started from the equilibrium and the mass went up, the cycle would not be when it first came back but the second time. The mass goes from equilibrium up, comes back, passes it, goes down and then comes back to equilibrium.

Since time is measured in seconds, units for the period are also seconds and we use the capital letter T as its symbol. If we turn that definition around and ask ourselves how many cycles take place in a period of time then we have a frequency. We label frequency with f and measure it in Herz. Here is the relation between period and frequency:

They have a reciprocal relationship. A period is a time for one full cycle to complete itself and a frequency is the number of cycles that are completed per time. We can easily get one from the other so to define any periodic motion we need only one of them. The other thing that is often mentioned in periodic motion is amplitude. Remember when I told you to imagine a mass starting from the bottom position? That is amplitude. The symbol we use is capital letter A and as any other distance, we measure it in meters.

Since we already said some basic things about the mass on a spring, this is a good time to introduce ourselves to the simple harmonic oscillator. It consists of a weight attached to one end of a spring while the other end is connected to a rigid support. When the system is left at rest at the equilibrium position then net force acting on the mass is zero. If the mass is displaced from the equilibrium position, the spring exerts a restoring elastic force that obeys Hooke's law.

Hooke's law states how the force (F) needed to extend or compress a spring by distance x scales linearly with respect to that distance. That force always has the opposite direction from the movement and displacement x because when the spring is compressed, the force wants to extend it and when it is extended, it wants to compress it. The force tends to restore the system to equilibrium. This is the explanation for the minus sign in the formula: F = −kx

F (in Newtons) in that formula is the restoring force, x is the displacement (in meters) and the k is the spring constant (N·m−1) that depends on the material from which the spring is made of.

If you remember that a body has potential energy when it is at rest and kinetic energy when it has speed, you will easily comprehend what happens with energy in this motion that we described. Let us look at the graph.

Keep in mind that the total energy is always the sum of kinetic and potential energies. We started in the amplitude position where the body is at rest before we have let it go. Since it is at rest, there is no kinetic energy and the entire energy is potential. When the body starts moving, its potential energy decreases and the kinetic one increases as it is picking more and more speed. In the equilibrium position, the potential energy is zero and the entire energy is kinetic. When the body continues to move to upper amplitude, that kinetic energy is decreasing and the potential one increasing until the position of amplitude where there is again no kinetic and the entire energy is potential. I think it is time for a review of what we have covered so far...

As you can see, displacement, speed, and acceleration are periodic. We can solve differential equations to get solution functions. Those functions are trigonometric. If this is too much for you, I won't mind if you scroll through this part. :)

Those are solutions to differential equations and ω = 2πf is the angular frequency. We can see the graphs of those function and notice how everything can be explained just by looking at them.

As you can see from these graphs, displacement starts from amplitude, velocity from zero because it is zero there and acceleration from its maximum value. Functions are periodical and they change in a way that we already explained.

We are now moving to an important question regarding the period and the frequency. What do they depend on? Will the movement change if we change the mass or will it change if we have the same mass but increase the amplitude?

It does not matter if you pull that mass down by just a couple of centimeters or by a meter, the period and the frequency will be the same. Let's see why that is. We know that angular momentum is ω = 2πf and we also know that angular moment depends on mass and the spring constant which is shown with these formulas:

As I said in the beginning, to get to sound we first need to discuss waves and to start talking about waves we needed vibration. In this physics explained post we have covered vibration and everything we will need regarding harmonic motion from displacement and acceleration to energies and frequency. Be patient, waves are coming soon, are you ready for them? Until next time

To learn more about this topic, be sure to check out these resources:

If you are interested in something physics (or science in general) related, tell me and I would be happy to make a post about it and explained it. Promoting science and making it understandable gives me great joy, I would be happy to help you.