Mass Spring Systems in Everyday LifeBy: Rachel Prawitt

Modeling the Motion of a Spring There is a weight attached to a string

that is suspended from a horizontal bar it is attached to.

Where is Equilibrium?Cosine

Cosine

Sine function also called Equilibrium

If we neglect any damping forces (such as air resistance etc.) then, the motion of the spring can be modeled by:

Where x(t) is the position of the object along the number line at time t. The other quantities are constants: w is a constant that depends on the stiffness of the spring and the mass of the weight, v0 is the initial velocity, and x0 is the initial position of the object.

Suppose a weight is set into motion from a position of 3 cm below equilibrium position with a downward velocity of 4cm/sec. Assume that the spring stiffness and mass of the weight means that w=2 for this system.

When I graphed this by hand it looks like this:

When I used my graphing calculator to graph this equation it looks like this:

How do I write an equation from the graph my calculator came up with in the form of x(t)=Acos[B(t-C)]

X(t)=3.60[2(t-0.27)]

What are the differences between my sketched graph and the calculators graph? Why were they different?

The graph I drew is only showing the Cosine and the Sine graph combined. The values for the period are the same.

The difference is that the amplitude ended up being 0.6 more than I had figured out previously because of the min & max locations of their slopes. This is really hard to determine without a graphing calculator.

You get the amplitude from adding the sine and cosine values together at a certain point on the graph. Because of the way the slopes of sine and cosine aren’t straight up or down they sort of progress this makes it possible for them to be added to equal 0.6 more than what the actual amplitude of the sine and cosine have individually. It is a small enough thing to be hard to notice when you are just hand sketching a graph like this.

Why do we care about Mass Spring Systems?

We care about this system because we use it everyday, most of the time we do this without even realizing it.

What are some simple examples of Mass Spring Systems we use?

Shocks in a vehicle Makes up many mattresses Used for a bungee jump cord Makes a trampoline fun to bounce on Makes up a pogo stick

Springs in a Mattress…

Shocks in a vehicle..

What is a complex example of a Mass Spring System that I would use in My field of Study?

I would love to one day design guns. I am studying to be an engineer and I think many guns have some sort of a mass spring system in them.

Example in a Gun..

Reflection…I never realized how much we use mass-spring systems in

everyday life. For example, vehicles have shocks or spring systems that enable us to experience a comfortable drive. What I mean by that is the shocks are able to absorb any bumps, dips, vibrations or whatever else we may hit in the road. Another example of this would be when I take my Razor up to the dunes and do some crazy jump, landing back on the ground after this big jump isn’t painful, most of the time, because of the shocks. The reason I say most of the time is because if the jump is way too big or if the shocks aren’t the best quality of stiffness then the amplitude of the bounce from landing the jump can increase.

The way the mass- spring system works is when the system is set in motion, the spring’s amplitude shifts higher or lower. The amplitude change, then causes the spring to obviously be displaced from its equilibrium. The spring then just fluctuates back and forth within the period until it reaches its equilibrium again. The frequency of this fluctuation happening or in my previous example, how often the razor is jumped at the dunes, doesn’t affect the way the fluctuation takes place. I think it is important for me to understand the way this system works because I am seeking a degree in Engineering and understanding this is important when trying to understand the physics of the world and the way things are engineered.

Reflection…

Another example of a mass-spring system I have discovered I use every day is in bed. Many mattresses have a mass-spring system in them, the damping and spring’s equilibrium is what varies in mattresses making it possible to provide the perfect mattress stiffness for every seeking individual. It is important for mattress manufactures to understand the way this mass-spring system works so that they can make and sale many different types of mattresses. It is also important for the rest of the world to understand this so we are able to understand the selling points that the mattress manufacture is making. Also, it’s helpful for us to understand why certain types of mattresses are more beneficial and comfortable to some people when they aren’t to others.

Reflection…At the beginning of this project I understood that

trigonometry is used in the real world. However, doing this signature project I came to find out that it is used a lot more often than I foreseen. Having to find specific examples in everyday life of when this mass-spring system is used was crazy to me to think how important it is. Specifically, if my car didn’t have shocks I couldn’t imagine how bumpy of a ride I would have to encounter. I never realized until I did this assignment and calculated a mass-spring system myself, that in order to have the shocks on my car work correctly, someone else had to understand this same mass-spring system to design them. I have a friend who had bad shocks on his car and he went through 2 sets of tires in just a couple of months before they figured out the issue. I also have recently learned that even if you are just some kid out going bungee jumping you need to understand this system because if you don’t understand it you may calculate the length of the bungee cord to be too long once the spring is displaced and not have the opportunity to go back and correct it because the displacement caused you to fall flat to the ground.

Modeling of a Spring System Assignment Math 1060 Trigonometry