MASS SPRINGS ARE EVERYWHERE YOU LOOK Gabrielle Pack Math 1060 Zeph Smith

MOTION OF A SPRING Simple Spring with A weighted endSimple Spring with A weighted end

SYSTEM OF A SPRING  When a mass-spring system is set in motion whether in the form of a trampoline or mattress the process can be quite difficult to understand. In this process the mass is displaced from its equilibrium position and oscillates back and forth until it reaches equilibrium again. It is this oscillation that creates a periodic motion which is independent of the frequency which is how many times it oscillates within the period and the amplitude which is how high/low the mass moves.

CAN YOU SEE THE MOTION OF A SPRING AROUND YOU?  The motion of a spring takes place all around us in various examples that we see daily. Mattress Trampoline Car shock absorption Bungee cord Pogo sticks And so on and so on

VISUAL EXAMPLE

 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. The equation would look like this: x(t) = (4cm/sec) / 2 sin(2t)+ 3cos(2t) = 2sin(2t) + 3cos(2t)

This is my attempt to graph without a calculator:

With a graphing calculator my graph turned out like this :

The process of how I got an equation from the graph on my calculator In the form of x(t)=Acos[B(t-C)] is as follows:

MY GRAPH VS A CALCULATOR The differences between my graph and the calculators graph are only a few but these few difference’s are notable enough: - The amplitudes were barely different by less that 0.5 which is a very small difference between the two graphs -The graph that I did by hand is only the sine and cosine graphs combined and they both had almost exactly the same period

PHARMACY Pharmacy is my current field of study and I work at a compounding pharmacy. In this pharmacy we use machines that bounce and spin the compounds we make in order to help mix and create a good consistency in our creams. This machine has a spring system in order to support the weight of the compounded creams and bounce them in a controlled motion.

REFLECTION Mass Spring systems are heavily relied on in our every day lives whether we realize it or not. On my drive to work every day I am very comfortable due to the shocks on the car that use a spring system to absorb all the dips and bumps that I drive over. I sleep every night on a mattress that uses a mass spring system to absorb my weight so that I can sleep effortlessly.

REFLECTION CONT. I depend on the action of the spring everyday in the compounding pharmacy at work as well. I was working on a machine a couple of weeks ago thinking about this project and what everyday examples I can use and it was literally right in front of my face. The machines that we so heavily rely on to make hundreds of compounds a day have a springing mechanism to them. They use the springs to bounce and spin the compound at rapid speeds to blend an smooth the compounded creams.

REFLECTION CONT. In the beginning of this project I knew that math was always applicable to the world around us. It was trying to look at the world with my trigonometry eyes that I started seeing mass spring mechanisms everywhere I turned. Whether it be the car I was in, the bouncing toy my daughter uses or the mattress that I fall asleep on every night. To be able to break down this mechanism on paper and understand how it works feels great. This class was very challenging at times and often took me a

REFLECTION CONT. long time to grasp some of these concepts. In the end I really appreciate the knowledge and determination that I ‘ve gained towards math. I only hope that I can keep this forward momentum going in my math education especially since I start pharmacy school in 3 months.