Mass Spring
Systems
in Everyday Life
By: 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
Sine function also
called Equilibrium
Cosine
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