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Physics of a Jumperoo™ |
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Thought Experiment
From
my lab book for Physics 211X we can repeat the mathematics behind an experiment
to find k of my daughter’s Jumperoo™ toy. There are three springs, so we can
just focus on one spring and apply these results to each of the others.
Hooke, Robert: Hooke’s law of elasticity of materials. [Photograph]. In Encyclopædia Britannica. Retrieved fromhttp://www.britannica.com/EBchecked/media/123664/Illustration-of-Hookes-law-of-elasticity-of-materials-showing-the
The
picture is from Robert Hooke’s book and shows how this experiment is going to
work. A spring is stretched due to the force of gravity, and k can be found due
to this.
I
will take the s-axis to be in the vertical position and call this new axis y.
Now I can find the force required to use this spring to be the force of gravity
multiplied by a mass, or weight. I will show this in terms of Newton’s second
law as 
so using 
I now know the force used to distend the
spring 
and can set this equal to 
.
Now
I could use some weights to extend the spring and determine, 
.
Using a few different weights and graphing the force vs. the change in position
of the spring, I predict that I will get a straight (enough) line, and
determine a slope from it getting a value for k.
Here
is the shocker, it’s a thought experiment and I know the weight rating for the
product at 11.3 kg. Finding k is just a matter of reverse engineering the
problem.
I
already know the final force that should be used to safely operate this product
its 
,
I just need to find ∆s. I could measure it, but I also know a maximum
height range for the product at 0.81 m. I will use a measurement of my daughter
who is at the higher end of the weight and height range for the product to find
that her feet touch the ground 16 inches from the seat while she stands up. 
.
I can solve two equations for sf, 
.
I found 
and then graphed the results for you.
Knowing
k is the slope of this line, 
.
This allows me to determine the amount of force my daughter can exert on her
springs in order for her to bounce up and land rather softly on the ground. I
can and will be said that there are many errors in my thought experiment, but
the principle behind Hooke’s Law applies.
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Uaf Physics 211X |
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