Extra credit
Assignment
This assignment is OPTIONAL can addUP TO 1.5 to your overall 4A grade equivalentto one la
The goal of this assignment is to better understandthe last question on theHooke's Law La
You may have observed that the oscillationwas symmetric about the mass equili
iumrather than the natural length You willshow graphically that when the object isat the mass equili
ium the potential energy ofsystem is minimized it is a stable equili
ium
We know that the force on the object thepotential energy of the system are related by
Ã‰ JU
which reduces to
Fy day
for one dimensional motion
My
Natural
length
Turning
Mass Xy
points
equili
ium On
1 Define the system to be mass springEarth Plot the gravitational potential
energy the spring potential energyUgly Usp y on the same set of axesU on the vertical axis y on horizontal
It is not necessary to scale the axessimply sketch the shapes of the plots
Let the x axis pass through the natural
length SO Ug o there
all
y o co
esponds
Fy
w hat length
sample axes
2 Find the total potential energy function Uly
y adding Ugly Usp y Plot u lyon another set of axes
Do this by hand no calculator Desmosshow work
Hint complete the square
3 U ly is parabolic Determine the coordinatesof the vertex label it on the
parabola Express the coordinates in terms of
M K g constant s
Do you recognize the y coordinate of thevertex This is a special vertical position
what is it
4 Label the regions on the plot where thenet force is directed up where the
net force is directed down
Hint Fy day
5 Add a horizontal line representing thetotal energy of the system to the plotThe line should be above the horizontal
axis
Label the turning points on the plot
Are the turning points symmetric aboutthe natural length or the massequili
ium
6 Does the choice of coordinate system mattershow that the y coordinate of Umin isindependent of coordinate system
For example you could let y o co
espondw mass equili
ium rather than
natural length repeat steps 1 3 above