Simple Harmonic Motion Principles:
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The period of oscillation is independent of the displacement. (It does not matter how far you stretch the string,
of displace the pendulum, the period does not change. That’s how grandfather clocks work.)
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The force that returns the system to equilibrium is directly proportional to the displacement. (The more you
stretch the spring, the greater the force).
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The P vs T graph makes a sine curve, as does the V vs. T curve, as does the A vs T .
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Note: A vs T is a sine curve, but A vs x is linear, with a constant negative slope.
Undamped, no energy lost in the system
Damping Conceptually accounts for friction, air resistance, heat loss due to spring deformation and string stretch,
ectara. Note the period does not change on the red graph.
Springs:
F=-kx
The negative gives the restoring force, opposite the direction of displacement. (If it was positive what
would happen to system?
The period is independent of ___gravity___ and ___displacement______ and ____orientation,
And depends on______mass___________ and _________spring constant k___
The period is linear with ________sqaure root (Mass ___ and inversely proportional to _______ root (k) _
K is measured in ___N/m__ units and can be thought of a measure of how __stiff____ or ___stretchy_ the spring is.
A truck suspension spring has a _____huge_______K and a slinky has a _____tiny________k
Energy= ½ kx2
Note energy is proportional to the square of the maximum displacement.
The mass moves fastest when________x=0, at in the equilibrium posistion____________________
The mass has a velocity of zero when ______min and max displacement. When it turns around_____
When the velocity is zero, all the energy is stored________in the spring______________________
When the velocity is max, the spring stores _________0__________ energy.
Pendulums:
The period is independent of ____mass____________ and _________max displacement______________
And depends on_______length_________________ and __________acceleration due to g________
The period is linear with _____root (length) __ and inversely proportional to _____root(gravity )_
This will be super useful in your life you ever need to _____set a grandfather clock___________________ or scarily
need to ________figure out what planet you are on or what acceleration you are experiencing________
The pendulum moves fastest when its________At the bottom_______________
The pendulum moves slowest when______________________At its max displacement. (theta is maxed)____
The potential energy of the pendulum is _____mgh____________________ This is proportional to θ2 . Can you prove it?
To derive the period equation, we assume ____sin(x)=x______________________ This is a good assumption when the
angle is small because of the picture below:
Why some much attention to spring and pendulums?
A power series can be made equal to _____any function you want with any accuracy you want______________
Considering energy we can make co =__0____________ . We can make c1 go away by _____using a local minimum, aka a
stable equilibrium_______________
c2 is more important than c3 because___for small displacements a small distanced cubed is even less than a small
distance squared. It is the term that dominates. ____________________________