Transcript 4.5 Elastic potential energy and Simple Harmonic Motion (SHM)

4.5 Elastic potential
energy and Simple
Harmonic Motion (SHM)
How does a rubber band reflect the
link between energy and forces?
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Think carefully about the behaviour of
stretchy objects like rubber bands and
springs
What happens to them?
Stretchy stuff responds to
forces by storing energy
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When you apply force to an elastic object like
a spring or elastic band
The force does work on the object because it
causes a displacement
This gives the spring energy
The spring releases this energy when it is
given the chance to return to equilibrium
length
http://www.sciencewithmrnoon.com/physics/
weightlab.swf
Equilibrium line
without mass
Equilibrium line with
mass
Fg Fe
Fg
V =Vmax
0 m/s
Fg Fe
Fg
Fe
V = 0 m/s
Fe
How do we know it releases
energy?
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Well, what were to happen if you placed
something in front of that spring?
Elastic potential energy
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Energy stored in springs are a type of
potential energy
Like gravity, once the energy is put into the
spring, it can be released if the conditions are
right
In fact, elastic potential energy is similar to
gravity
You can compare pulling on a spring to Eg
In gravitational potential energy, stored energy
from raised objects is released when the
gravity pulls the object back down – similar to
what the spring does when it “pulls” the mass
back to equilibrium point.
Ideal spring
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An ideal spring is one that doesn’t deform
when stretched or depressed
That means it doesn’t get damaged and can
return back to normal shape
An ideal spring that we study also assumes
that external forces like friction do not
interfere with it – nor does the spring
experience any internal forces
Reality vs. ideal conditions
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What happens eventually to a bouncing mass
on a spring in real conditions?
If you had an ideal spring, what happens
eventually to the bouncing mass?
Hooke’s Law
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An ideal spring follows Hooke’s Law, which
states that the force required to deform a
spring per unit length is always constant
Where: F = -kx
F = force applied in Newtons (N)
k = spring constant in N/m
x = position of spring relative to equilibrium in
metres (m)
Why the negative?
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The negative is supposed to make up for the
relativity in direction of forces
Hooke’s law is written from the spring’s point
of view
The F value in the equation refers to the force
that the spring is applying on the mass
Hooke’s Law is written from
the point of view of the spring
Simple Harmonic Motion
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SHM is created when the force (and therefore
acceleration) is proportional to the
displacement
In the previous animation of the spring’s
motion, notice that the net force on the mass
is the greatest when the displacement is
greatest from equilibrium
Fg Fe
Fg
V =Vmax
0 m/s
Fg Fe
Fg
Fe
V = 0 m/s
Fe
SHM creates periodic motion
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This relationship is periodic – like waves, it
exhibits regular, repeated motion that can be
described using many of the same properties
that you use to describe wave functions
Imagine the spring that we discussed earlier,
but this time it moves along a track
The mass is attached to a writing device that
can sketch out the path of the mass
Mathematically…
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In order to make a clear relationship between
the spring and the forces associated with it,
you can compare the movement of the
bouncing mass to a handle on a rotating disc
Direction of rotation
Direction of rotation
Direction of rotation
Direction of rotation
Direction of rotation
Conservation of energy still
applies
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If you analyze the motion of a mass on a spring,
the speed gained by the mass and lost by the
mass is traceable back to the total amount of
elastic potential energy put into the system
A spring with a mass that is stretched or
compressed will store energy
Once released, the mass will move
As the kinetic energy of the mass increases, the
energy in the spring decreases, and vice versa
Dampened Harmonic Motion
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DAMPENED HARMONIC MOTION occurs
when the spring system loses energy over
time causing the displacement to decrease in
the spring
This energy is dissipated to other forms
This is desirable in some mechanical
systems like shock absorbers – or else your
car would continue to bounce up and down
after the shock absorber is depressed