When a restoring
force has a mathematical relationship of F
= - kx then the motion created is referred to as
SIMPLE HARMONIC MOTION
or SHM.
Both F (force in Newtons)and
x (displacement from equilibrium
in meters) are VECTORS.
The relationship
F = -kx is referred to as Hooke's
Law. The "k"
is a spring constant measured in Newtons
/ meter. The negative sign indicates
that the direction of the restoring force
"F" is always opposite that
of the stretch or compression,
"x"of the spring.
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SHM
& CIRCULAR MOTION
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When
an object travels in a circular path, its "shadow"
travels along
a straight line in Simple Harmonic Motion.
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This
means that there is a connection between circular
motion and SHM.
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Remember
that a = v^2 /R for circular motion.
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The horizontal
component of that acceleration vector is cosine
theta=(cos theta) v^2/ R.
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Please
study the diagrams below.
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OBSERVE:
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The SHM
velocity is greatest at the center of the circle--vertical
lines are spaced farthest apart.
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The SHM
acceleration is greatest at the endpoints--the biggest
change in the spacing of the
vertical lines occurs at the endpoints!
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Look at the above diagrams
and note that the shadow of the ball moving in uniform circular
motion appears on the horizontal line.
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It is this part of the
motion which is defined as SIMPLE HARMONIC MOTION (SHM).This
relationship allows us to derive an equation for the PERIOD (T)
of oscillation. See below!
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ONCE AGAIN PLEASE NOTE:
The SHM velocity is greatest at the center
of the circle--vertical lines are spaced farthest apart.
The SHM acceleration is greatest at the endpoints
where there is the biggest change in the spacing of the vertical
lines.
Assignment
2-- SHM Design Lab
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THE ENERGY
STORED IN A SPRING
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The maximum position
of displacement from equilibrium is known as AMPLITUDE.
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Since x measures the
position from equilibrium, the amplitude is a very special
x value
designated by xo.
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The MAXIMUM
acceleration is given by a
= k xo / m and depends on "x".
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The Work
Energy Theorem states that work done
by all forces equals change in Kinetic
Energy. Remember that the spring starts at rest when stretched
and the restoring force will pull it back toward the "rest
position" where it will be moving the fastest at any point
of oscillation!
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The work done by the
spring is conservative
so that
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The area under the Force,
Position graph of F = kx gives us work done
on the spring to compress it.
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This
work equals the energy stored in the spring. Stored
energy is also known as Potential Energy.
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The
area under this graph is the area of a triangle. So A = 1/2
(b)(h)= 1/2(x)(F)= 1/2 (x)(kxo). This gives the work
done by the external force as 1/2 (k xo^2).
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The SLOPE
of this graph determines the"k"
(or spring constant).
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The AREA
of this graph yields the work done
on the spring!
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