Math 334 Lecture #18
§
3.7: Mechanical Vibrations
MassSpring Systems.
A spring is attached to a ﬁxed support, an object is suspended
on the spring, and a dashpot is placed about the object.
[Sketch picture of this conﬁguration.]
Let
u
(
t
) be the position of the center of the object at time
t
, with
u
= 0 being the
position of the object when it is at rest, and
u >
0 is the
downward
direction.
The principle governing the motion of the mass is Newton’s second law:
ma
= sum of forces acting on the mass
,
where
m
is the mass of the object and
a
is its acceleration.
The relationship between
a
and
u
is
a
=
u
00
.
[We will consider four forces that act on the object.]
Force due to Gravity
. The weight of the object is
F
g
=
mg
, where
g
is the gravitation
constant.
Force due to the Spring
. Attaching the object to the spring stretches the spring by a
distance
L
, called the
elongation
.
[Sketch picture of spring before and after the object is suspended on the spring.]
Hooke’s Law states that the force exerted by the spring is negatively proportional to the
elongation:

kL
where
k >
0.
When the object is at rest (i.e. is in equilibrium), the weight of the object and the force
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 Spring '11
 Smith
 Math, Force, Mass, IVP

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