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10.4 The Simple Pendulum
A
pendulum
is simply a mass attached to a string/rope (or rigid rod) that, once set
to motion will swing back and forth under the influence of gravity
into motion, will swing back and forth under the influence of gravity.
Clearly, the motion of the pendulum is
periodic, i.e. it repeats in time.
But is the motion of the pendulum
simple
harmonic motion
(SHM)???
Let’s draw the FBD for the pendulum.
T
x
y
θ
L
W
Break the forces down into their
x
 and
y

omponents
W
x
W
y
Assume it is swinging on the way up (ccw).
components.
Notice, the force acting along the direction of motion is the
x
component of the
weight, and it acts in the –
x
direction.
For SHM, we need the force (
W
x
) to be
W
W
x
−
=
sin
sin
mg
W
x
−
=
⇒
proportional to
, but in the opposite
direction.
The force is not proportional to
, but to sin
.
Thus, the pendulum motion is
strictly not SHM.
But, for small angles, sin
θ≈θ
, so that:
mg
W
x
−
=
So, in a smallangle approximation, pendulum motion is SHM!!!
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mg
F
−
=
for the pendulum in the smallangle approximation.
The displacement of the pendulum mass, which is the distance it travels along
its arc, is given by:
L
s
=
mgs
−
mg
⎞
⎛
L
F
=
)
(
s
L
⎟
⎠
⎜
⎝
−
=
ent)
(Displacem
Constant)
(
×
−
=
kx
−
=
Thus, for the pendulum,
L
mg
k
=
,
m
k
=
ω
Since
we find the following result for the frequency of pendulum motion:
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 Fall '08
 SPRUNGER
 Physics, Gravity, Mass

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