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Unformatted text preview: f SHM. It tells us that the angular
acceleration α of the pendulum is proportional to the angular displacement θ but opposite in sign. Thus, as
the pendulum bob moves to the right, its acceleration to the left increases until the bob stops and begins
moving to the left. Then, when it is to the left of the equilibrium position, its acceleration to the right
tends to return to the right, and so on, as it swings back and forth in SHM. More precisely, the motion of
a simple pendulum swinging though only small angles is approximately SHM.
Comparing equations (41) and (22), we see that the angular frequency of the pendulum is (42)
Next, if we combine equation (42) into the following equation 7 2 2 (43) Then, we see that the period of the pendulum may be written as (44) 2 All mass of a simple pendulum is concentrated in the mass m of the particle-like bob, which is at radius L,
from the pivot point. Thus, we can use the fact that (45)
to write (46)
for the rotational inertia of the pendulum. Substituting equation (46) into equation (44) and simplifying
we get (47) 2
This equation is ONLY FOR A SIMPLE PENDULUM WITH SMALL ANGLES.
2.5.2 The Physical Pendulum A real pendulum, usually called a physical
pendulum, can have a complicated distribution of
mass, much different from that of a simple pendulum.
Does a physical pendulum also undergo SHM? If so,
what is its period?
Figure 8 shows an arbitrary physical pendulum
displaced to one end by angle θ. The gravitational
force acts at its center of mass m at a distance h from
the pivot point. Comparing Figures 7 and 8 reveals
only one important difference between an arbitrary
physical pendulum and a simple pendulum. For a
physical pendulum the restoring component Fg sin θ
of the gravitation force has a moment arm of distance
h. In all other respects, an analysis of the physical
pendulum would duplicate out analysis of the simple
pendulum up to equation (44). Again, (for small θm)
we would find that the motion is approximately
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