Transcript The Simple Pendulum

THE SIMPLE PENDULUM
UNIT 8.2
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The Simple Pendulum
As you have seen, the
periodic motion of a
mass-spring system
is one example of
simple harmonic
motion.
Now consider the
trapeze acrobats.
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The Simple Pendulum
Like the vibrating mass-spring system, the
swinging motion of a trapeze acrobat is a
periodic vibration.
Is a trapeze acrobat’s motion an example of
simple harmonic motion?
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The Simple Pendulum
To answer this question, we
will use a simple pendulum
as a model of the acrobat’s
motion, which is a physical
pendulum.
A simple pendulum consists of
a mass called a bob, which
is attached to a fixed string.
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The Simple Pendulum
When working with a simple pendulum, we
assume that the mass of the bob is
concentrated at a point and that the mass
of the string is negligible.
Furthermore, we disregard the effects of
friction and air resistance.
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The Simple Pendulum
For a physical pendulum, on the other hand,
the distribution of the mass must be
considered, and friction and air resistance
also must be taken into account.
To simplify our analysis, we will disregard
these complications and use a simple
pendulum to approximate a physical
pendulum in all of our examples.
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The Simple Pendulum
To see whether the pendulum’s motion is
simple harmonic, we must first examine
the forces exerted on the pendulum’s bob
to determine which force acts as the
restoring force.
If the restoring force is proportional to the
displacement, then the pendulum’s
motion is simple harmonic.
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The Simple Pendulum
Let us select a coordinate system in which
the x-axis is tangent to the direction of
motion and the y-axis is perpendicular to
the direction of motion.
Because the bob is always changing its
position, these axes will change at each
point of the bob’s motion.
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The Simple Pendulum
The forces acting on the bob at any point
include the force exerted by the string and
the bob’s weight.
The force exerted by the string always acts
along the y-axis, which is along the string.
At any point other than the equilibrium
position, the bob’s weight can be resolved
into two components along the chosen
axes.
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The Simple Pendulum
Because both the force exerted by the string
and the y-component of the bob’s weight
are perpendicular to the bob’s motion, the
x-component of the bob’s weight is the net
force acting on the bob in the direction of
its motion.
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The Simple Pendulum
In this case, the xcomponent of the bob’s
weight always pushes or
pulls the bob toward its
equilibrium position and
hence is the restoring
force.
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The Simple Pendulum
As with a mass-spring system, the restoring
force of a simple pendulum is not
constant.
Instead, the magnitude of the restoring force
varies with the bob’s distance from the
equilibrium position.
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The Simple Pendulum
The magnitude of the restoring force
decreases as the bob moves toward the
equilibrium position and becomes zero at
the equilibrium position.
When the angle of displacement is relatively
small (<15°), the restoring force is
proportional to the displacement.
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The Simple Pendulum
For such small angles of displacement, the
pendulum’s motion is simple harmonic.
We will assume small angles of displacement
unless otherwise noted.
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The Simple Pendulum
Because a simple pendulum vibrates with
simple harmonic motion, many of our
earlier conclusions for a mass-spring
system apply here.
At maximum displacement, the restoring
force and acceleration reach a maximum
while the velocity becomes zero.
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The Simple Pendulum
Conversely, at equilibrium the restoring
force and acceleration become zero and
velocity reaches a maximum.
Table 12-1 on the following page illustrates
the analogy between a simple pendulum
and a mass-spring system.
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The Simple Pendulum
As with the mass-spring system, the
mechanical energy of a simple pendulum is
conserved in an ideal (frictionless) system.
However, the spring’s potential energy is
elastic, while the pendulum’s potential
energy is gravitational.
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The Simple Pendulum
We define the gravitational potential energy
of a pendulum to be zero when it is at the
lowest point of its swing.
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The Simple Pendulum
The figure below illustrates how a
pendulum’s mechanical energy changes as
the pendulum oscillates.
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The Simple Pendulum
At maximum displacement from equilibrium,
a pendulum’s energy is entirely
gravitational potential energy.
As the pendulum swings toward equilibrium,
it gains kinetic energy and loses potential
energy.
At the equilibrium position, its energy
becomes solely kinetic.
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The Simple Pendulum
As the pendulum swings past its equilibrium
position, the kinetic energy decreases
while the gravitational potential energy
increases.
At maximum displacement from equilibrium,
the pendulum’s energy is once again
entirely gravitational potential energy.
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