There are many vibrating systems in the world around us: water sloshing in a tub, a glass
that rings after it has been struck, or a rocking chair. They have in common that there is a
central rest position, where the system would be if we hadn't disturbed it, and a force that
is trying to return the system to the rest position. However, in disturbing the system, we
have given it energy, and when the system arrives at the rest position, this energy is in
the form of kinetic energy -- which means it keeps moving. The moving system has mass, which
confers inertia, so it can't just stop.
The pendulum is a slightly special case, because the restoring force is due to
the weight of the pendulum bob; this has the implication that increasing the mass will
increase both the size of the force and the size of the inertial effect, with the result
that the period of vibration does not change. When the restoring force is due to
bending or stretching a spring, increasing the mass causes the period to get longer
(for example, the ringing glass will make a lower note when there is some water in it,
because the water becomes part of the moving mass).
In other respects, all vibrating systems behave the same way.
The graph below represents the displacement of some vibrating object, and how it
depends on time. This could be any kind of vibrating system, but we will describe it as
a pendulum.
This particular pendulum has period 4 seconds, for convenience
of the discussion. The straight central line represents the position of the
undisturbed pendulum bob.
At times 1, 5, 9, ... the bob has reached its largest displacement
in one direction; at times 3, 7, ... it is at its maximum displacement in the opposite
direction. At times 0, 2, 4, 6, ... it is at the center (where it would be if we had
not disturbed it), but sometimes it is moving one direction and sometimes the other,
as can be seen by the alternating upwards and downwards slopes of the curve. The
speed is largest at these times (the slope is largest), while at the maximum displacement
points the bob has just ceased moving one direction and is about to move the other,
and at this instant is not moving at all. However, the bob is accelerating, since
the velocity is changing. In fact this is where the acceleration is largest, because
the curvature is largest. At the same time, the force on the bob is largest, because
the displacement is largest. At this and all other times
the force and the acceleration are related by the
Law of Force and Acceleration:
Force = mass x acceleration.
The vibrating system has both kinetic energy and potential energy. The
potential energy of a pendulum is gravitational energy, while for the mass
on a spring there is energy stored in the stretched (or bent) spring (there
may also be gravitational energy, if the motion causes change in height).
The potential energy is large when the displacement is large (in either
direction), which the kinetic energy is large when the speed is high.
The graph of the displacement given above then translates into this graph
of the two kinds of energy:
The solid line is the potential energy, and the dotted line is the
kinetic energy
At all times the sum of kinetic energy and potential energy is the same
constant value,
because no energy is leaving or entering the system.
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