The
Pendulum
A pendulum is a point
mass - sometimes called a 'bob' - connected to a string, rod or rope,
that experiences simple harmonic motion as it swings back and forth. The
mathematical treatment assumes the mass is a 'point' so that it is not
experiencing air resistance, it also assumes that it swings without friction
at the pivot point and that the string is inextensible.
The equilibrium position
of the pendulum is the position when the mass is hanging directly downward.
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Consider a point pendulum
bob connected to a massless rope or rod that is held at an angle
q from the horizontal. If you release the mass, then the system
will swing to position and back again.
Historic
Setting
It is recorded that
a chance observation of a swinging church lamp led Galileo to find that
a pendulum made every swing in the same time, independent of the size
of the arc. He used this discovery in measuring time in his astronomical
studies. His experiments showed that the longer the pendulum, the longer
is the time of its swing.
Christiaan Huygens
determined the mathematical relationship between the length of the pendulum
and the period of vibration when the arc of swing is small. He arrived
at the formula:
where
- T is the period,
or time for one complete swing,
- l is the length
- the distance from the point of suspension to the center of gravity
of the bob. Care has to be taken that the point of suspension is a
point - this can be achieved by clamping the string frimly between
two pieces of card.
- g is the acceleration
of gravity.
In 1673, Huygens devised
a practicable means of making a pendulum control the speed with which
a clock mechanism runs. This led not only in the development of many types
of clock, but also in the application of pendulum control to other mechanisms.
Problems and how
they were overcome
Metal pendulums expand
when heated (longer - therefore period increases!); to counteract the
effect of temperature changes, compensation pendulums have been devised,
many of them operating by the opposite expansion of different metals in
compound rods.
Forces acting on the
bob, such as air resistance and friction at the pivot point, affect its
swing - great care is taken to minimise friction or movement at the pivot
point and a pendulum in a vaccuum will not suffer air resistance (but
most pendulum clocks do not need such a great expense!)
Forces
acting on the pendulum
When you release
the pendulum bob it will accelerate toward the equilibrium position. As
it passes through the equilibrium position, it will slow down until it
reaches a position -Q, and then accelerate
back. At any given moment, the velocity of the pendulum bob will be perpendicular
to the rope. The pendulum’s path follows an arc of a circle, where
the rope is a radius of the circle and the bob’s velocity is a line
tangent to the circle.
To calculate the forces
acting on the pendulum at any given point in its trajectory it will be
most convenient to choose a y-axis that runs parallel to the rope. The
x-axis then runs parallel to the instantaneous velocity of the bob so
that, at any given moment, the bob is moving along the x-axis.
Two forces act on
the bob:
- the force of gravity,
F = mg, pulling the bob straight downward and
- the tension of
the rope FT pulling
the bob upward along the y-axis.
The gravitational
force can be broken down into an x-component, mg sin, and a y-component,
mg cos.
The y
component balances out the force of tension—the pendulum bob doesn’t
accelerate along the y-axis—so the tension in the rope must also
be mg cosQ.
Therefore,
the tension force is maximum for the equilibrium position and decreases
with Q.
The restoring
force is mg sinQ , so the restoring force is
greatest at the endpoints of the oscillation, and is zero when the pendulum
passes through its equilibrium position.
N.B.
The restoring force for the pendulum, mg sin, is not directly proportional
to the displacement of the pendulum bob which makes calculating the various
properties of the pendulum very difficult. Pendula, however, usually only
oscillate at small angles, where sinQ=Q
. In such cases, we can derive straightforward formulae, which are only
approximations but work well enough in practice.
Energy
The mechanical energy
of the ideal pendulum is a conserved. The potential energy of the pendulum,
mgh, increases with the height of the bob, therefore the potential energy
is minimized at the equilibrium point and is maximized at the extreme
positions. Conversely, the kinetic energy and velocity of the pendulum
are maximized at the equilibrium point and minimized at the extremes..
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