Everything about Pendulum totally explained
A
pendulum is a weight that's attached to a pivot, from which it can swing freely. This object is subject to a restoring force due to
gravity that will accelerate it toward an equilibrium position. When the pendulum is displaced from its place of rest, the restoring force will cause the pendulum to
oscillate about the equilibrium position.
A basic example is the
simple gravity pendulum or
bob pendulum. This is a weight (or
bob) on the end of a massless string, which, when initially displaced, will swing back and forth under the influence of gravity over its central (lowest) point.
The regular motion of the pendulum can be used for time keeping, and pendulums are used to regulate
pendulum clocks.
History
As recorded in the 4th century
Chinese Book of Taha hasan, one of the earliest uses of the pendulum was in the
seismometer device of the
Han Dynasty (
202 BC -
220 AD) scientist and inventor
Zhang Heng (
78-
139). Its function was to sway and activate a series of levers after being disturbed by the tremor of an
earthquake far away. After this was triggered, a small ball would fall out of the urn-shaped device into a metal toad's mouth below, signifying the
cardinal direction of where the earthquake was located (and where government aid and assistance should be swiftly sent).
Among his scientific studies,
Galileo Galilei performed a number of observations of all the properties of pendulums. His interest in the pendulum may have been sparked by looking at the swinging motion of a chandelier in the Pisa cathedral. He began serious studies of the pendulum around 1602. Galileo noticed that period of the pendulum is independent of the bob mass or the amplitude of the swing. He also found a direct relationship between the square of the period and the length of the arm. The
isochronism of the pendulum suggested a practical application for use as a
metronome to aid musical students, and possibly for use in a clock.
Perhaps based upon the ideas of Galileo, in 1656 the
Dutch scientist
Christiaan Huygens patented a mechanical
clock that employed a pendulum to regulate the movement. This approach proved much more accurate than previous time pieces, such as the
hourglass.
Following an illness, in 1665 Huygens made a curious observation about pendulum clocks. Two such clocks had been placed on his
fireplace mantel, and he noted that they'd acquired an opposing motion. That is, they were beating in unison but in the opposite direction—an anti-phase motion. Regardless of how the two clocks were adjusted, he found that they'd eventually return to this state, thus making the first recorded observation of a coupled
oscillator.
During his
Académie des Sciences expedition to
Cayenne,
French Guiana in 1671,
Jean Richer demonstrated that the periodicity of a pendulum was slower at Cayenne than at
Paris. From this he deduced that the force of gravity was lower at Cayenne. Huygens reasoned that the
centripetal force of the
Earth's rotation modified the weight of the pendulum bob based on the
latitude of the observer.
In his 1673 opus
Horologium Oscillatorium sive de motu pendulorum, Christiaan Huygens published his theory of the pendulum. He demonstrated that for an object to descend a curve under gravity in the same time interval, regardless of the starting point, it must follow a
cycloid (rather than the circular arc of a pendulum). This confirmed the earlier observation by
Marin Mersenne that the period of a pendulum does vary with amplitude, and that Galileo's observation was accurate only for small swings in the neighborhood of the center line.
The English scientist
Robert Hooke devised the
conical pendulum, consisting of a pendulum that's free to swing in both directions. By analyzing the circular movements of the pendulum bob, he used it to analyze the orbital motions of the planets. Hooke would suggest to
Isaac Newton in 1679 that the components of orbital motion consisted of inertial motion along a tangent direction plus an attractive motion in the radial direction. Isaac Newton was able to translate this idea into a mathematical form that described the movements of the planets with a central force that obeyed an
inverse square law—
Newton's law of universal gravitation. Robert Hooke was also responsible for suggesting (as early as 1666) that the pendulum could be used to measure the force of gravity.
In 1851,
Jean-Bernard-Leon Foucault suspended a pendulum (later named the
Foucault pendulum) from the dome of the
Panthéon in
Paris. It was the third Foucault pendulum he constructed, the first one was constructed in his basement and the second one was a demonstration model with a length of 11 meters. The mass of the pendulum in Pantheon was 28 kg and the length of the arm was 67 m. The Foucault pendulum was a worldwide sensation: it was the first demonstration of the Earth's rotation with a purely indoors experiment. Once the Paris pendulum was set in motion the plane of motion was observed to
precess about 270° clockwise per day. A pendulum located at either of the poles will precess 360° relative to the ground it's suspended above. There is a mathematical relation between the latitude where a Foucault pendulum is deployed and its precession; the lengthening of the period of the precession is inversely proportional to the sine of the latitude.
The
National Institute of Standards and Technology based the U.S. national time standard on the Riefler Clock from 1904 until 1929. This
pendulum clock maintained an accuracy of a few hundredths of a second per day. It was briefly replaced by the double-pendulum W. H. Shortt Clock before the NIST switched
to an electronic time-keeping system.
Basic principles
Simple pendulum
If and only if the pendulum swings through a small
angle (in the range where the function
sin(
θ) can be approximated as
θ) the motion may be approximated as
simple harmonic motion. The period of a simple pendulum is significantly affected only by its
length and the acceleration of
gravity. The period of motion is independent of the
mass of the bob or the angle at which the arm hangs at the moment of release. The period of the pendulum is the time taken for one complete swing (left to right and back again) of the pendulum. The formula for the period,
T, is
»
Double pendulum
A double pendulum consists of one pendulum attached to the free end of another pendulum. The behaviour of this system is significantly more complicated than that of a single simple or physical pendulum. For small angles of displacement this system is approximately linear and can be modelled by the theory of
normal modes. As the angles increase, however, the double pendulum can exhibit
chaotic motion that's sensitive to the initial conditions.
A special case of the compound double pendulum, known as
Rott's Pendulum, is that where the pivots of the two pendulums are horizontal when the system is in static equilibrium, and the periods of the pendulums are a factor of two apart. In this case, the coupling between the pendulums is to first order
non-linear which — for small angles — leads to periodic behaviour with a period much larger than that of either pendulum.
Use for measurement
The most widespread application is for
timekeeping. A pendulum whose time period is 2 seconds is called the
seconds pendulum since most clock
escapements move the seconds hands on each swing. Clocks that keep time with the use of pendulums lose accuracy due to friction. Pendulums are also widely used as metronomes for pianists.
The presence of
g as a variable in the periodicity equation for a pendulum means that the frequency is different at various locations on Earth. So, for example, when an accurate pendulum clock in
Glasgow,
Scotland, (
g = 9.815 63 m/s
2) is transported to
Cairo,
Egypt, (
g = 9.793 17 m/s
2) the pendulum must be shortened by 0.23% to compensate. The pendulum can therefore be used in
gravimetry to measure the local
gravity at any point on the surface of the Earth. Note that
g = 9.8 m/s² is a safe standard for acceleration due to gravity if locational accuracy isn't a concern.
A pendulum in which the rod isn't vertical but almost horizontal was used in early
seismometers for measuring earth tremors. The bob of the pendulum doesn't move when its mounting does and the difference in the movements is recorded on a drum chart.
Problems
Pendulums in air are affected by atmospheric and mechanical
drag. These effects can be compensated for if they're known and constant. Atmospheric drag is affected by the density of air, which is in turn affected by its moisture content, temperature, and barometric pressure. Precise clocks used for the timing of astronomic observations were improved by operating the pendulum in a partially evacuated and temperature controlled chamber. Since the drag is proportional to the square of the velocity, a long pendulum or a pendulum with a high rotational moment of inertia about its pivot, which both produce slow oscillation, will be less affected by atmospheric drag than is a faster pendulum.
Simple pendulums in everyday clocks are affected by the ambient temperature, which
thermal expansion of the material holding the bob will change the period of the pendulum. This change of length can be minimized by using special materials for the pendulum rod which exhibit little change with temperature or by using a more complex
gridiron pendulum, sometimes called a "banjo" pendulum for its similarity in appearance to the
musical instrument.
it can be also affected by the type of the spring/ rope. so the more stretchy the spring the larger distance it covers and the more time it takes to complete one complete oscillation.
Other applications
Schuler tuning
As first explained by
Maximilian Schuler in his classic 1923 paper, a pendulum whose period exactly equals the orbital period of a hypothetical satellite orbiting just above the surface of the earth (about 84 minutes) will tend to remain pointing at the center of the earth when its support is suddenly displaced. This is the basic principle of
Schuler tuning that must be included in the design of any
inertial guidance system that will be operated near the earth, such as in ships and aircraft.
Religious practice
Pendulum motion appears in religious ceremonies as well. The swinging
incense burner called a
censer, also known as a
thurible, is an example of a pendulum.
Further Information
Get more info on 'Pendulum'.
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