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Pendulum

A simple gravity pendulum (plural ''pendulums'' or ''pendula''), also called a '''bob pendulum''', is a weight on the end of a rigid rod (or a string/rope), which, when given an initial push, will swing back and forth under the influence of gravity over its central (lowest) point. A '''torsion pendulum''' consists of a body suspended by a fine wire or elastic fiber in such a way that it executes rotational oscillations as the suspending wire or fiber twists and untwists. Another variety of a '''torsion pendulum''' is a fixed elastic coil connected to a rod-like object; once moved off its resting position, the coil will set the rod into an oscillatory motion. The pendulum was discovered by Ibn Yunus Al-Masri during the 10th century, who was the first to study and document its oscillatory motion. Its value for use in clocks was introduced by physicists during the 15th century. Analysis of a simple gravity pendulum To begin, we shall make three assumptions about the simple pendulum The rod/string/cable on which the bob is swinging is massless and always remains taut; The bob is a Point Mass ; Motion occurs in a 2 dimensional plane, i.e. pendulum does not swing in and out of the page. Consider Figure 2. The blue arrow is the Gravitational Force acting on the bob, violet arrows are that same force resolved into components parallel and perpendicular to the bob's instantaneous motion, the motion along the red axis, which is always perpendicular to the cable/rod. Newton's second law :$F=ma\backslash ,$ where $F$ is the force acting on mass $m$, causing it to accelerate at $a$ meters per second2. Because the bob is constrained to move on the green circular arc, there is no need to consider any force other than the one responsible for instantaneous acceleration parallel to the known path, the short violet arrow in our case :$F\; =\; mg\backslash sin\; heta\; =\; ma\backslash ,$ :$a\; =\; g\; \backslash sin\; heta\backslash ,$ Linear acceleration $a$ along the red axis can be related to the change in angle $heta$ by the arc length formula :$s\; =\; \backslash ell\; heta\backslash ,$ :$v\; =\; \{ds\backslash over\; dt\}\; =\; \backslash ell\{d\; heta\backslash over\; dt\}$ :$a\; =\; \{d^2s\backslash over\; dt^2\}\; =\; \backslash ell\{d^2\; heta\backslash over\; dt^2\}$ This, however, is not the acceleration we seek because the gravitational force on the bob causes a ''decrease'' in angle $heta$. One accounts for that by placing a negative sign in front of $a$, thus: :$\backslash ell\{d^2\; heta\backslash over\; dt^2\}\; =\; -\; g\; \backslash sin\; heta$ :$\{d^2\; heta\backslash over\; dt^2\}+\{g\backslash over\; \backslash ell\}\; \backslash sin\; heta=0\; \backslash quad\backslash quad\backslash quad\; (1)$ This is the equal to that which it lost to the fall. In other words, Gravitational Potential energy is converted into kinetic energy. Change in potential energy is given by :$\backslash Delta\; U\; =\; mgh\backslash ,$ change in kinetic energy (body started from rest) is given by :$\backslash Delta\; K\; =\; \{1\backslash over2\}mv^2$ Since no energy is lost, those two must be equal :$\{1\backslash over2\}mv^2\; =\; mgh$ :$v\; =\; \backslash sqrt\{2gh\}\backslash ,$ Using the arc length formula above, this equation can be re-written in favor of $\{d\; heta\backslash over\; dt\}$ :$\{d\; heta\backslash over\; dt\}\; =\; \{1\backslash over\; \backslash ell\}\backslash sqrt\{2gh\}$ but what is $h$? It is the vertical distance the pendulum fell. Consider Figure 3. If the pendulum starts its swing from some initial angle $heta\_0$, then $y\_0$, the vertical distance from the screw, is given by :$y\_0\; =\; \backslash ell\backslash cos\; heta\_0\backslash ,$ similarly, for $y\_1$, we have :$y\_1\; =\; \backslash ell\backslash cos\; heta\backslash ,$ then $h$ is the difference of the two :$h\; =\; \backslash ell\backslash left(\backslash cos\; heta-\backslash cos\; heta\_0\; ight)$ substituting this into the equation for $\{d\; heta\backslash over\; dt\}$ gives   :<math> Heta(t) heta_0\cos\left(\sqrt{g\over \ell}t ight) \quad\quad\quad\quad heta_0 \ll 1</math>

the period of a complete oscillation can be easily found, and we have obtained Huygens 's law:

:$$
T_0 = 2\pi\sqrt{\frac{\ell}{g}}

:$T\_0\; =\; 2\backslash pi\backslash sqrt\{\backslash ell\backslash over\; g\}\backslash quad\backslash quad\backslash quad\backslash quad\; |\; heta\_0|\; \backslash ll\; 1$

Arbitrary-amplitude period


For amplitudes beyond the small angle approximation, one can compute the exact period by inverting equation (2)

:$\{dt\backslash over\; d\; heta\}\; =\; \{1\backslash over\backslash sqrt\{2\}\}\backslash sqrt\{\backslash ell\backslash over\; g\}\{1\backslash over\backslash sqrt\{\backslash cos\; heta-\backslash cos\; heta\_0\}\}$

and integrating over one complete cycle,

:$T\; =\; heta\_0\; ightarrow0\; ightarrow-\; heta\_0\; ightarrow0\; ightarrow\; heta\_0$
or twice the half-cycle

:$T\; =\; 2\backslash left(\; heta\_0\; ightarrow0\; ightarrow-\; heta\_0\; ight)$

or 4 times the quarter-cycle

:$T\; =\; 4\backslash left(\; heta\_0\; ightarrow0\; ight)$

which leads to

:$T\; =\; 4\{1\backslash over\backslash sqrt\{2\}\}\backslash sqrt\{\backslash ell\backslash over\; g\}\backslash int^\{\; heta\_0\}\_0\; \{1\backslash over\backslash sqrt\{\backslash cos\; heta-\backslash cos\; heta\_0\}\}d\; heta$

Alas, this integral cannot be evaluated in terms of elementary functions. It can be re-written in the form of the Elliptic Function of the first kind, which gives one very little advantage for it is a redundant exercise of expressing one insoluble integral in terms of another

:$T\; =\; 4\backslash sqrt\{\backslash ell\backslash over\; g\}F\backslash left(\{\; heta\_0\backslash over\; 2\},\backslash csc^2\{\; heta\_0\backslash over2\}\; ight)\backslash csc\; \{\; heta\_0\backslash over\; 2\}$

or more concisely,

:$T\; =\; 4\backslash sqrt\{\backslash ell\backslash over\; g\}E\backslash left(\{\backslash sin\; heta\_0\backslash over\; 2\},\; \{\backslash pi\; \backslash over\; 2\}\; ight)$

where $E(k,\backslash phi)$ is Legendre's elliptic function of the first kind

$E(k,\backslash phi)\; =\; \backslash int^\{\backslash phi\}\_0\; \{1\backslash over\backslash sqrt\{1-k^2\backslash sin^2\{\; heta\}\}\}d\; heta$

The value of the elliptic function can be also computed using the following series:

:$T\; =\; T\_0\; \backslash left(\; 1+\; \backslash left(\; \backslash frac\{1\}\{2\}\; ight)^2\; \backslash sin^2\backslash left(\backslash frac\{\; heta\_0\}\{2\}\; ight)\; +\; \backslash left(\; \backslash frac\{1\; \backslash cdot\; 3\}\{2\; \backslash cdot\; 4\}\; ight)^2\; \backslash sin^4\backslash left(\backslash frac\{\; heta\_0\}\{2\}\; ight)\; +\; \backslash left(\; \backslash frac\; \{1\; \backslash cdot\; 3\; \backslash cdot\; 5\}\{2\; \backslash cdot\; 4\; \backslash cdot\; 6\}\; ight)^2\; \backslash sin^6\backslash left(\backslash frac\{\; heta\_0\}\{2\}\; ight)\; +\; \backslash cdots\; ight)$

Figure 4 shows the deviation of $T$ from $T\_0$, the period obtained from small-angle approximation.

For a swing of $180^\backslash circ$ the bob is balanced over its pivot point and so $T=\backslash infty$ (keep in mind the pendulum is made of a rigid rod).

For example, the period of a 1m pendulum at initial angle 10 degrees is $4\backslash sqrt\{1\backslash over\; g\}E\backslash left(\{\backslash sin\; 10\backslash over\; 2\},\{\backslash pi\backslash over2\}\; ight)\; =\; 2.0102$ seconds, whereas the approximation $2\backslash pi\; \backslash sqrt\{1\backslash over\; g\}\; =\; 2.0064$ that's about 1 second per swing (both examples use g = 9.80665).


Applications


As first explained by M. 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.

The presence of ''g'' in the equation means that the pendulum frequency is different at different places on earth. So for example if you have an accurate pendulum clock in Glasgow (''g'' = 9.815 63 m/s²) and you take it to Cairo (''g'' = 9.793 17 m/s²), you must shorten the pendulum by 0.23%.

Two coupled pendulums form a Double Pendulum . A pendulum whose time period is two seconds is called the second pendulum.

A pendulum is also used for finding acceleration due to gravity(g), formula is T = 4π&²l/g. It is also used in seismographs.


Torsion pendulums


If ''I'' is the Moment Of Inertia of a body with respect to its axis of oscillation, and if ''K'' is the Torsion Coefficient of the fiber ( Torque required to twist it through an angle of one Radian ), then the period of oscillation of a torsion pendulum is given by

:$T\; =\; 2\; \backslash pi\; \backslash sqrt\{\backslash frac\{I\}\{K\}\}$

Both ''I'' and ''K'' may have to be determined by experiment. This can be done by measuring the period ''T'' and then adding to the suspended body another body of known moment of inertia ''I''', giving a new period of oscillation ''T'''

:$T\text{'}\; =\; 2\backslash pi\; \backslash sqrt\{\backslash frac\{I+I\text{'}\}\{K\}\}$

and then solving the two equations to get

:$K\; =\; \backslash frac\{4\backslash pi^2I\text{'}\}\{T\text{'}^2\; -\; T^2\}$

:$I\; =\; \backslash frac\{T^2I\text{'}\}\{T\text{'}^2\; -\; T^2\}$

The oscillating balance wheel of a watch is in effect a torsion pendulum, with the suspending fiber replaced by hairspring and pivots. The watch is regulated, first roughly by adjusting ''I'' (the purpose of the screws set radially into the rim of the wheel) and then more accurately by changing the free length of the hairspring and hence the torsion coefficient ''K''.


Damped pendulum


The pendulum equation does not take into account the effects of friction and dissipation. While these effects can be very complicated to model, a good approximation is to add a term proportional to the velocity:

:$\backslash ell\; \backslash frac\{d^2\; heta\}\{dt^2\}=-g\; \backslash sin\; heta\; -\; \backslash gamma\; \backslash frac\{d\; heta\}\{dt\}$

The positive constant γ is the viscous damping parameter.
A system described by this equation is called a damped pendulum.


Pendulums for divination and dowsing


Pendulums (these may be a crystal suspended on a chain, or a metal weight) are often used for Divination and Dowsing . There exist many different techniques. One widely used form is the following. The user will first determine which direction (left-right, up-down) determines "yes" and which "no," before proceeding to ask the pendulum specific questions. In another form of divination, the pendulum is used with a pad or cloth that may have yes and no, but also other words written in a circle. The person holding the pendulum aims to hold it as steadily as possible over the center. An interviewer may pose questions to the person holding the pendulum, and it swings by minute unconscious bodily movement in the direction of the answer. In the practice of Radiesthesia a pendulum is used for medical diagnosis. However all these uses of pendulums are not scientifically tested or supported.


See also

• Pendulum Clock


• Simple Harmonic Motion


• Foucault Pendulum


• Spherical Pendulum


• Double Pendulum


• Kater's Pendulum


• Harmonograph


• Metronome