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# Pendulum

A gravity pendulum is a weight on the end of a rigid rod, which, when given some initial lift from the vertical position, 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.

Galileo Galilei discovered that pendula exhibit regular periodic motion, a feature which he correctly speculated would make them useful for timekeeping devices such as clocks. This has led to their frequent metaphorical use as a way of describing the passage of time, or the experience of it. The mass of the weight on the pendulum does not affect the outcome, just as it does not affect the outcome of dropped objects.

For smaller displacements, the movement of an ideal pendulum can be described mathematically as simple harmonic motion, as the change in potential energy at the bottom of a circular arc is nearly proportional to the square of the displacement. Real pendulums do not have infinitesimal displacements, so their behaviour is actually of a non-linear kind. Real pendulums will also lose energy as they swing, and so their motion will be damped, with the size of the oscillation decreasing approximately exponentially with time.

The case of a pendulum with a point mass swinging on a massless rigid rod of length l is the simple gravity pendulum (simple pendulum). If θ is the angle between the rod and the vertical, the acceleration is given by $-g\cdot\sin\theta$ and is equal to the angular acceleration multiplied by the length of the rod. So we have the following differential equation:

$\ell\cdot\frac{d^2\theta}{dt^2}=-g\cdot \sin\theta$

When the amplitude of the swing is small, $\sin\theta \approx \theta$. If the pendulum is initially still at an angle θ0 which is also the maximum angle, the function which solves the differential equation above is the following harmonic law:

$\theta = \theta_0\cdot \cos\left(\sqrt{\frac{g}{\ell}}\cdot t\right)$

The term $\sqrt{\frac{g}{\ell}}$ is a pulsation, which is equal to $\frac{2\pi}{T_0}$,

where T0 is the period of a complete oscillation (outward and return).

Since $\omega = \sqrt{\frac{g}{\ell}} = \frac{2\pi}{T_0}$,

the period of a complete oscillation can be easily found, and it is given by Huygens's law:

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

When the amplitude of the swing is no more negligible, the differential equation cannot be simplified anymore, and the period T must be corrected with the following expression, which is the result of a complete elliptic integral of the first kind:

$T = T_0 \left( 1+ \left( \frac{1}{2} \right)^2 \sin^2\left(\frac{\theta_0}{2}\right) + \left( \frac{1 \cdot 3}{2 \cdot 4} \right)^2 \sin^4\left(\frac{\theta_0}{2}\right) + \left( \frac {1 \cdot 3 \cdot 5}{2 \cdot 4 \cdot 6} \right)^2 \sin^6\left(\frac{\theta_0}{2}\right) + \cdots \right)$

where θ0 is the semi-amplitude of the oscillation, that is the maximum angle between the rod of the pendulum and the vertical.

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

The following table shows the correction of the period T for wide amplitudes.

 θ0 T / T0 0 1.0000 15 1.0043 30 1.0174 45 1.0399 60 1.0732 75 1.1189 90 1.1803 105 1.2622 120 1.3729 135 1.5279 150 1.7622 165 2.1854 180 $+\infty$

Notice that the equation for T is independent of the bob's mass. This can be considered a consequence of the equivalence of gravitational mass and inertial mass: a heavier bob has a stronger restoring force, but on the other hand has more inertia and the two effects cancel.

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.

## 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

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

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'

$T' = 2\pi \sqrt{\frac{I+I'}{K}}$

and then solving the two equations to get

$K = \frac{4\pi^2I'}{T'^2 - T^2}$
$I = \frac{T^2I'}{T'^2 - T^2}$

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.