# Pendulum

A gravity pendulum (plural pendula) 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 vertical 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

$F=ma\,$

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

$F = mg\sin\theta = ma\,$
$a = g \sin\theta\,$

Linear acceleration a along the red axis can be related to the change in angle θ by the arc length formula

$s = \ell\theta\,$
$v = {ds\over dt} = \ell{d\theta\over dt}$
$a = {d^2s\over dt^2} = \ell{d^2\theta\over dt^2}$

This, however, is not the acceleration we seek because the gravitational force on the bob causes a decrease in angle θ. One accounts for that by placing a negative sign in front of a, thus:

$\ell{d^2\theta\over dt^2} = - g \sin\theta$
${d^2\theta\over dt^2}+{g\over \ell} \sin\theta=0 \quad\quad\quad (1)$

This is the differential equation which, when solved for θ(t), will yield the motion of the pendulum. It can also be obtained via the conservation of mechanical energy principle: any given object which fell a vertical distance h starting from rest, would have acquired kinetic energy 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

$\Delta U = mgh\,$

change in kinetic energy (body started from rest) is given by

$\Delta K = {1\over2}mv^2$

Since no energy is lost, those two must be equal

${1\over2}mv^2 = mgh$
$v = \sqrt{2gh}\,$

Using the arc length formula above, this equation can be re-written in favor of ${d\theta\over dt}$

${d\theta\over dt} = {1\over \ell}\sqrt{2gh}$

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 θ0, then y0, the vertical distance from the screw, is given by

$y_0 = \ell\cos\theta_0\,$

similarly, for y1, we have

$y_1 = \ell\cos\theta\,$

then h is the difference of the two

$h = \ell\left(\cos\theta-\cos\theta_0\right)$

substituting this into the equation for ${d\theta\over dt}$ gives

${d\theta\over dt} = \sqrt{{2g\over \ell}\left(\cos\theta-\cos\theta_0\right)}\quad\quad\quad (2)$

This equation is known as the first integral of motion, it gives the velocity in terms of the location and includes an integration constant related do the initial displacement (θ0). We can differentiate, by applying the chain rule, with respect to time to get the acceleration

${d\over dt}{d\theta\over dt} = {d\over dt}\sqrt{{2g\over \ell}\left(\cos\theta-\cos\theta_0\right)}$
${d^2\theta\over dt^2} = {1\over 2}{-(2g/\ell) \sin\theta\over\sqrt{(2g/\ell) \left(\cos\theta-\cos\theta_0\right)}}{d\theta\over dt} = {1\over 2}{-(2g/\ell) \sin\theta\over\sqrt{(2g/\ell) \left(\cos\theta-\cos\theta_0\right)}}\sqrt{{2g\over \ell} \left(\cos\theta-\cos\theta_0\right)} = -{g\over \ell}\sin\theta$
${d^2\theta\over dt^2} = -{g\over \ell}\sin\theta$

same as obtained through force analysis.

## Small angle approximation

The problem with the equations developed in the previous section is that they are unintegrable. To shed some light on the behavior of the pendulum we shall make another approximation. Namely, we restrict the motion of the pendulum to a relatively small amplitude, that is, relatively small θ. How small? Small enough that the following approximation is true within some desirable tolerance

$\sin\theta\approx\theta$

iff

$|\theta|\ll 1$

Substituting this approximation into (1) yields

${d^2\theta\over dt^2}+{g\over \ell}\theta=0$

Solution to this equation is a well-known, and quite expected, oscillatory function

$\theta(t) = \theta_0\cos\left(\sqrt{g\over \ell}t\right) \quad\quad\quad\quad |\theta_0| \ll 1$

where θ0 is the largest angle attained by the pendulum. Period, the time for one complete oscillation (time for the bob to return to its starting position), is given by divided by whatever is multiplying the time in the argument of the cosine

$T_0 = 2\pi\sqrt{\ell\over g}\quad\quad\quad\quad |\theta_0| \ll 1$

## Arbitrary-amplitude period

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

${dt\over d\theta} = {1\over\sqrt{2}}\sqrt{\ell\over g}{1\over\sqrt{\cos\theta-\cos\theta_0}}$

and integrating over one complete cycle,

$T = \theta_0\rightarrow0\rightarrow-\theta_0\rightarrow0\rightarrow\theta_0$

or twice the half-cycle

$T = 2\left(\theta_0\rightarrow0\rightarrow-\theta_0\right)$

or 4 times the quarter-cycle

$T = 4\left(\theta_0\rightarrow0\right)$

$T = 4{1\over\sqrt{2}}\sqrt{\ell\over g}\int^{\theta_0}_0 {1\over\sqrt{\cos\theta-\cos\theta_0}}d\theta$

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

$T = 4\sqrt{\ell\over g}F\left({\theta_0\over 2},\csc^2{\theta_0\over2}\right)\csc {\theta_0\over 2}$

or more concisely,

$T = 4\sqrt{\ell\over g}E\left({\sin\theta_0\over 2}, {\pi \over 2} \right)$

where E(k,φ) is Legendre's elliptic function of the first kind

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

for example, the period of a 1m pendium at initial angle 20 degrees is $4\sqrt{1\over g}E\left({\sin 10},{\pi\over2}\right) = 2.0102$ seconds, whereas the approximation $2\pi \sqrt{1\over g} = 2.0064$ that's about 1 second per swing (both examples use g = 9.80665).

Figure 4 shows the deviation of T from T0, the period obtained from small-angle approximation.

## 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.

## Torsion pendula

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.

## 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:

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

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

## Pendula for divination and dowsing

Pendula (this 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 also other words written in a circle. The person holding the pendulum aims to hold it as steadfast as possible in 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.