Pendulum (mechanics)


A pendulum is a body suspended from a fixed support that freely swings back and forth under the influence of gravity. When a pendulum is displaced sideways from its resting, equilibrium position, it is subject to a restoring force due to gravity that will accelerate it back towards the equilibrium position. When released, the restoring force acting on the pendulum's mass causes it to oscillate about the equilibrium position, swinging it back and forth. The mathematics of pendulums are in general quite complicated. Simplifying assumptions can be made, which in the case of a [|simple pendulum] allow the equations of motion to be solved analytically for small-angle oscillations.

Simple gravity pendulum

A simple gravity pendulum is an idealized mathematical model of a real pendulum. It is a weight on the end of a massless cord suspended from a pivot, without friction. Since in the model there is no frictional energy loss, when given an initial displacement it swings back and forth with a constant amplitude. The model is based on the assumptions:
The differential equation which governs the motion of a simple pendulum is
where is the magnitude of the gravitational field, is the length of the rod or cord, and is the angle from the vertical to the pendulum.

Small-angle approximation

The differential equation given above is not easily solved, and there is no solution that can be written in terms of elementary functions. However, adding a restriction to the size of the oscillation's amplitude gives a form whose solution can be easily obtained. If it is assumed that the angle is much less than 1 radian, or
then substituting for into using the small-angle approximation,
yields the equation for a harmonic oscillator,
The error due to the approximation is of order .
Let the starting angle be. If it is assumed that the pendulum is released with zero angular velocity, the solution becomes
The motion is simple harmonic motion where is the amplitude of the oscillation. The corresponding approximate period of the motion is then
which is known as Christiaan Huygens's law for the period. Note that under the small-angle approximation, the period is independent of the amplitude ; this is the property of isochronism that Galileo discovered.

Rule of thumb for pendulum length

gives
If SI units are used, and assuming the measurement is taking place on the Earth's surface, then, and .
Therefore, relatively reasonable approximations for the length and period are:
where is the number of seconds between two beats, and is measured in metres.

Arbitrary-amplitude period

For amplitudes beyond the small angle approximation, one can compute the exact period by first inverting the equation for the angular velocity obtained from the energy method,
and then integrating over one complete cycle,
or twice the half-cycle
or four times the quarter-cycle
which leads to
Note that this integral diverges as approaches the vertical
so that a pendulum with just the right energy to go vertical will never actually get there.
This integral can be rewritten in terms of elliptic integrals as
where is the incomplete elliptic integral of the first kind defined by
Or more concisely by the substitution
expressing in terms of,
Here is the complete elliptic integral of the first kind defined by
For comparison of the approximation to the full solution, consider the period of a pendulum of length 1 m on Earth at an initial angle of 10 degrees is
The linear approximation gives
The difference between the two values, less than 0.2%, is much less than that caused by the variation of with geographical location.
From here there are many ways to proceed to calculate the elliptic integral.

Legendre polynomial solution for the elliptic integral

Given and the Legendre polynomial solution for the elliptic integral:
where denotes the double factorial, an exact solution to the period of a simple pendulum is:
Figure 4 shows the relative errors using the power series. is the linear approximation, and to include respectively the terms up to the 2nd to the 10th powers.

Power series solution for the elliptic integral

Another formulation of the above solution can be found if the following Maclaurin series:
is used in the Legendre polynomial solution above.
The resulting power series is:
more fractions available in the On-Line Encyclopedia of Integer Sequences with having the numerators and having the denominators.

Arithmetic-geometric mean solution for elliptic integral

Given and the arithmetic–geometric mean solution of the elliptic integral:
where is the arithmetic-geometric mean of and.
This yields an alternative and faster-converging formula for the period:
The first iteration of this algorithm gives
This approximation has the relative error of less than 1% for angles up to 96.11 degrees. Since the expression can be written more concisely as
The second order expansion of reduces to
A second iteration of this algorithm gives
This second approximation has a relative error of less than 1% for angles up to 163.10 degrees.

Approximate formulae for the nonlinear pendulum period

Though the exact period can be determined, for any finite amplitude rad, by evaluating the corresponding complete elliptic integral, where, this is often avoided in applications because it is not possible to express this integral in a closed form in terms of elementary functions. This has made way for research on simple approximate formulae for the increase of the pendulum period with amplitude, though the deviation with respect to the exact period increases monotonically with amplitude, being unsuitable for amplitudes near to rad. One of the simplest formulae found in literature is the following one by Lima :, where.
  • ‘Very large-angle’ formulae, i.e. those which approximate the exact period asymptotically for amplitudes near to rad, with an error that increases monotonically for smaller amplitudes. One of the better such formulae is that by Cromer, namely:.
Of course, the increase of with amplitude is more apparent when, as has been observed in many experiments using either a rigid rod or a disc. As accurate timers and sensors are currently available even in introductory physics labs, the experimental errors found in ‘very large-angle’ experiments are already small enough for a comparison with the exact period, and a very good agreement between theory and experiments in which friction is negligible has been found. Since this activity has been encouraged by many instructors, a simple approximate formula for the pendulum period valid for all possible amplitudes, to which experimental data could be compared, was sought. In 2008, Lima derived a weighted-average formula with this characteristic:
where, which presents a maximum error of only 0.6%.

Arbitrary-amplitude angular displacement

The Fourier series expansion of is given by
where is the elliptic nome, and the angular frequency.
If one defines
can be approximated using the expansion
. Note that for, thus the approximation is applicable even for large amplitudes.
Equivalently, the angle can be given in terms of the Jacobi elliptic function with modulus
For small,, and, so the solution is well-approximated by the solution given in Pendulum #Small-angle approximation.

Examples

The animations below depict the motion of a simple pendulum with increasing amounts of initial displacement of the bob, or equivalently increasing initial velocity. The small graph above each pendulum is the corresponding phase plane diagram; the horizontal axis is displacement and the vertical axis is velocity. With a large enough initial velocity the pendulum does not oscillate back and forth but rotates completely around the pivot.

Compound pendulum

A compound pendulum is one where the rod is not massless, and may have extended size; that is, an arbitrarily shaped rigid body swinging by a pivot. In this case the pendulum's period depends on its moment of inertia around the pivot point.
The equation of torque gives:
where:
is the angular acceleration.
is the torque
The torque is generated by gravity so:
where:
  • is the total mass of the rigid body
  • is the distance from the pivot point to the system's centre-of-mass
  • is the angle from the vertical
Hence, under the small-angle approximation, ,
where is the moment of inertia of the body about the pivot point.
The expression for is of the same form as the conventional simple pendulum and gives a period of
And a frequency of
If the initial angle is taken into consideration, then the expression for becomes:
and gives a period of:
where is the maximum angle of oscillation and is the complete elliptic integral of the first kind.
An important concept is the equivalent length,, the length of a simple pendulums that has the same angular frequency as the compound pendulum:
Consider the following cases:
  • The simple pendulum is the special case where all the mass is located at the bob swinging at a distance from the pivot. Thus, and, so the expression reduces to: . Notice, as expected.
  • A homogeneous rod of mass and length swinging from its end has and, so the expression reduces to: . Notice, a homogeneous rod oscillates as if it were a simple pendulum of two-thirds its length.
  • A heavy simple pendulum: combination of a homogeneous rod of mass and length swinging from its end, and a bob at the other end. Then the system has a total mass of, and the other parameters being and, so the expression reduces to:
Where. Notice these formulae can be particularized into the two previous cases studied before just by considering the mass of the rod or the bob to be zero respectively. Also notice that the formula does not depend on both the mass of the bob and the rod, but actually on their ratio,. An approximation can be made for :
Notice how similar it is to the angular frequency in a spring-mass system with effective mass.