pendulum

Simple gravity pendulum

A simple gravity pendulum is an idealized mathematical model of a real pendulum. It is a weight (or bob) 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 rod or cord is massless, inextensible and always remains under tension.

The bob is a point mass.

The motion occurs in two dimensions.

The motion does not lose energy to external friction or air resistance.

The gravitational field is uniform.

The support is immobile.

The differential equation which governs the motion of a simple pendulum is

where g is the magnitude of the gravitational field, ℓ is the length of the rod or cord, and θ 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 (often cited as less than 0.1 radians, about 6°), or

θ

≪

1

,

{\displaystyle \theta \ll 1,}

then substituting for sin θ into Eq. 1 using the small-angle approximation,

sin

⁡

θ

≈

θ

,

{\displaystyle \sin \theta \approx \theta ,}

yields the equation for a harmonic oscillator,

d

2

θ

d

t

2

+

g

ℓ

θ

=

0.

{\displaystyle {\frac {d^{2}\theta }{dt^{2}}}+{\frac {g}{\ell }}\theta =0.}

Rule of thumb for pendulum length

T

0

=

2

π

ℓ

g

{\displaystyle T_{0}=2\pi {\sqrt {\frac {\ell }{g}}}}

gives

ℓ

=

g

π

2

T

0

2

4

.

{\displaystyle \ell ={\frac {g}{\pi ^{2}}}{\frac {T_{0}^{2}}{4}}.}

If SI units are used (i.e. measure in metres and seconds), and assuming the measurement is taking place on the Earth's surface, then g ≈ 9.81 m/s2, and ⁠g/π2⁠ ≈ 1 m/s2 (0.994 is the approximation to 3 decimal places).

Therefore, relatively reasonable approximations for the length and period are:

ℓ

≈

T

0

2

4

,

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 (Eq. 2),

d

t

d

θ

=

ℓ

2

g

1

cos

⁡

θ

−

cos

⁡

θ

0

{\displaystyle {\frac {dt}{d\theta }}={\sqrt {\frac {\ell }{2g}}}{\frac {1}{\sqrt {\cos \theta -\cos \theta _{0}}}}}

and then integrating over one complete cycle,

T

=

t

(

θ

0

→

0

→

−

θ

Legendre polynomial solution for the elliptic integral

Given Eq. 3 and the Legendre polynomial solution for the elliptic integral:

K

(

k

)

=

π

2

∑

n

=

0

∞

(

(

2

n

−

1

)

!

!

(

2

n

)

!

!

k

n

)

2

{\displaystyle K(k)={\frac {\pi }{2}}\sum _{n=0}^{\infty }\left({\frac {(2n-1)!!}{(2n)!!}}k^{n}\right)^{2}}

where n!! denotes the double factorial, an exact solution to the period of a simple pendulum is:

Power series solution for the elliptic integral

Another formulation of the above solution can be found if the following Maclaurin series:

sin

⁡

θ

0

2

=

1

2

θ

0

−

1

48

θ

0

3

+

1

3

840

θ

0

5

−

1

645

120

θ

0

7

+

⋯

.

{\displaystyle \sin {\frac {\theta _{0}}{2}}={\frac {1}{2}}\theta _{0}-{\frac {1}{48}}\theta _{0}^{3}+{\frac {1}{3\,840}}\theta _{0}^{5}-{\frac {1}{645\,120}}\theta _{0}^{7}+\cdots .}

Arithmetic-geometric mean solution for elliptic integral

Given Eq. 3 and the arithmetic–geometric mean solution of the elliptic integral:

K

(

k

)

=

π

2

M

(

1

−

k

,

1

+

k

)

,

{\displaystyle K(k)={\frac {\pi }{2M(1-k,1+k)}},}

where M(x,y) is the arithmetic-geometric mean of x and y.

This yields an alternative and faster-converging formula for the period:

T

=

2

π

M

(

1

,

cos

Approximate formulae for the nonlinear pendulum period

Though the exact period

T

{\displaystyle T}

can be determined, for any finite amplitude

θ

0

<

π

{\displaystyle \theta _{0}<\pi }

rad, by evaluating the corresponding complete elliptic integral

K

(

k

)

{\displaystyle K(k)}

, where

k

≡

sin

⁡

(

θ

0

/

2

)

{\displaystyle k\equiv \sin(\theta _{0}/2)}

, 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 (useful in introductory physics labs, classical mechanics, electromagnetism, acoustics, electronics, superconductivity, etc. The approximate formulae found by different authors can be classified as follows:

Arbitrary-amplitude angular displacement

The Fourier series expansion of

θ

(

t

)

{\displaystyle \theta (t)}

is given by

θ

(

t

)

=

8

∑

n

≥

1

odd

(

−

1

)

⌊

n

/

2

⌋

n

q

n

/

2

1

+

q

n

Examples

The animations below depict the motion of a simple (frictionless) 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 (or physical 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

O

{\displaystyle O}

. In this case the pendulum's period depends on its moment of inertia

I

O

{\displaystyle I_{O}}

around the pivot point.

The equation of torque gives:

τ

=

I

α

{\displaystyle \tau =I\alpha }

where:

α

{\displaystyle \alpha }

is the angular acceleration.

τ

{\displaystyle \tau }

is the torque

The torque is generated by gravity so:

τ

=

−

m

g

r

Damped, driven pendulum

The above discussion focuses on a pendulum bob only acted upon by the force of gravity. Suppose a damping force, e.g. air resistance, as well as a sinusoidal driving force acts on the body. This system is a damped, driven oscillator, and is chaotic.

Equation (1) can be written as

m

l

2

d

2

θ

d

t

2

=

−

m

g

l

sin

⁡

θ

{\displaystyle ml^{2}{\frac {d^{2}\theta }{dt^{2}}}=-mgl\sin \theta }

(see the Torque derivation of Equation (1) above).

A damping term and forcing term can be added to the right hand side to get

m

l

2

d

2

θ

d

Physical interpretation of the imaginary period

The Jacobian elliptic function that expresses the position of a pendulum as a function of time is a doubly periodic function with a real period and an imaginary period. The real period is, of course, the time it takes the pendulum to go through one full cycle. Paul Appell pointed out a physical interpretation of the imaginary period: if θ0 is the maximum angle of one pendulum and 180° − θ0 is the maximum angle of another, then the real period of each is the magnitude of the imaginary period of the other.

Coupled pendula

Coupled pendulums can affect each other's motion, either through a direction connection (such as a spring connecting the bobs) or through motions in a supporting structure (such as a tabletop). The equations of motion for two identical simple pendulums coupled by a spring connecting the bobs can be obtained using Lagrangian mechanics.

The kinetic energy of the system is:

E

K

=

1

2

m

L

2

(

θ

˙

1

2

+

θ

˙

2

2

)

{\displaystyle E_{\text{K}}={\frac {1}{2}}mL^{2}\left({\dot {\theta }}_{1}^{2}+{\dot {\theta }}_{2}^{2}\right)}

where

m

{\displaystyle m}

is the mass of the bobs,

L

{\displaystyle L}

is the length of the strings, and