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 (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.}
The error due to the approximation is of order θ3 (from the Taylor expansion for sin θ). Let the starting angle be θ0. If it is assumed that the pendulum is released with zero angular velocity, the solution becomes
The motion is simple harmonic motion where θ0 is the amplitude of the oscillation (that is, the maximum angle between the rod of the pendulum and the vertical). 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 θ0; this is the property of isochronism that Galileo discovered.
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 , T 0 ≈ 2 ℓ {\displaystyle {\begin{aligned}\ell &\approx {\frac {T_{0}^{2}}{4}},\\T_{0}&\approx 2{\sqrt {\ell }}\end{aligned}}}
where T0 is the number of seconds between two beats (one beat for each side of the swing), and l 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 (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 0 → 0 → θ 0 ) , {\displaystyle T=t(\theta _{0}\rightarrow 0\rightarrow -\theta _{0}\rightarrow 0\rightarrow \theta _{0}),}
or twice the half-cycle
T = 2 t ( θ 0 → 0 → − θ 0 ) , {\displaystyle T=2t(\theta _{0}\rightarrow 0\rightarrow -\theta _{0}),}
or four times the quarter-cycle
T = 4 t ( θ 0 → 0 ) , {\displaystyle T=4t(\theta _{0}\rightarrow 0),}
which leads to
T = 4 ℓ 2 g ∫ 0 θ 0 d θ cos θ − cos θ 0 . {\displaystyle T=4{\sqrt {\frac {\ell }{2g}}}\int _{0}^{\theta _{0}}{\frac {d\theta }{\sqrt {\cos \theta -\cos \theta _{0}}}}.}
Note that this is an improper integral because the integrand has singularities at θ = ± θ 0 + 2 π Z {\displaystyle \theta =\pm \theta _{0}+2\pi \mathbb {Z} } , but these singularities are integrable as long as 0 < θ 0 < π {\displaystyle 0<\theta _{0}<\pi } . At θ 0 = ± π {\displaystyle \theta _{0}=\pm \pi } , the singularities become non-integrable, implying that the integral's value diverges as the maximum swing angle approaches the vertical
lim θ 0 → π T = ∞ , {\displaystyle \lim _{\theta _{0}\to \pi }T=\infty ,}
so that a pendulum with just the right energy to go vertical will never actually get there. (Conversely, a pendulum close to its maximum can take an arbitrarily long time to fall down.) This integral can be rewritten in terms of elliptic integrals as
T = 4 ℓ g F ( π 2 , sin θ 0 2 ) {\displaystyle T=4{\sqrt {\frac {\ell }{g}}}F\left({\frac {\pi }{2}},\sin {\frac {\theta _{0}}{2}}\right)}
where F is the incomplete elliptic integral of the first kind defined by
F ( φ , k ) = ∫ 0 φ d u 1 − k 2 sin 2 u . {\displaystyle F(\varphi ,k)=\int _{0}^{\varphi }{\frac {du}{\sqrt {1-k^{2}\sin ^{2}u}}}\,.}
Or more concisely by the substitution
sin u = sin θ 2 sin θ 0 2 {\displaystyle \sin {u}={\frac {\sin {\frac {\theta }{2}}}{\sin {\frac {\theta _{0}}{2}}}}}
expressing θ in terms of u,
Here K is the complete elliptic integral of the first kind defined by
K ( k ) = F ( π 2 , k ) = ∫ 0 π 2 d u 1 − k 2 sin 2 u . {\displaystyle K(k)=F\left({\frac {\pi }{2}},k\right)=\int _{0}^{\frac {\pi }{2}}{\frac {du}{\sqrt {1-k^{2}\sin ^{2}u}}}\,.}
For comparison of the approximation to the full solution, consider the period of a pendulum of length 1 m on Earth (g = 9.80665 m/s2) at an initial angle of 10 degrees is
4 1 m g K ( sin 10 ∘ 2 ) ≈ 2.0102 s . {\displaystyle 4{\sqrt {\frac {1{\text{ m}}}{g}}}\ K\left(\sin {\frac {10^{\circ }}{2}}\right)\approx 2.0102{\text{ s}}.}
The linear approximation gives
2 π 1 m g ≈ 2.0064 s . {\displaystyle 2\pi {\sqrt {\frac {1{\text{ m}}}{g}}}\approx 2.0064{\text{ s}}.}
The difference between the two values, less than 0.2%, is on the same order of magnitude as the maximum variation of g with geographical location. From here there are many ways to proceed to calculate the elliptic integral.
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:
T = 2 π ℓ g ( 1 + ( 1 2 ) 2 sin 2 θ 0 2 + ( 1 ⋅ 3 2 ⋅ 4 ) 2 sin 4 θ 0 2 + ( 1 ⋅ 3 ⋅ 5 2 ⋅ 4 ⋅ 6 ) 2 sin 6 θ 0 2 + ⋯ ) = 2 π ℓ g ⋅ ∑ n = 0 ∞ ( ( ( 2 n ) ! ( 2 n ⋅ n ! ) 2 ) 2 ⋅ sin 2 n θ 0 2 ) . {\displaystyle {\begin{alignedat}{2}T&=2\pi {\sqrt {\frac {\ell }{g}}}\left(1+\left({\frac {1}{2}}\right)^{2}\sin ^{2}{\frac {\theta _{0}}{2}}+\left({\frac {1\cdot 3}{2\cdot 4}}\right)^{2}\sin ^{4}{\frac {\theta _{0}}{2}}+\left({\frac {1\cdot 3\cdot 5}{2\cdot 4\cdot 6}}\right)^{2}\sin ^{6}{\frac {\theta _{0}}{2}}+\cdots \right)\\&=2\pi {\sqrt {\frac {\ell }{g}}}\cdot \sum _{n=0}^{\infty }\left(\left({\frac {(2n)!}{(2^{n}\cdot n!)^{2}}}\right)^{2}\cdot \sin ^{2n}{\frac {\theta _{0}}{2}}\right).\end{alignedat}}}
Figure 4 shows the relative errors using the power series. T0 is the linear approximation, and T2 to T10 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:
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 .}
is used in the Legendre polynomial solution above. The resulting power series is:
T = 2 π ℓ g ( 1 + 1 16 θ 0 2 + 11 3 072 θ 0 4 + 173 737 280 θ 0 6 + 22 931 1 321 205 760 θ 0 8 + 1 319 183 951 268 147 200 θ 0 10 + 233 526 463 2 009 078 326 886 400 θ 0 12 + ⋯ ) , {\displaystyle T=2\pi {\sqrt {\frac {\ell }{g}}}\left(1+{\frac {1}{16}}\theta _{0}^{2}+{\frac {11}{3\,072}}\theta _{0}^{4}+{\frac {173}{737\,280}}\theta _{0}^{6}+{\frac {22\,931}{1\,321\,205\,760}}\theta _{0}^{8}+{\frac {1\,319\,183}{951\,268\,147\,200}}\theta _{0}^{10}+{\frac {233\,526\,463}{2\,009\,078\,326\,886\,400}}\theta _{0}^{12}+\cdots \right),}
more fractions available in the On-Line Encyclopedia of Integer Sequences with OEIS: A223067 having the numerators and OEIS: A223068 having the denominators.
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 θ 0 2 ) ℓ g . {\displaystyle T={\frac {2\pi }{M\left(1,\cos {\frac {\theta _{0}}{2}}\right)}}{\sqrt {\frac {\ell }{g}}}.}
The first iteration of this algorithm gives
T 1 = 2 T 0 1 + cos θ 0 2 . {\displaystyle T_{1}={\frac {2T_{0}}{1+\cos {\frac {\theta _{0}}{2}}}}.}
This approximation has the relative error of less than 1% for angles up to 96.11 degrees. Since 1 2 ( 1 + cos ( θ 0 2 ) ) = cos 2 θ 0 4 , {\textstyle {\frac {1}{2}}\left(1+\cos \left({\frac {\theta _{0}}{2}}\right)\right)=\cos ^{2}{\frac {\theta _{0}}{4}},} the expression can be written more concisely as
T 1 = T 0 sec 2 θ 0 4 . {\displaystyle T_{1}=T_{0}\sec ^{2}{\frac {\theta _{0}}{4}}.}
The second order expansion of sec 2 ( θ 0 / 4 ) {\displaystyle \sec ^{2}(\theta _{0}/4)} reduces to T ≈ T 0 ( 1 + θ 0 2 16 ) . {\textstyle T\approx T_{0}\left(1+{\frac {\theta _{0}^{2}}{16}}\right).}
A second iteration of this algorithm gives
T 2 = 4 T 0 1 + cos θ 0 2 + 2 cos θ 0 2 = 4 T 0 ( 1 + cos θ 0 2 ) 2 . {\displaystyle T_{2}={\frac {4T_{0}}{1+\cos {\frac {\theta _{0}}{2}}+2{\sqrt {\cos {\frac {\theta _{0}}{2}}}}}}={\frac {4T_{0}}{\left(1+{\sqrt {\cos {\frac {\theta _{0}}{2}}}}\right)^{2}}}.}
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 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:
‘Not so large-angle’ formulae, i.e. those yielding good estimates for amplitudes below π / 2 {\displaystyle \pi /2} rad (a natural limit for a bob on the end of a flexible string), though the deviation with respect to the exact period increases monotonically with amplitude, being unsuitable for amplitudes near to π {\displaystyle \pi } rad. One of the simplest formulae found in literature is the following one by Lima (2006): T ≈ − T 0 ln a 1 − a {\textstyle T\approx -\,T_{0}\,{\frac {\ln {a}}{1-a}}} , where a ≡ cos ( θ 0 / 2 ) {\displaystyle a\equiv \cos {(\theta _{0}/2)}} . ‘Very large-angle’ formulae, i.e. those which approximate the exact period asymptotically for amplitudes near to π {\displaystyle \pi } rad, with an error that increases monotonically for smaller amplitudes (i.e., unsuitable for small amplitudes). One of the better such formulae is that by Cromer, namely: T ≈ 2 π T 0 ln ( 4 / a ) {\textstyle T\approx {\frac {2}{\pi }}\,T_{0}\,\ln {(4/a)}} . Of course, the increase of T {\displaystyle T} with amplitude is more apparent when π / 2 < θ 0 < π {\displaystyle \pi /2<\theta _{0}<\pi } , 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:
T ≈ r a 2 T Lima + k 2 T Cromer r a 2 + k 2 , {\displaystyle T\approx {\frac {r\,a^{2}\,T_{\text{Lima}}+k^{2}\,T_{\text{Cromer}}}{r\,a^{2}+k^{2}}},}
where r = 7.17 {\displaystyle r=7.17} , which presents a maximum error of only 0.6% (at θ 0 = 95 ∘