# Pendulum
A pendulum is a weight, called a bob, suspended from a pivot so that it can swing back and forth under gravity. For small displacements a pendulum is a simple harmonic oscillator, and its regular period made it the regulator of mechanical clocks from the 17th century until electronic timekeeping, as well as a standard instrument for measuring the local acceleration due to gravity.[1](https://openstax.org/books/college-physics-2e/pages/16-4-the-simple-pendulum)
| Key fact | Detail |
|---|---|
| Simple pendulum | A point mass (bob) on a string of length L with negligible mass; only gravity and string tension act on the bob[2](https://openstax.org/books/university-physics-volume-1/pages/15-4-pendulums) |
| Restoring force | Net force tangent to the arc equals \( -mg\sin\theta \); for angles under about 15° this is proportional to displacement, giving simple harmonic motion[1](https://openstax.org/books/college-physics-2e/pages/16-4-the-simple-pendulum) |
| Period (small amplitude) | \( T = 2\pi\sqrt{L/g} \), independent of the bob's mass and nearly independent of amplitude[2](https://openstax.org/books/university-physics-volume-1/pages/15-4-pendulums) |
| Physical pendulum | \( T = 2\pi\sqrt{I/(mgL)} \), where I is the moment of inertia about the pivot and L the distance from pivot to center of mass[2](https://openstax.org/books/university-physics-volume-1/pages/15-4-pendulums) |
| First pendulum clock | Built for Christiaan Huygens in 1656; daily error improved from about 15 minutes (foliot clocks) to seconds[3](https://www.nist.gov/si-redefinition/second/second-past) • [4](https://museum.seiko.co.jp/en/knowledge/MechanicalTimepieces03/) |
| Large amplitudes | The exact period involves elliptic integrals; the true period is always longer than the small-angle value[5](https://mypage.cuhk.edu.cn/academics/wkshum/sage/pendulum/pendulum.html) • [6](https://galileo.phys.virginia.edu/classes/152.mf1i.spring02/Pendulum.htm) |
| Gravity measurement | A pendulum of precisely known length can determine g; a 75.000 cm pendulum with period 1.7357 s gives g = 9.8281 m/s²[2](https://openstax.org/books/university-physics-volume-1/pages/15-4-pendulums) |
## The simple pendulum and its period
The simple pendulum is an idealization: a small-diameter bob of mass m on a light, unstretchable string of length L. Gravity pulls the bob downward with weight mg, which has a component \( mg\cos\theta \) along the string, opposed by tension, and a component \( mg\sin\theta \) tangent to the arc.[2](https://openstax.org/books/university-physics-volume-1/pages/15-4-pendulums) This tangent component points back toward the equilibrium position at \( \theta = 0 \), so it is a restoring force.[1](https://openstax.org/books/college-physics-2e/pages/16-4-the-simple-pendulum)
For angles below about 15°, \( \sin\theta \) and \( \theta \) (in radians) differ by less than 1%, so the restoring force is approximately proportional to the arc displacement s. The motion is then simple harmonic, with angular frequency \( \omega = \sqrt{g/L} \) and period[1](https://openstax.org/books/college-physics-2e/pages/16-4-the-simple-pendulum) • [2](https://openstax.org/books/university-physics-volume-1/pages/15-4-pendulums)
\[ T = 2\pi\sqrt{\frac{L}{g}}. \]
**What the formula says.** Only the length and the local acceleration due to gravity enter the period; the bob's mass and, to a good approximation, the amplitude do not. This mass independence follows because the inertial mass in \( T = 2\pi\sqrt{m/k} \) cancels the gravitational mass in the effective force constant \( k = mg/L \).[1](https://openstax.org/books/college-physics-2e/pages/16-4-the-simple-pendulum) For the approximation to be better than the precision of a careful length and period measurement, the maximum displacement should be kept below about 0.5°.[2](https://openstax.org/books/university-physics-volume-1/pages/15-4-pendulums)
## Large amplitudes and the nonlinear pendulum
Beyond the small-angle regime the equation of motion is nonlinear,
\[ \frac{d^2\theta}{dt^2} + \frac{g}{L}\sin\theta = 0, \]
and the period depends on amplitude. The period lengthens as amplitude grows because the restoring force \( -mg\sin\theta \) increases more slowly than \( -mg\theta \).[6](https://galileo.phys.virginia.edu/classes/152.mf1i.spring02/Pendulum.htm) The effect is measurable at ordinary angles: a clock regulated by a pendulum swinging at 10° would gain about four seconds per hour, and a one-radian swing lengthens the period by a little under 10%, two radians by about 35%, and three radians almost threefold.[6](https://galileo.phys.virginia.edu/classes/152.mf1i.spring02/Pendulum.htm)
The exact solution is written with the Jacobi elliptic function sn and the incomplete elliptic integral of the first kind, with the modulus \( k = \sin(\phi_{\max}/2) \) set by the release angle; there is no elementary closed form.[5](https://mypage.cuhk.edu.cn/academics/wkshum/sage/pendulum/pendulum.html) The period can also be expanded as a series in the amplitude, and the true period is always longer than the small-angle value \( T_0 = 2\pi\sqrt{L/g} \).[7](https://sites.science.oregonstate.edu/~gibsonn/Teaching/MTH323-010W19/Supplements/nonlin-pendulum.pdf)
## Physical and torsional pendulums
Any rigid body swinging about a horizontal axis is a physical pendulum. Taking torques about the pivot and applying the small-angle approximation gives \( \omega = \sqrt{mgL/I} \) and[2](https://openstax.org/books/university-physics-volume-1/pages/15-4-pendulums)
\[ T = 2\pi\sqrt{\frac{I}{mgL}}, \]
where I is the moment of inertia about the pivot and L the distance from the pivot to the center of mass. For a simple pendulum \( I = mL^2 \) and this reduces to the simple-pendulum result. Physical pendulums serve as tuned-mass dampers: several companies place large pendulums atop skyscrapers, where they swing opposite the building's sway and reduce it.[2](https://openstax.org/books/university-physics-volume-1/pages/15-4-pendulums)
A torsional pendulum is a rigid body suspended by a wire; the restoring torque \( \tau = -\kappa\theta \) comes from the twisting of the wire, with torsion constant \( \kappa \), and the period is \( T = 2\pi\sqrt{I/\kappa} \).[2](https://openstax.org/books/university-physics-volume-1/pages/15-4-pendulums)
## Pendulums as clocks
Mechanical clocks from around 1300 were regulated by a foliot balance, a rotating bar whose speed was set by sliding weights; such clocks were accurate to only about 15 minutes per day.[4](https://museum.seiko.co.jp/en/knowledge/MechanicalTimepieces03/) • [3](https://www.nist.gov/si-redefinition/second/second-past) [Galileo Galilei](https://www.edgepedia.org/galileo-galilei) discovered the isochronism of the pendulum in 1583 and conceived a pendulum clock in 1637 but never completed one. In 1656 [Christiaan Huygens](https://www.edgepedia.org/christiaan-huygens) (1629–1695) used a pendulum in a weight-driven clock with a crown wheel escapement, inventing the first pendulum clock; the Dutch clockmaker Salomon Coster received a patent from the States General on June 16, 1657, and made the first examples.[4](https://museum.seiko.co.jp/en/knowledge/MechanicalTimepieces03/) • [8](https://www.metmuseum.org/art/collection/search/205602?showOnly=openAccess) Huygens published the invention in *Horologium* (1658) and his full theory of pendulum motion in *Horologium Oscillatorium* (1673).[9](https://www.metmuseum.org/de/essays/european-clocks-in-the-seventeenth-and-eighteenth-century)
**Escapements.** The crown wheel escapement forced amplitudes above 30°, destroying isochronism. [Robert Hooke](https://www.edgepedia.org/robert-hooke)'s recoil anchor escapement reduced the swing to two to five degrees, and William Clement improved it around 1671, enabling a long pendulum of over one meter. Clement's Royal Pendulum Clock, a longcase clock with a two-second cycle (0.5 Hz) and accuracy measured in seconds, was installed at the Royal Greenwich Observatory; the anchor escapement then displaced the crown wheel, and the second hand became meaningful.[4](https://museum.seiko.co.jp/en/knowledge/MechanicalTimepieces03/) The technology spread quickly: the English maker Ahasuerus Fromanteel sent his son John to Coster's workshop from September 1657 to May 1658, and advertised pendulum clocks in London in 1658.[8](https://www.metmuseum.org/art/collection/search/205602?showOnly=openAccess)
**Limits at sea and in standards.** Pendulum clocks cannot tolerate a ship's motion, so the longitude problem was solved instead by [John Harrison](https://www.edgepedia.org/john-harrison)'s marine chronometer, which kept time to within one-fifth of a second per day after 31 years of work funded under the [Longitude Act](https://www.edgepedia.org/longitude-act).[3](https://www.nist.gov/si-redefinition/second/second-past) The Royal Society of London proposed defining a length standard as the pendulum length that takes one second per half-swing, but French scientists found that the required length depends on altitude, because gravity affects the swing rate.[3](https://www.nist.gov/si-redefinition/second/second-past) Pendulum clocks remained the most advanced timekeepers into the 20th century; when the National Bureau of Standards was established in 1901, the leading clock was the Riefler, whose pendulum was encased in a vacuum chamber to reduce friction.[3](https://www.nist.gov/si-redefinition/second/second-past)
## Measuring gravity
Because the period depends on g, a pendulum of precisely known length measures local gravitational acceleration through \( g = 4\pi^2 L/T^2 \).[2](https://openstax.org/books/university-physics-volume-1/pages/15-4-pendulums) The quantity measured, little g, is approximately 9.8 m/s² at Earth's surface and 1.62 m/s² on the Moon, and it varies with location and altitude; NIST lists the pendulum among the standard techniques for measuring it, alongside free-fall methods and gravimeters.[10](https://www.nist.gov/how-do-you-measure-it/how-do-you-measure-strength-gravity) Reversible pendulums with interchangeable knife edges were used for absolute gravity determinations, such as the 1905 measurement at Potsdam, before falling out of common use in modern determinations.[11](https://nvlpubs.nist.gov/nistpubs/Legacy/SP/nbsspecialpublication420.pdf)
## References
1. [The Simple Pendulum, College Physics 2e, OpenStax](https://openstax.org/books/college-physics-2e/pages/16-4-the-simple-pendulum)
2. [Pendulums, University Physics Volume 1, OpenStax](https://openstax.org/books/university-physics-volume-1/pages/15-4-pendulums)
3. [Second: The Past, NIST](https://www.nist.gov/si-redefinition/second/second-past)
4. [The invention of the pendulum clock, Seiko Museum Ginza](https://museum.seiko.co.jp/en/knowledge/MechanicalTimepieces03/)
5. [Pendulum equation, W. K. Shum, CUHK](https://mypage.cuhk.edu.cn/academics/wkshum/sage/pendulum/pendulum.html)
6. [The Pendulum, University of Virginia](https://galileo.phys.virginia.edu/classes/152.mf1i.spring02/Pendulum.htm)
7. [The Nonlinear Pendulum, Oregon State University](https://sites.science.oregonstate.edu/~gibsonn/Teaching/MTH323-010W19/Supplements/nonlin-pendulum.pdf)
8. [Clockmaker: Ahasuerus I Fromanteel, The Metropolitan Museum of Art](https://www.metmuseum.org/art/collection/search/205602?showOnly=openAccess)
9. [European Clocks in the Seventeenth and Eighteenth Centuries, The Metropolitan Museum of Art](https://www.metmuseum.org/de/essays/european-clocks-in-the-seventeenth-and-eighteenth-century)
10. [How Do You Measure the Strength of Gravity?, NIST](https://www.nist.gov/how-do-you-measure-it/how-do-you-measure-strength-gravity)
11. [The International Bureau of Weights and Measures 1875–1975, NBS Special Publication 420](https://nvlpubs.nist.gov/nistpubs/Legacy/SP/nbsspecialpublication420.pdf)
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*Topic: Encyclopedia › Physical world and mathematics › Physics › Classical physics › Mechanics › Motion, forces, and dynamics › Newtonian dynamics of particles › Newton's laws of motion*
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