Pendulum Calculator
Period T (seconds)
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Frequency (Hz)
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Angular Frequency (rad/s)
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How a Pendulum Works
A simple pendulum is a weight (called a bob) suspended from a fixed point by a string or rod that swings back and forth under the influence of gravity. The period of a simple pendulum — the time for one complete oscillation (swing out and back) — depends only on its length and the local gravitational acceleration. According to Encyclopaedia Britannica, the isochronous property of pendulums (equal time for each swing regardless of amplitude) was first observed by Galileo Galilei around 1602, reportedly while watching a swinging chandelier in the Pisa Cathedral.
The relationship T = 2 pi times the square root of (L/g) is one of the most elegant equations in classical mechanics. It reveals that the period is independent of the bob's mass and, for small angles, independent of the amplitude. This was a revolutionary insight that led directly to the development of the pendulum clock by Christiaan Huygens in 1656, which remained the most accurate timekeeping technology for nearly 300 years. According to the National Institute of Standards and Technology (NIST), the best pendulum clocks of the early 20th century achieved accuracy within 1 second per year.
Beyond timekeeping, pendulums serve important roles in science and engineering. Geophysicists use gravimeters (precision pendulum instruments) to measure local variations in gravitational acceleration, which reveal subsurface geological structures. Seismometers use pendulum principles to detect earthquake waves. The Foucault pendulum, first demonstrated publicly in 1851 at the Pantheon in Paris, provides a direct visual proof of Earth's rotation. For related oscillation calculations, see our spring constant calculator and work and energy calculator.
The Pendulum Period Formula
The formula for the period of a simple pendulum is: T = 2 pi x sqrt(L / g), where T is the period in seconds, L is the pendulum length in meters (measured from the pivot point to the center of mass of the bob), and g is the gravitational acceleration in m/s^2 (9.81 m/s^2 at Earth's surface). The frequency is f = 1/T (oscillations per second, in Hz), and the angular frequency is omega = 2 pi / T (in radians per second).
Worked Example: A pendulum with L = 1.5 meters on Earth (g = 9.81 m/s^2). Period T = 2 x 3.14159 x sqrt(1.5 / 9.81) = 6.2832 x sqrt(0.15291) = 6.2832 x 0.39103 = 2.457 seconds. Frequency f = 1 / 2.457 = 0.407 Hz. Angular frequency omega = 2 x 3.14159 / 2.457 = 2.558 rad/s. If this same pendulum were on the Moon (g = 1.62 m/s^2): T = 6.2832 x sqrt(1.5/1.62) = 6.2832 x 0.9623 = 6.045 seconds — about 2.5 times slower.
Key Terms You Should Know
- Period (T): The time for one complete oscillation — one full swing from starting position to the opposite side and back. Measured in seconds. Not the same as a half-swing, which is T/2.
- Frequency (f): The number of complete oscillations per second, measured in hertz (Hz). Frequency is the reciprocal of the period: f = 1/T.
- Angular Frequency (omega): The rate of change of the phase angle, measured in radians per second. Related to frequency by omega = 2 pi f. Used extensively in physics and engineering oscillation analysis.
- Small-Angle Approximation: The assumption that sin(theta) is approximately equal to theta (in radians) for small angles, typically below 15 degrees. This simplification makes the pendulum equation solvable in closed form and is accurate to within 1% at 15 degrees.
- Simple Pendulum: An idealized model consisting of a point mass suspended by a massless, inextensible string from a frictionless pivot. Real pendulums approximate this model when the bob is heavy relative to the string and the string length is much greater than the bob's diameter.
Pendulum Period on Different Planets
The period of a pendulum changes with gravitational acceleration, making pendulums useful for measuring local gravity. The table below shows how a 1-meter pendulum's period varies across different celestial bodies, using gravitational values from NASA.
| Location | g (m/s^2) | Period (s) | Frequency (Hz) | Relative to Earth |
|---|---|---|---|---|
| Earth | 9.81 | 2.006 | 0.499 | 1.00x |
| Moon | 1.62 | 4.934 | 0.203 | 2.46x slower |
| Mars | 3.72 | 3.259 | 0.307 | 1.62x slower |
| Jupiter | 24.79 | 1.263 | 0.792 | 1.59x faster |
| Venus | 8.87 | 2.112 | 0.473 | 1.05x slower |
| ISS (microgravity) | ~0 | N/A | N/A | Does not oscillate |
Practical Examples
Example 1 — Grandfather Clock Design: A clock requires a 2-second period (1 second per half-swing). Rearranging the formula: L = g x (T / (2 pi))^2 = 9.81 x (2 / 6.2832)^2 = 9.81 x 0.10132 = 0.994 meters. This confirms that grandfather clocks use a pendulum of approximately 1 meter, or 39.1 inches — a fact of physics that has dictated the size of these clocks since the 17th century.
Example 2 — Measuring Local Gravity: A student times 50 complete oscillations of a 0.75-meter pendulum and measures a total time of 87.1 seconds. Period T = 87.1 / 50 = 1.742 seconds. Solving for g: g = 4 pi^2 x L / T^2 = 39.478 x 0.75 / 3.035 = 9.76 m/s^2. This is close to the standard value of 9.81 m/s^2, with the small difference attributable to local geological variations and measurement uncertainty. See our acceleration calculator for more physics problems.
Example 3 — Foucault Pendulum at 45 Degrees Latitude: The Foucault pendulum's apparent rotation rate is 360 x sin(latitude) degrees per day. At 45 degrees N: 360 x sin(45) = 360 x 0.7071 = 254.6 degrees per day. A full apparent rotation takes 360 / 254.6 x 24 = 33.9 hours. At the North Pole (90 degrees latitude), the rotation completes in exactly 24 hours. At the equator (0 degrees), there is no apparent rotation at all.
Tips and Strategies
- Keep swing angles small for accurate results. The simple period formula is accurate to within 1% for angles up to about 15 degrees (0.26 radians). Beyond this, the actual period increases progressively — at 30 degrees the error is about 2%, and at 90 degrees the error exceeds 18%.
- Measure length to the center of the bob. The effective pendulum length is measured from the pivot point to the center of mass of the bob, not to the bottom or top of the weight. For a spherical bob, measure to its center.
- Time multiple oscillations for accuracy. Rather than timing a single swing, time 20-50 complete oscillations and divide by the count. This reduces the relative timing error significantly. Use a stopwatch or phone timer with at least 0.01-second precision.
- Account for the string or rod mass in real pendulums. A physical (compound) pendulum where the string or rod has non-negligible mass behaves slightly differently. The effective length is shorter than the total length, and the period formula includes a moment of inertia term.
- Temperature affects pendulum length. Metal pendulum rods expand and contract with temperature changes. A steel rod pendulum gains approximately 0.5 seconds per day for each 10 degrees C increase in temperature. Precision pendulum clocks use compensating mechanisms (bimetallic strips, invar alloys) to counteract this effect.
Frequently Asked Questions
What is the pendulum period formula?
The period of a simple pendulum is calculated using T = 2 times pi times the square root of (L/g), where T is the period in seconds, L is the pendulum length in meters, and g is the gravitational acceleration (9.81 m/s squared on Earth). This formula assumes small oscillation angles (less than about 15 degrees) where the approximation sin(theta) is approximately equal to theta holds true. A 1-meter pendulum on Earth has a period of approximately 2.006 seconds, which is why grandfather clocks traditionally use meter-long pendulums for a convenient 2-second period.
Does mass affect a pendulum's period?
No, for an ideal simple pendulum undergoing small oscillations, the period depends only on the pendulum length and the local gravitational acceleration, not on the mass of the bob. This counterintuitive result was first observed by Galileo Galilei around 1602 and is a direct consequence of the fact that gravitational force and inertia both scale linearly with mass, causing mass to cancel out of the equation of motion. In practice, a heavier bob may swing slightly longer than a lighter one because it is less affected by air resistance, but the difference is negligible for typical pendulum experiments.
Why do longer pendulums swing slower?
The period of a pendulum is proportional to the square root of its length. A pendulum 4 times longer has a period 2 times longer because the square root of 4 equals 2. Physically, a longer pendulum travels a greater arc distance during each swing while experiencing the same gravitational restoring force per unit displacement (when measured as an angle). The longer path takes more time to traverse, resulting in a longer period. This square root relationship means that doubling the length increases the period by a factor of approximately 1.414 (the square root of 2), not by a factor of 2.
What is a Foucault pendulum?
A Foucault pendulum is a long, heavy pendulum designed to demonstrate the rotation of the Earth. First demonstrated publicly by French physicist Leon Foucault in 1851 at the Pantheon in Paris using a 67-meter (220-foot) wire and a 28-kg (62-lb) iron bob, it swings back and forth while the Earth rotates beneath it. At the North or South Pole, the pendulum's swing plane appears to complete a full 360-degree rotation in 24 hours. At other latitudes, the rotation rate equals 360 degrees times the sine of the latitude per day. Many science museums worldwide display Foucault pendulums as a visual proof of Earth's rotation.
What happens to a pendulum at large angles?
At swing angles greater than about 15 degrees, the small-angle approximation breaks down and the simple formula T = 2 times pi times the square root of (L/g) becomes increasingly inaccurate. The true period of a pendulum at large angles is longer than the small-angle formula predicts. For example, at 45 degrees, the actual period is approximately 4% longer, and at 90 degrees it is about 18% longer. The exact solution involves an elliptic integral that cannot be expressed in simple closed form. For most practical applications and physics problems, keeping the angle below 15 degrees ensures the simple formula is accurate to within 1%.
How are pendulums used in clocks?
Pendulum clocks use the consistent period of a pendulum to regulate timekeeping. The pendulum swings at a fixed rate determined by its length, and an escapement mechanism counts the swings and advances the clock's gears by one tick per swing. A 1-meter pendulum with a 2-second period (1 second each way) produces 30 complete oscillations per minute. Pendulum clocks dominated timekeeping from their invention by Christiaan Huygens in 1656 until quartz clocks replaced them in the 1930s-1940s. The most accurate pendulum clocks achieved precision within 1 second per year by compensating for temperature changes that affect pendulum length.