Simple Pendulum Calculator finds the period and frequency of a pendulum from its length, or solves for length or gravity from timed swings, with the true period, speed and tension.
T = 2π √(L ÷ g). The mass of the bob does not change the period.
Angle, Mass, Bob, Units
Angle sets the true period. Mass sets the forces. Bob diameter adds its radius to the string.
Step-by-Step Calculation
- T = 2π × √(L ÷ g) = 2π × √(2 ÷ 9.80665) = 2.83749 s = 2.837 s
- Frequency: f = 1 ÷ T = 1 ÷ 2.83749 = 0.352424 Hz
- Swing angle: θ = 10° = 0.174533 rad
- Exact period factor: 1 ÷ AGM(1, cos(θ ÷ 2)) = 1 ÷ AGM(1, 0.996195) = 1.00191
- True period: 2.83749 × 1.00191 = 2.8429 s
- Height gained: h = L × (1 − cos θ) = 2 × (1 − 0.984808) = 0.0303845 m
- Top speed: v = √(2 × g × h) = √(2 × 9.80665 × 0.0303845) = 0.771972 m/s
- Bottom tension: F = m × g × (3 − 2 cos θ) = 0.5 × 9.80665 × 1.03038 = 5.05231 N
Period by Swing Angle
Same Pendulum on Other Worlds
Length for a Chosen Period
Pendulum Diagram
Released at 10°, the bob rises 30.38 mm at each end. The swing runs at its true period of 2.843 s.
Simple Pendulum Calculator and Period Formula
The Simple Pendulum Calculator finds the period and frequency of a pendulum from its length and the local gravity. It also works backward, from a timed swing to the length or to gravity itself.
A simple pendulum is a small heavy bob hanging from a light string that swings freely from a fixed point. It is called simple because it is an idealized model. The bob is treated as a single point, the string has no mass and does not stretch, and there is no friction.
The simple pendulum period formula links the time of one full swing to the length L and gravity g. One full swing means out and back to the starting point.
$$T = 2\pi \sqrt{\frac{L}{g}}$$
To calculate a simple pendulum, divide the length by gravity, take the square root, and multiply by 2π. The Simple Pendulum Calculator opens with a 2 m pendulum under standard gravity of 9.80665 m/s². Its period is 2.837 s, its frequency is 0.3524 Hz, and it makes 21.15 full swings a minute.
The mass of the bob is not in the formula, so a heavy bob and a light bob on the same string keep the same time. The square root matters too. A pendulum four times as long has twice the period, not four times.
| Length | Period on Earth | Frequency |
|---|---|---|
| 0.25 m | 1.003 s | 0.9968 Hz |
| 0.5 m | 1.419 s | 0.7048 Hz |
| 1 m | 2.006 s | 0.4984 Hz |
| 2 m | 2.837 s | 0.3524 Hz |
| 4 m | 4.013 s | 0.2492 Hz |
So the period of a 1 m pendulum is 2.006 s, and the period of a pendulum whose length is 4 m is 4.013 s. The OpenStax section on pendulums derives the formula and gives its limit, a swing of less than about 15°.
Pendulum Frequency Formula
The frequency of a simple pendulum is the number of full swings it makes each second. It is one divided by the period, so the pendulum frequency formula is the period formula turned upside down.
$$f = \frac{1}{T} = \frac{1}{2\pi} \sqrt{\frac{g}{L}}$$
One oscillation takes exactly one period, so the two words count the same thing. A frequency of 1 Hz means one full cycle every second. By the same rule 300 Hz means 300 cycles a second, 2.5 GHz is 2.5 billion cycles a second, and 50 Hz is 3,000 cycles a minute, or 3,000 RPM.
The first card of the Simple Pendulum Calculator also gives the angular frequency, ω = 2π f, which is 2.214 rad/s for the default pendulum. The Frequency Calculator and the Angular Frequency Calculator convert these into other units.
Length of a Pendulum From Its Period
To calculate the length of a simple pendulum, turn the period formula around. Choose “Length from the period” and the Simple Pendulum Calculator solves for L.
$$L = g \left(\frac{T}{2\pi}\right)^2$$
The period is entered the way it is measured, as a total time and a number of swings. The default is 56.8 s for 20 swings, a period of 2.84 s, which needs a length of 2.004 m. Set the number of swings to 1 to type a period directly, or switch the unit to hertz to enter a frequency. With a Bob Diameter entered, the result also says what length to cut the string to.
Two common textbook cases work the same way. A pendulum that takes 42 seconds for 20 oscillations has a time period of 2.1 s and is 1.095 m long. A pendulum that makes 10 oscillations in 5 seconds has a period of 0.5 s, a frequency of 2 Hz and a length of 62.1 mm.
The “Length for a Chosen Period” panel lists the length for round periods at the gravity in use. These are its rows for standard gravity.
| Period | Length | Frequency |
|---|---|---|
| 0.5 s | 62.1 mm | 2 Hz |
| 1 s | 24.84 cm | 1 Hz |
| 2 s | 99.36 cm | 0.5 Hz |
| 3 s | 2.236 m | 0.3333 Hz |
| 5 s | 6.21 m | 0.2 Hz |
| 10 s | 24.84 m | 0.1 Hz |
A pendulum with a period of 2 seconds is called a seconds pendulum, because each swing one way takes one second. It is 99.36 cm long, just under a meter. A pendulum with a period of 1 second is much shorter, at 24.84 cm.
The pendulum with the longest time period is the longest one, or the one in the weakest gravity. Nothing else changes the period of a small swing.
Finding g With a Simple Pendulum
The simple pendulum experiment measures gravity with a string, a bob and a stopwatch. Choose “Gravity from the period” and enter the length and the timing.
$$g = \frac{4\pi^2 L}{T^2}$$
With the default 2 m length and 56.8 s for 20 swings, the Simple Pendulum Calculator gives 9.789 m/s². That is 0.18% below standard gravity. The second card then corrects for the swing angle. At 10° each swing runs slightly long, and the corrected value is 9.827 m/s².
Timing many swings gives a steadier answer than timing one. A reaction error of 0.2 s on the stopwatch is 7% of a single 2.84 s swing but only 0.35% of 20 swings.
A better lab method times the pendulum at several lengths and plots T² against L. The points fall on a straight line whose slope is 4π² ÷ g. The “Gravity from several trials” option does that fit for two to four trials. Its sample data is four lengths, each timed for 20 swings.
| Length | Time for 20 swings | Period | T² |
|---|---|---|---|
| 40 cm | 25.4 s | 1.27 s | 1.613 s² |
| 60 cm | 31.1 s | 1.555 s | 2.418 s² |
| 80 cm | 35.9 s | 1.795 s | 3.222 s² |
| 100 cm | 40.1 s | 2.005 s | 4.02 s² |
The fitted slope is 4.013 s² per meter, which gives g = 9.838 m/s². The first card also shows a length offset of +2.333 mm. A large offset means every length was measured short by that amount, which is what happens when the radius of the bob is left out.
Gravity is the only thing that changes when the same pendulum is taken somewhere else. The “Same Pendulum on Other Worlds” panel uses the surface acceleration from the NASA planetary fact sheets. The 2 m pendulum takes 6.981 s on the Moon, 4.626 s on Mars and 1.848 s on Jupiter.
Swing Angle and the True Period
The simple formula assumes a very small swing. A wider swing takes a little longer, and the second card shows the exact figure. For the default 2 m pendulum released at 10°, the true period is 2.843 s, which is 0.1907% longer than the formula gives.
A good estimate for moderate angles uses the release angle θ in radians. The Simple Pendulum Calculator does not stop at this series. It uses the exact result, so it stays correct at any angle below 180°.
$$T \approx 2\pi \sqrt{\frac{L}{g}} \left(1 + \frac{\theta^2}{16} + \frac{11\,\theta^4}{3072}\right)$$
The “Period by Swing Angle” panel lists the true period at fixed angles. These rows are for the 2 m pendulum, where the simple formula gives 2.837 s.
| Swing angle | True period | Longer by |
|---|---|---|
| 5° | 2.839 s | 0.04762% |
| 10° | 2.843 s | 0.1907% |
| 15° | 2.85 s | 0.4301% |
| 20° | 2.859 s | 0.7669% |
| 30° | 2.887 s | 1.741% |
| 45° | 2.951 s | 3.997% |
| 60° | 3.045 s | 7.318% |
| 90° | 3.349 s | 18.03% |
The error looks small, but a clock adds it up. A pendulum clock set by the simple formula and swinging at 10° would lose 164.5 s a day. Past 90° a string goes slack at the ends of the swing, so the Simple Pendulum Calculator notes that such a swing needs a rigid rod.
A simple pendulum is in simple harmonic motion only for small swings, where the pull back toward the center is proportional to the distance from it. For the position, velocity and acceleration at any moment of that motion, use the Simple Harmonic Motion Calculator.
Pendulum Force, Speed and Energy
The swing angle and bob mass under “Swing Angle, Mass, Units” turn the Simple Pendulum Calculator into a pendulum force calculator as well. The mass does not change the period, but it sets every force on the string.
$$F = m g (3 – 2 \cos\theta)$$
That is the tension at the bottom of the swing for a bob released from rest at angle θ. With the default 0.5 kg bob at 10°, the bottom tension is 5.052 N, a little more than the 4.903 N weight. At each end of the swing the tension drops to m g cos θ, which is 4.829 N.
The third card gives the motion. The bob rises 30.38 mm at each end and passes the bottom at 0.772 m/s, from v = √(2 g h). The swing is 69.46 cm wide from end to end, and it carries 149 mJ of energy.
For the energy at any point of the swing, the Kinetic Energy of a Pendulum Calculator splits it into kinetic and potential parts. The extra pull at the bottom is the force that bends the path into a circle, which the Centripetal Force Calculator works out on its own.
Simple Pendulum Questions From School Science
The simple pendulum is taught from middle school onward, first as a way to measure time and later as a way to measure gravity. These answers cover the points that come up in class.
What are the four laws of a simple pendulum?
The law of length says the period is proportional to the square root of the length. The law of gravity says it is inversely proportional to the square root of g. The law of mass says it does not depend on the mass or material of the bob. The law of isochronism says it does not depend on the size of the swing, as long as the swing is small.
How can a Class 7 student make a simple pendulum?
Tie a small metal ball or a stone to about one meter of strong thread. Fix the other end to a stand or a hook so the bob hangs freely. Pull the bob a little to one side and let go without pushing. Start a stopwatch as it passes the middle, count 20 full oscillations, and stop the watch.
What is the time period formula for Class 7 and Class 8?
At that level the time period of a simple pendulum is the total time divided by the number of oscillations. If 20 oscillations take 40 seconds, the time period is 2 seconds. The Simple Pendulum Calculator uses the same division in its “Time for N Swings” and “Number of Swings” fields.
What is the effective length, and how is it measured?
The effective length is the distance from the point of suspension to the center of the bob. It is the length of the thread plus the radius of the bob. Measure the thread with a meter scale and the diameter of the bob with vernier calipers, then add half the diameter to the thread length. In the Simple Pendulum Calculator, type the thread length and fill the optional Bob Diameter field, and the radius is added for you.
What is an example of a simple pendulum?
A metal ball on a thin thread in a school lab is the closest real example. The pendulum of a wall clock and a child on a playground swing behave in a similar way. Their mass is spread out along the arm, though, so they are only roughly simple pendulums.
What is a physical pendulum?
A physical pendulum is any rigid body that swings about a pivot, such as a rod, a hoop or a clock pendulum with a heavy arm. Its period depends on its moment of inertia and on the distance from the pivot to its center of mass. The Physical Pendulum Calculator handles that case.
Does the calculator work on a phone?
Yes. It runs in any phone browser like a simple pendulum calculator app, and the Copy Results, Share and Print / PDF buttons save a run. The Pendulum Diagram panel also animates the bob, swinging at its true period. More tools for swinging and turning bodies are on the Rotational Motion Calculators hub.