Find a pendulum's period, frequency, angular frequency, or required length, and compare the small-angle shortcut with a more exact large-angle result.
Advanced options
How to use our Simple Pendulum Calculator
- Choose an option in Calculate for the quantity you want to find.
- Enter Pendulum length (m) if you are solving for period, frequency, or angular frequency.
- Enter Period (s) if you are solving for Required pendulum length, or if you want the matching frequency and angular frequency from a known period.
- Check Gravitational acceleration (m/s^2), or use Gravity preset in Advanced options to fill a common value quickly.
- Optional: enter Initial angle (deg) to compare the basic small-angle result with the more exact large-angle period.
- Optional: choose Display format and Decimal places if you want fixed decimals or scientific notation.
- Click Calculate to show the result cards.
- Sanity-check the output: a longer pendulum should give a longer Period, small-angle formula, and a larger starting angle should make Period, exact large-angle model slightly longer than the small-angle result.
Definitions
Calculate: The main quantity you want the tool to solve for, such as period, frequency, angular frequency, or length.
Pendulum length (m): The distance from the pivot point to the bob's center of mass.
Gravitational acceleration (m/s^2): How strongly gravity pulls at your location. Standard Earth gravity is 9.80665 m/s^2 [1].
Period (s): The time for one full back-and-forth swing.
Frequency, small-angle formula: The number of full swings each second. It equals 1 divided by the period.
Angular frequency, small-angle formula: A physics way to describe how fast the motion repeats, measured in radians per second.
Initial angle (deg): The starting angle away from straight down. Larger angles make the simple shortcut less accurate.
Period, exact large-angle model: A more exact period that accounts for large starting angles using a complete elliptic integral [3].
Large-angle difference: The percent difference between the exact large-angle period and the small-angle period.
Model note: A short message that tells you whether the small-angle approximation is probably good enough for the angle you entered.
Common mistakes and quick fixes
Mistake: Entering 0 or a negative number for Pendulum length (m) when solving for period, frequency, or angular frequency.
Fix: Use a positive length measured from the pivot to the bob's center of mass.
Mistake: Leaving Period (s) at 0 or typing a negative value when solving for Required pendulum length .
Fix: Enter a positive time for one full back-and-forth swing.
Mistake: Typing a nonpositive value for Gravitational acceleration (m/s^2) .
Fix: Enter a value greater than 0, or use Gravity preset to fill a standard value.
Mistake: Entering an Initial angle (deg) near or above 180.
Fix: Keep the angle in the range from 0 up to less than 180 degrees, and preferably below about 170 degrees for stable classroom use.
Mistake: Reading Large-angle difference as an error in the calculator.
Fix: Treat it as the percent by which the exact large-angle period is longer than the small-angle estimate.
Mistake: Forgetting that Gravity preset can change Gravitational acceleration (m/s^2) and therefore all time-based outputs.
Fix: Recheck the gravity value before comparing Period, small-angle formula across Earth, Moon, Mars, or Jupiter.
Limitations & Key Assumptions / Boundary Conditions
- This calculator uses the ideal simple-pendulum model, so it assumes a point-like bob, a massless string or rod, no air resistance, and no friction at the pivot.
- The small-angle outputs assume the starting angle is small. They are usually most reliable for angles around 10 to 15 degrees or less.
- The exact large-angle comparison changes the period estimate, but it still does not include real-world losses such as damping or a moving support.
- Pendulum length (m) must be greater than 0, Gravitational acceleration (m/s^2) must be greater than 0, and Period (s) must be greater than 0 when used.
- Initial angle (deg) must stay below 180 degrees for this model. Very large angles near 180 degrees can make the exact calculation numerically unstable and physically less classroom-friendly.
- If you use a different local gravity than standard Earth gravity, real measurements can still vary slightly because of location, altitude, and setup details.
Methodology
Core equations
The calculator starts with the standard small-angle simple-pendulum model.
T = 2*pi*sqrt(L/g)
Here, T is period, L is pendulum length, and g is gravitational acceleration.
f = 1/T
f is frequency in hertz, meaning cycles per second.
omega = sqrt(g/L) = 2*pi/T
omega is angular frequency in radians per second.
L = g*(T/(2*pi))^2
That last equation is used when solving for required pendulum length from a target period.
Large-angle comparison
If you enter Initial angle (deg), the calculator also computes a more exact period for a larger starting swing. The angle is first converted from degrees to radians.
theta0_rad = theta0_deg*pi/180
T_exact = 4*sqrt(L/g)*K(sin(theta0_rad/2)^2)
Here, K(m) is the complete elliptic integral of the first kind [2]. This exact-model period gets longer as the starting angle gets larger [2].
percent_diff = ((T_exact - T)/T)*100
A positive percent means the exact large-angle period is longer than the small-angle estimate.
Worked mini-example
Suppose L = 1 m, g = 9.80665 m/s^2, and Initial angle = 5 deg.
T = 2*pi*sqrt(1/9.80665) = 2.0064 s
f = 1/2.0064 = 0.4984 Hz
omega = 2*pi/2.0064 = 3.1316 rad/s
With the exact large-angle model at 5 degrees, the period is about 2.0074 s, so the difference is about 0.0476%. That is tiny, which tells you the small-angle shortcut is very good here.
How the calculator decides what to show
If you solve for period, frequency, or angular frequency, the tool uses Pendulum length (m) and Gravitational acceleration (m/s^2). If you solve for length, it uses Period (s) and gravity. The large-angle comparison is only shown when the needed length-based period can be evaluated.
Assumptions behind the math
The small-angle formula assumes ideal pendulum motion and ignores air drag, friction, and large bob size. Standard Earth gravity is taken as 9.80665 m/s^2 when that preset is used [1]. Real lab results can differ if the amplitude is large, the length is measured incorrectly, or the pendulum is not close to the ideal simple-pendulum setup.