The simple pendulum calculator above gives the familiar period from length and gravity, and then does the thing almost no other pendulum page does: it computes the exact period as well, and tells you the percentage error in the approximation at the amplitude you entered. That matters because the standard formula is not a law, it is a small-angle simplification, and its error grows quickly once the swing gets wide.
Arb Digital builds free calculators that show where a model stops being accurate instead of pretending it never does. A pendulum swinging through 15 degrees runs about 0.43 per cent slow against the approximation. At 60 degrees the discrepancy is over 7 per cent, which for a clock would be nearly two hours a day.
What This Simple Pendulum Calculator Does
You give it a length, a value of gravity and a swing amplitude. The headline result is the period from the standard small-angle formula, because that is the number most people come here for. The grid then gives the frequency, the exact period computed without the approximation, the size of the error between them, and the maximum speed of the bob as it passes through the lowest point.
Gravity is both a preset menu and an editable field, because a pendulum is genuinely sensitive to it. Local gravity varies by about half a per cent between the equator and the poles, and this was historically how that variation was measured — a pendulum clock carried to a different latitude visibly changes rate.
Bob mass is an input only because it feeds the energy figures. It has no effect whatsoever on the period, which is one of the more surprising and useful facts about pendulums and is discussed below.
How to Use It
- Measure the length to the centre of mass. This is from the pivot to the centre of the bob, not to its top surface. For a spherical bob it is the pivot-to-surface distance plus the radius.
- Set gravity for your location. Standard gravity is fine for most purposes. If you are matching a measured period to within a fraction of a per cent, use a local value instead.
- Enter the amplitude as the angle from vertical. Not the total swept angle. A pendulum swinging 30 degrees from side to side has an amplitude of 15 degrees.
- Compare the two period figures. The hero is the approximation, the grid holds the exact result. If the error is under about 0.1 per cent the approximation is fine for anything you are doing; above one per cent it is not.
- Read the speed at the bottom if you need energy or tension. That figure comes from energy conservation and does not depend on the approximation at all.
The Formula: How the Pendulum Period Is Calculated
The restoring torque on a pendulum displaced by an angle θ is proportional to sin θ, not to θ. That makes the exact equation of motion non-linear and not solvable in elementary functions. The standard treatment replaces sin θ with θ, which is valid when θ is small in radians, and the equation collapses into simple harmonic motion with period T = 2π√(L ÷ g).
OpenStax University Physics Volume 1, section 15.4 on pendulums, states the boundary explicitly: for angles below about 15 degrees, roughly 0.26 radians, the sine and the angle differ by less than one per cent, and the approximation is used on that basis.
Work the defaults. A one-metre pendulum at standard gravity of 9.80665 m/s² — the exact value fixed by the NIST CODATA entry for standard acceleration of gravity — gives T = 2π√(1 ÷ 9.80665) = 2π × 0.31933 = 2.0064 s. The frequency is 0.4984 Hz, so a one-metre pendulum takes almost exactly one second per swing in each direction.
The exact period can be written using the complete elliptic integral of the first kind, and this tool evaluates it with the arithmetic-geometric mean, which converges in a handful of iterations: Texact = T ÷ AGM(1, cos(θ0 ÷ 2)). At the default 15 degrees that gives 2.0150 s against the approximate 2.0064 s, a real difference of 0.43 per cent.
How Big the Small-Angle Error Actually Gets
The correction grows roughly as the square of the amplitude at first, and then faster. The useful figures to have in mind are these: at 1 degree the error is about 0.002 per cent, at 5 degrees about 0.05 per cent, at 10 degrees about 0.19 per cent, at 15 degrees about 0.43 per cent, at 30 degrees about 1.7 per cent, at 60 degrees about 7.3 per cent, and at 90 degrees about 18 per cent. A pendulum released from horizontal takes nearly a fifth longer per swing than the standard formula predicts.
Whether that matters depends entirely on what you are doing. For a physics experiment measuring gravity to three significant figures, a 10-degree swing already introduces an error larger than your other uncertainties, and dropping to 5 degrees is worth the effort. For a demonstration of the general behaviour, 30 degrees is fine.
For timekeeping the tolerance is brutal. A pendulum clock running 0.43 per cent slow loses about six minutes a day. That is why precision pendulum clocks used very small amplitudes, typically a degree or two, and why their escapements were designed to disturb the swing as little as possible. The approximation was not a convenience for them; keeping the amplitude small enough for it to hold was a design requirement.
Why Mass Does Not Affect the Period
Increasing the mass of the bob increases the restoring force acting on it in exact proportion, and also increases its inertia in exact proportion. The two cancel, and the period is unchanged. It is the same cancellation that makes all objects fall at the same rate in a vacuum, which the free fall calculator covers directly.
This means a heavy pendulum and a light one of the same length keep the same time. In practice a heavier bob is still preferred for clocks, not because it changes the period but because it carries more energy relative to the frictional and air-resistance losses, so its amplitude decays more slowly and the small-angle condition holds for longer between windings.
The same independence does not extend to length, which is the dominant term. Period goes as the square root of length, so quadrupling the length doubles the period. To make a pendulum with a two-second period at standard gravity you need a length of 0.9937 m — which is very nearly one metre, and is no coincidence, since one early proposal for defining the metre was the length of a pendulum beating seconds.
What a Simple Pendulum Is Not
The word simple in the name is a technical restriction, not a description of difficulty. A simple pendulum is a point mass on a massless, inextensible string. Every real pendulum departs from that in at least one way.
A rod with a bob on the end is a compound or physical pendulum, and its period depends on its moment of inertia and the distance from the pivot to its centre of mass, not simply on a length. For a uniform rod pivoted at one end, the period matches a simple pendulum of two-thirds the rod's length. If your pendulum's supporting structure has significant mass, this formula will not match your measurement.
Air resistance and pivot friction do not change the period appreciably in light damping, but they do decay the amplitude. Since the exact period depends on amplitude, a real pendulum's period changes slowly as it dies away — it speeds up toward the small-angle value. That is a genuine effect and it is why clock pendulums need constant amplitude, not merely constant length.
Where This Sits Next to Our Other Oscillation Tools
This page is the gravitational oscillator. The simple harmonic motion calculator is the mass-spring version, and the difference between them is exactly the point of the error discussion here: a spring's restoring force is genuinely proportional to displacement, so its period is exactly amplitude-independent with no approximation involved. A pendulum's is only approximately so.
For the angular quantities describing the swing, the angular velocity calculator converts between radians per second and revolutions per minute. To express the frequency in other units use the frequency converter, and to convert a length measured in feet or inches before entering it use the length converter. If you are interested in why gravity differs between locations in the first place, the gravitational force calculator covers the underlying inverse-square law.
Arb Digital builds free tools like this one because useful pages earn attention. If you want tools, calculators or content built for your own audience, we can help.
Browse All Free Tools Talk to Arb DigitalCommon Mistakes to Avoid
- Measuring length to the top of the bob — it must be to the centre of mass. On a 1 m pendulum with a 50 mm bob that is a 2.5 per cent length error and a 1.2 per cent period error.
- Entering the total swept angle as the amplitude — amplitude is measured from vertical to one extreme, so it is half the side-to-side angle.
- Using the standard formula at large angles — at 60 degrees it is more than 7 per cent out, which is not a rounding difference.
- Expecting a heavier bob to swing more slowly — mass cancels out completely. A heavier bob only helps by resisting damping, not by changing the period.
- Applying this to a rod or a bar — that is a compound pendulum with a different formula. A uniform rod pivoted at one end behaves like a simple pendulum of two-thirds its length.
Related Free Tools From Arb Digital
Compare with the spring oscillator using the simple harmonic motion calculator, and see the same mass cancellation in the free fall calculator. Convert the swing rate with the angular velocity calculator or the frequency converter, and convert a length before entering it with the length converter. For the origin of the gravity values used here, the gravitational force calculator covers the inverse-square law. Browse everything at the free online tools hub.
Frequently Asked Questions
Because the restoring torque is proportional to the sine of the angle, not to the angle itself, which makes the exact equation non-linear. Replacing the sine with the angle turns it into simple harmonic motion. That substitution is accurate to better than one per cent below about 15 degrees.
At 10 degrees the error is about 0.19 per cent, at 15 degrees about 0.43 per cent, at 30 degrees about 1.7 per cent, at 60 degrees about 7.3 per cent and at 90 degrees about 18 per cent. This calculator shows the exact period and the error alongside the approximate result.
No. A heavier bob experiences a proportionally larger restoring force and has proportionally more inertia, and the two cancel exactly. Heavier bobs are used in clocks only because they lose amplitude more slowly against friction and air resistance.
From the pivot point to the centre of mass of the bob. Measuring to the top of the bob rather than its centre is a common error, and on a metre-long pendulum with a fifty millimetre bob it shifts the period by more than one per cent.
Because gravity is not the same everywhere. It ranges from about 9.780 metres per second squared at the equator to about 9.832 at the poles, a difference of roughly half a per cent, and it also falls with altitude. A pendulum clock visibly changes rate when moved between latitudes.
Not directly. A rod is a compound pendulum whose period depends on its moment of inertia. A uniform rod pivoted at one end swings like a simple pendulum of two-thirds its length, so enter that value if you want an approximate answer from this page.
The exact period involves a complete elliptic integral of the first kind, which this tool evaluates using the arithmetic-geometric mean. The exact period equals the approximate one divided by the arithmetic-geometric mean of 1 and the cosine of half the amplitude.
This tool is provided for educational and study use. It models an idealised point mass on a massless string with no air resistance or pivot friction, so a real pendulum will lose amplitude over time and its period will drift toward the small-angle value.
