If you have ever wondered why your grandfather's clock was so tall, you will find the answer on our pendulum period calculator! Keep reading our article to discover the basics of one of the favorite gadgets of every physicist.

Here you will learn:

  • What is a pendulum;
  • What is the period of a pendulum;
  • How to find the period of a pendulum in the small angles approximation;
  • The formula for the period of a simple pendulum with ample initial angles;
  • The effects of the gravitational acceleration on the pendulum's period.

Tick...tock: a short introduction to pendulums

Pendulums are elementary objects: a heavy mass connected to a rigid swing, free to oscillate around a pivot. In most of their descriptions, pendulums act only under the force of gravity: when this happens, we call them simple pendulums.

These devices, unexpectedly, contain more physics than many others and can be used to prove fundamental properties of the natural world we live in. Moreover, they have a broad set of uses (though it's slowly dwindling): their irreplaceable presence in time-tracking devices quite literally showed us the way during the exploration of our planet.

What is the period of a pendulum and how to calculate the duration of an oscillation

The period of a pendulum is the time required by the ensemble mass (bob) plus swing to complete one oscillation: with this, we mean that the mass returns in the same position and moves in the same direction as the ones of the initial states.

The formula for the period of a pendulum is:

T=2πLgT = 2\cdot\pi\cdot\sqrt{\frac{L}{g}}

Where:

  • TT is the period of the pendulum in seconds;
  • LL is the length of the swing (in meters or feet); and
  • gg is the acceleration due to gravity (g9.81 m/s2g\approx 9.81\ \text{m}/\text{s}^2).

This simple equation for the period of a pendulum applies only in what physicists call the small angle approximation, a regime of the pendulum deriving from the trigonometric functions which model its behavior. The approximation asserts that, for angles θ1 rad\theta\ll1\ \text{rad}:

sin(θ)θ\sin{(\theta)}\approx\theta

In degrees we set the maximum value for which this approximation is valid to 10°10\degree-15°15\degree. Up to this value, the period of the pendulum equation shows us that the time required to perform the oscillation is independent of the initial angular displacement. The calculations in this approximation are much easier.

How to find the period of a pendulum outside the small angle approximation

If you need to consider oscillations larger than a few degrees, the small angle approximation won't give you the correct results: we need to introduce a nonlinearity in the form of a trigonometric function.

The formula for the period of a pendulum for any initial angle is:

T ⁣= ⁣2 ⁣ ⁣π ⁣ ⁣ ⁣Lg ⁣ ⁣(n=0((2 ⁣ ⁣n)!(2n ⁣ ⁣n!)2 ⁣)2 ⁣ ⁣ ⁣sin(θ02)2n)\begin{align*} T & \!=\! 2\!\cdot\!\pi\!\cdot\!\sqrt{\!\frac{L}{g}}\!\cdot\!\left(\sum_{n=0}^{\infty}\left(\frac{(2\!\cdot\! n)!}{\left(2^{ n}\!\cdot\! n!\right)^2}\!\right)^2\!\!\cdot\!\right.\\ &\left.\sin{\left(\frac{\theta_0}{2}\right)^{2\cdot n}}\right) \end{align*}

Our period of a pendulum calculator implements this series up to n=20n=20. This is enough for most applications!

Pendulums at sea: why your grandfather clock would run slow during your Caribbean holidays

The overbearing effect of gravity on the operation of pendulums was apparent to physicists as soon as they started studying them. However, the actual implications deriving from that small constant gg in the formula for the period were understood fully only later, when expeditions started carrying around the globe clocks with ever-increasing precision. Clocks close to the equator run slower than clocks at higher latitudes.

This is due to variations in the acceleration due to gravity, which, in turn, come from the imperfect shape of our planet (even though its shape may not be perfect, we still like Earth a lot!). The equatorial bulge — the technical name for the effect of rotation on a planet — causes gg to assume different values:

  • At the poles, g=9.863 m/s2g = 9.863\ \text{m}/\text{s}^2; while
  • At the equator, g=9.798 m/s2g = 9.798\ \text{m}/\text{s}^2.

We computed these values using our gravitational force calculator.

This difference caused an oscillation period to increase at the equator: an effect to consider since, back in those days, clocks were used to calculate a ship's position in the oceanic expanses.

More than periods: pendulum calculator for every needs

Now you know: pendulums are simple yet fascinating devices. We know this at Omni, which is why we created a small suite of pendulum calculators including this period of pendulum calculator!

Check them out:

FAQ

How do I calculate the period of a pendulum?

To find the period of a simple pendulum, you often need to know only the length of the swing. The equation for the period of a pendulum is:
T = 2π × sqrt(L/g)
This formula is valid only in the small angles approximation.

What is the period of a pendulum with length l = 1 m?

The period of such a pendulum is about 2 seconds.

To calculate this quantity, follow these steps:

  1. Find the value of your local acceleration due to gravity. A safe bet is g = 9.81 m/s².
  2. Substitute the value of g and l in the equation for the period of a pendulum: T = 2π × sqrt(L/g) =2π × sqrt(1/9.81) = 2.006.
  3. You are all set.

Observe how this length returns one-second-long "half oscillations": this is the preferred length for pendulum clocks.

What is the small angles approximation?

The small angle approximation is a mathematical approximation used in trigonometry. For small values of the arguments of trigonometric functions, we can use the following set of approximations:

  • sin(x) ≈ x;
  • cos(x) ≈ 1; and
  • tan(x) ≈ x.

You can derive these expressions using the series expansion of the functions: high order terms would result negligible for small values of x.

Why does gravity affects the period of a pendulum?

The gravitational force is responsible for the "return force" experienced by the pendulum's bob. It accelerates the mass when it moves toward the center, and slows until it stops on the way back.

The value of the acceleration due to gravity affects this force due to Newton's second law, and since at the poles g is 1% higher than at the equator, the period of the pendulum changes accordingly.

Davide Borchia
T = 2π√(L/g)
Acceleration of gravity (g)
g
Pendulum length (L)
ft
Initial angle (θ₀)
deg
Pendulum period (T)
sec
Period (T)
sec
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