Calculator of Alexandria

Pendulum period calculator

Period of a simple pendulum from its length, or the length for a given period, with T = 2π√(L/g) derived from the small-angle approximation.

Solve for

Time for one full swing, there and back.

From the pivot to the centre of the bob.

Formula

T=2πLgT = 2\pi\sqrt{\frac{L}{g}}
How it's derived

The formula

T=2πLgT = 2\pi\sqrt{\frac{L}{g}}

TT is the period (one full swing out and back), LL the length from the pivot to the centre of the bob and gg the acceleration due to gravity. Choose any one to solve for.

Where it comes from

When the bob is pulled aside by angle θ\theta, gravity’s pull mgmg splits into a part along the string, which the string cancels, and a part along the arc, which pulls the bob back toward the bottom:

F=−mgsin⁡θF = -mg\sin\theta

The bob moves along an arc of length s=Lθs = L\theta. Newton’s second law along that arc gives:

mLd2θdt2=−mgsin⁡θmL\frac{d^2\theta}{dt^2} = -mg\sin\theta

For small angles, sin⁡θ≈θ\sin\theta \approx \theta (in radians). At 10° the error is only 0.5%. The mass cancels and:

d2θdt2=−gL θ\frac{d^2\theta}{dt^2} = -\frac{g}{L}\,\theta

This is the equation of simple harmonic motion: acceleration proportional to displacement and pointing back toward the centre. Its solution is θ=θ0cos⁡(ωt)\theta = \theta_0\cos(\omega t), a steady oscillation with angular frequency ω=g/L\omega = \sqrt{g/L}. One full cycle is 2π2\pi radians of ωt\omega t, so the period is:

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

Worked example

A 1 m pendulum on Earth:

T=2π19.81=2.006 sT = 2\pi\sqrt{\frac{1}{9.81}} = 2.006\ \text{s}

Run it backward for a “seconds pendulum” that ticks once per second, a full period of 2 s:

L=9.81×(22π)2=0.994 mL = 9.81 \times \left(\frac{2}{2\pi}\right)^2 = 0.994\ \text{m}

That’s why tall grandfather clocks are about a meter from pivot to bob.

Watch out for

  • Large swings run slow. The formula assumes small angles. At 30° the true period is about 1.7% longer, and at 60° about 7% longer.
  • Mass doesn’t matter, length does. A heavier bob swings at the same rate. To double the period, make the pendulum four times longer.
  • Measure to the centre of mass. For a heavy bob or a swinging rod, LL is to the centre of mass, and a rod needs a different formula entirely.

Sources

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