2.006067
s
0.498488
Hz
3.132092
rad/s
2.009893
s
0.54596
m/s
0.015192
m
0.149036
J
0.149036
J
2.006067
s
0.498488
Hz
3.132092
rad/s
2.009893
s
0.54596
m/s
0.015192
m
0.149036
J
0.149036
J
The Pendulum Calculator provides a complete analysis of a simple pendulum’s motion, including the period, frequency, angular frequency, maximum speed, height, and energy. It also includes an amplitude correction for improved accuracy at larger swing angles.
The simple pendulum—a point mass suspended by a massless, inextensible string—is one of the most studied systems in physics. Galileo first observed around 1602 that the period of a pendulum is approximately independent of its amplitude (isochronism), a discovery that led to the development of pendulum clocks by Christiaan Huygens in 1656.
For small oscillations, the period of a simple pendulum depends only on its length and the local gravitational acceleration: $$T = 2\pi\sqrt{\frac{L}{g}}$$ This remarkable result is independent of both the mass of the bob and the amplitude of swing (for small angles). The frequency and angular frequency follow as: $$f = \frac{1}{2\pi}\sqrt{\frac{g}{L}}, \quad \omega = \sqrt{\frac{g}{L}}$$
For larger amplitudes, the small-angle approximation breaks down and the exact period involves an elliptic integral. This calculator uses the first two correction terms of the series expansion: $$T \approx T_0\left(1 + \frac{1}{4}\sin^2\frac{\theta_0}{2} + \frac{9}{64}\sin^4\frac{\theta_0}{2}\right)$$ This gives accuracy better than 0.1% for amplitudes up to about 70°.
The energy analysis uses conservation of energy. At the maximum displacement, all energy is gravitational potential energy. At the lowest point, all energy is kinetic. The maximum height is $$h = L(1 - \cos\theta_0)$$ and by energy conservation, the maximum speed at the bottom is $$v_{max} = \sqrt{2gh}$$
Pendulums have extraordinary scientific importance. They have been used to measure gravitational acceleration, detect the rotation of the Earth (Foucault pendulum), define the second, and regulate timekeeping for centuries. This calculator serves physics students, educators, engineers working with pendulum-based systems, and anyone curious about this elegant oscillating system.
The calculator uses the standard equations of pendulum motion:
Small-angle period:
$$T_0 = 2\pi\sqrt{\frac{L}{g}}$$
Frequency and angular frequency:
$$f = \frac{1}{T_0}, \quad \omega = \sqrt{\frac{g}{L}}$$
Amplitude-corrected period:
$$T \approx T_0\left(1 + \frac{1}{4}\sin^2\frac{\theta_0}{2} + \frac{9}{64}\sin^4\frac{\theta_0}{2}\right)$$
Maximum height: $$h = L(1 - \cos\theta_0)$$
Maximum speed: $$v_{max} = \sqrt{2gh}$$
Maximum energies: $$PE_{max} = KE_{max} = mgh$$
The period is the time for one complete back-and-forth swing. The small-angle period is accurate to within 1% for angles up to about 23°. The corrected period accounts for the nonlinear effect of larger amplitudes. The maximum speed occurs at the lowest point of the swing, where all potential energy has converted to kinetic energy. The energies are equal because energy is conserved (in the absence of friction and air resistance).
Inputs
Results
A pendulum length of 0.994 m gives a period of almost exactly 2 seconds—the “seconds pendulum” used in grandfather clocks. With a small 5° amplitude, the corrected period is negligibly different.
Inputs
Results
A 3 m swing at 45° amplitude. The corrected period (3.59 s) is about 3.2% longer than the small-angle estimate (3.47 s). The maximum speed at the bottom is 4.14 m/s.
No. For an ideal simple pendulum, the period depends only on length and gravitational acceleration, not on mass. This is because the gravitational force and inertia both increase with mass, and these effects cancel exactly. In practice, heavier bobs maintain energy longer against air resistance, but the period itself is unchanged.
The simple formula T = 2π√(L/g) uses the small-angle approximation sinθ ≈ θ. At larger angles, sinθ < θ, so the restoring force is weaker than the linear approximation predicts. This means the pendulum takes longer to return, increasing the period. At 90° amplitude, the true period is about 18% longer than the small-angle prediction.
A seconds pendulum has a period of exactly 2 seconds (1 second per half-swing), which corresponds to a length of about 0.994 m at standard gravity. It was historically important for timekeeping and was once proposed as the basis for defining the meter. Grandfather clocks typically use seconds pendulums.
The period is inversely proportional to the square root of gravitational acceleration. Stronger gravity (larger g) means a shorter period (faster swinging). This means a pendulum clock runs faster at the poles (g ≈ 9.832 m/s²) than at the equator (g ≈ 9.780 m/s²), and would be extremely slow on the Moon (g ≈ 1.62 m/s²).
A simple pendulum is an idealization: a point mass on a massless string. A compound (physical) pendulum is a real rigid body swinging about a pivot. The compound pendulum’s period involves its moment of inertia: T = 2π√(I/(mgh)), where I is the moment of inertia about the pivot and h is the distance from pivot to center of mass.
Yes, this is one of the most precise traditional methods. By measuring the length and period of a pendulum, g can be calculated from g = 4π²L/T². Henry Kater’s reversible pendulum (1817) achieved accuracy of about 0.01%. Modern methods use atomic interferometry, but pendulum experiments remain a standard physics laboratory exercise.
Roboculator Team
The Roboculator Team explains calculations, planning tools, and practical formulas in clear language for real-life situations.
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