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Pendulum period

How do string length, gravity, and amplitude change the period? Compare small amplitude and large amplitude.

    P05 · Nonlinear pendulum and small-angle approximation0.00 s · Ready
    A simple pendulum experiment scene where a pendulum hanging from a fixed point oscillates left and right.

    Physics/Mechanics/Principles and Exploration

    Pendulum period — principles and investigation

    If you make the swing longer, how will the time for one round trip change? The simple pendulum experiment is an activity that measures the period by changing the string length, gravitational acceleration, and initial angle. Check the square root relationship that holds at small amplitudes, and compare the degree to which it deviates from the small-angle approximation when shaken large.

    How does it work?

    The longer the length and the smaller the gravity, the longer the period. At large amplitudes, the period increases compared to the small-angle approximation.

    Key relationship

    θ″ = −(g/L)sinθ · T₀ = 2π√(L/g) (small angle approximation)

    Try this investigation

    1. Drag and drop the weight or press the run button. Measure the time between passing the lowest point in the same direction.
    2. Compare the lengths 0.5 m, 1 m, and 1.5 m at a gravity of 9.8 m/s² and an amplitude of 10°. Measure 3 cycles each time and record the average.
    3. Compare the amplitudes of 10° and 60° at a length of 1 m, and record only the gravity at an amplitude of 10° by converting it to the value of the moon.

    Interpreting your results

    Why does quadrupling the length at small amplitudes double the period?

    Since T₀ = 2π√(L/g), √4 = twice.

    Assumptions and limitations

    The weight is a 1 kg mass, and the string has no mass and does not stretch. Ignore air resistance, axial friction, and damping. The burnout formula is only accurate for small amplitudes. RK4 is at 1/240 s intervals and the period is measured by interpolating the lowest point passed.