How the change of amplitude effects the time of one cycle of a pendulum?

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Denote the length of the pendulum by L and the amplitude by A. The period (time of one cycle) is given by:

T = 2 pi sqrt(L/g) [1 + 1/16 (A/L)^2 + term of order (A/L)^4]

So, increasing the amplitude will make the period longer. The effect is proportional to (A/L)^2 and thus becomes very small if the length is much larger than the amplitude.

To understand how the change in amplitude affects the time of one cycle of a pendulum, let's break down the process step by step.

The time period of a pendulum is determined by its length, gravity, and the amplitude of its swing. The time period is the time taken for the pendulum to complete one full swing back and forth.

Now, let's consider the effect of amplitude on the time period. The amplitude of a pendulum is the maximum angle it swings away from its equilibrium position (the center). The larger the amplitude, the greater the distance the pendulum travels during a swing.

As the pendulum swings with a larger amplitude, it covers more distance in the same amount of time compared to a swing with a smaller amplitude. This means that the pendulum moves faster as the amplitude increases.

According to pendulum theory, the time period of a simple pendulum (neglecting air resistance) is primarily determined by the length of the pendulum and the acceleration due to gravity, but it is not affected by the amplitude. This principle is known as "isochronism of small amplitudes," which means that for small angles of swing, the time period remains constant.

However, when the amplitude of the pendulum exceeds a certain limit (typically around 20 degrees), the time period starts to deviate slightly from this constant value. At larger amplitudes, the time period becomes longer than the time period at smaller amplitudes.

So, to summarize, for small amplitudes (less than 20 degrees), the change in amplitude does not significantly affect the time of one cycle of a pendulum. However, for larger amplitudes, the time period starts to increase slightly, meaning the pendulum takes longer to complete one full swing as the amplitude increases.

To further explore this relationship, you can conduct an experiment by measuring the time period of a pendulum at different amplitudes and observe the changes in the results.