For IB Physics SL and HL, calculate the period of a simple pendulum using . Here, is the pendulum length and is the gravitational field strength.
For IB Physics SL and HL, calculate the period of a simple pendulum using . Here, is the pendulum length and is the gravitational field strength.
The period is the time taken for one complete oscillation. For a simple pendulum undergoing small oscillations, its motion approximates simple harmonic motion because the restoring acceleration is approximately proportional to and opposite to the displacement.
The tangential restoring acceleration is . For small angles, , and arc displacement is , giving . Comparing this with the defining SHM equation gives . Since , the pendulum equation follows.
Use:
| Quantity | Meaning and unit |
|---|---|
| Period in seconds, | |
| Distance from the pivot to the bob's centre of mass, in metres, | |
| Gravitational field strength, normally near Earth's surface |
For example, consider a pendulum of length :
Therefore, one complete oscillation takes approximately .
The equation shows that and . A longer pendulum therefore has a greater period, while a stronger gravitational field produces a shorter period. The bob's mass does not affect the period.
A key assumption is that the angular displacement is small, so when is measured in radians. Air resistance must also be negligible, and the string should be light and inextensible. The formula does not apply accurately at large amplitudes, because the small-angle approximation fails and the true period becomes slightly longer.
If 10 oscillations take , then . Timing several oscillations reduces the percentage uncertainty caused by human reaction time. During experiments, release the bob without pushing it.
For a Calculate question, show the equation, substitution and final value. A common misconception is measuring only to the surface of the bob: it must be measured from the pivot to the bob's centre of mass. In practical questions, improve precision by timing many oscillations and dividing the total time by the number of oscillations.