How to Calculate the Fraction of Time a Pendulum Spends Between Two Angles

Question:

A simple pendulum oscillates with an amplitude of 10.0°. What fraction of the time does it spend between 5.4° and -5.4°? Assume simple harmonic motion (SHM).

Answer:

The fraction of time a pendulum spends between two angles is equal to the ratio of the difference between the angles and the total range of motion. In this case, the pendulum spends 54% of the time between 5.4° and -5.4°.

Explanation: The fraction of time a simple pendulum spends between two angles is equal to the ratio of the difference between the two angles and the total range of motion of the pendulum.

In this case, the pendulum has an amplitude of 10.0°, which means it swings 10.0° to the left and 10.0° to the right of its equilibrium position. The total range of motion is therefore 20.0°.

To find the fraction of time the pendulum spends between 5.4° and -5.4°, we need to calculate the difference between these two angles and divide it by the total range of motion:

Fraction of time = (5.4° - (-5.4°)) / 20.0°

= 10.8° / 20.0°

= 0.54

Therefore, the pendulum spends 54% of the time between 5.4° and -5.4°.