Question

A simple pendulum is oscillation with an angular amplitude large enough that acceleration of the bob at an instant of time is four times of its minimum value. Determine the range of values of the angular amplitude.

Answer 1

No such range of angular amplitude exists for a simple pendulum oscillating with a taut string.

Answer 2

The condition that the maximum acceleration is four times the minimum acceleration leads to $\tan(\theta_0/2) = 2$.

This implies $\theta_0 = 2\arctan(2)$.

However, for a simple pendulum to truly oscillate (meaning the string remains taut throughout the motion), the angular amplitude $\theta_0$ cannot exceed $90^\circ$. If $\theta_0 > 90^\circ$, the string slackens.

Since $2\arctan(2) \approx 126.87^\circ$, this value is greater than $90^\circ$.

Therefore, under the standard definition of a simple pendulum's oscillation, there is no such angular amplitude. The range of values is an empty set. If the question implies that the amplitude is the maximum angular displacement regardless of slackening, then the amplitude is $2\arctan(2)$.