When the length of a pendulum string is changing linearly while keeping the period constant, the amplitude of the swing increases exponentially. This phenomenon can be explained intuitively by considering the relationship between the length of the pendulum and the gravitational force acting on it.
The period of a pendulum is primarily determined by the gravitational acceleration and the length of the pendulum string. When the length of the string decreases, the gravitational force acting on the pendulum increases. This increase in gravitational force provides more energy to the pendulum, causing it to swing with a larger amplitude.
Imagine a pendulum as a swinging weight attached to a string. As the length of the string becomes shorter, the distance the weight has to travel during each swing decreases. Since the period remains constant, the weight needs to cover this reduced distance in the same amount of time. To accomplish this, the weight accelerates more rapidly, reaching higher velocities and swinging with a larger amplitude.
The exponential increase in amplitude occurs because the energy of the pendulum is proportional to the square of its velocity. When the gravitational force increases due to a shorter string, the pendulum gains more energy, resulting in higher velocities during the swing. This, in turn, leads to a larger amplitude of the swing.
In summary, as the length of the pendulum string decreases linearly while maintaining a constant period, the increase in gravitational force provides more energy to the pendulum. This increased energy leads to higher velocities, causing the pendulum to swing with a larger amplitude.