A hollow sphere is filled with water. There is a hole at the...

A hollow sphere is filled with water. There is a hole at the bottom of this sphere. This sphere is suspended with a string from a rigid support and given an oscillation. During oscillation, the hole is opened up and the periodic time of this oscillating system is measured. The periodic time of the system $\ldots \ldots \ldots \ldots$ (A) will remain constant (B) Will increase upto a certain time (C) Increases initially and then decreases to attain its initial periodic time (D) Initially decreases and then will attain the initial periodic time value.

A hollow sphere is filled with water. There is a hole at the bottom of this sphere. This sphere is suspended with a string from a rigid support and given an oscillation. During oscillation, the hole is opened up and the periodic time of this oscillating system is measured. The periodic time of the system $\ldots \ldots \ldots \ldots$
(A) will remain constant
(B) Will increase upto a certain time
(C) Increases initially and then decreases to attain its initial periodic time
(D) Initially decreases and then will attain the initial periodic time value.

Key Concepts

Variable Mass Distribution in Oscillatory Systems

When the distribution of mass within an oscillating system changes over time, as in cases where fluid is draining, both the total mass and its spatial distribution, including the moment of inertia and the effective lever arm from the pivot to the center of mass, are affected. These variations can lead to non-monotonic changes in the oscillation period until a new equilibrium or state is reached.

Center of Mass Dynamics

The center of mass is the point at which the mass of a system can be considered to be concentrated for the purpose of analyzing gravitational forces. In oscillatory motion, any variation in the position of the center of mass relative to the pivot directly affects the restoring torque and thereby the oscillation period.

Moment of Inertia

The moment of inertia quantifies how mass is distributed with respect to the axis of rotation. It plays a critical role in determining the oscillatory period of a system, as a change in mass distribution (for example, due to fluid movement or drainage) alters the moment of inertia and thus the dynamics of the oscillation.

Physical Pendulum

A physical pendulum refers to an extended rigid body oscillating about a horizontal axis under the influence of gravity. Unlike a simple pendulum, its period depends not only on the length but also on the body's moment of inertia and the location of its center of mass relative to the pivot point.

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