A problem about spherical pendulum that filled with water

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Discussion Overview

The discussion revolves around the dynamics of a hollow spherical pendulum filled with water, specifically focusing on how the period of oscillation changes as the water empties from the sphere. The conversation includes theoretical considerations and mathematical reasoning related to the properties of the pendulum and the behavior of the center of mass over time.

Discussion Character

  • Exploratory
  • Technical explanation
  • Mathematical reasoning

Main Points Raised

  • Some participants note that the mass and inertia of the sphere change as water empties from it.
  • One participant suggests that the problem can be simplified by treating the pendulum as a point mass suspended by a massless string, ignoring the moment of inertia of the sphere.
  • Another participant emphasizes the need to determine how the position of the center of mass of the sphere varies with time as the water level changes.
  • There is a suggestion to find the effective length of the pendulum, defined as the distance from the point of attachment of the string to the center of mass.
  • One participant expresses difficulty in reaching a definitive answer, noting that the period initially increases and then decreases, complicating the analysis if the moment of inertia is not ignored.
  • A later reply provides a hint to consider the position of the center of mass at different water levels (full, half, and empty) to aid in solving the problem.
  • One participant claims to have solved the problem, stating that the period first increases and then returns to its original value.

Areas of Agreement / Disagreement

Participants express varying viewpoints on how to approach the problem, with some agreeing on the simplification to a point mass model while others highlight the complexities introduced by the moment of inertia. The discussion remains unresolved regarding the best method to analyze the changing period of the pendulum.

Contextual Notes

There are limitations in the discussion regarding the assumptions made about the moment of inertia and the dependence on the definitions of effective length and center of mass. Unresolved mathematical steps may affect the clarity of the conclusions drawn.

amirghaderi
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We have a hollow spherical pendulum that is filled with water. The sphere has a hole under it. If it starts to oscillate and at the beginning it was full of water, how it’s period would be by passing time and empting sphere?
 
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amirghaderi said:
We have a hollow spherical pendulum that is filled with water. The sphere has a hole under it. If it starts to oscillate and at the beginning it was full of water, how it’s period would be by passing time and empting sphere?
Welcome to PF amirghaderi,

Well I suppose the first question to ask yourself is that what property changes as the water empties from the sphere?
 
the mass and the inertia of the sphere changes
 
amirghaderi said:
the mass and the inertia of the sphere changes
Indeed it does. However, the problem is greatly simplified because usually a spherical pendulum is seen as a generalisation of a simple pendulum in three-space, which means that we can ignore the moment of inertia of the sphere and just consider the pendulum as a point mass suspended by a massless string.

EDIT: So, now you need to decide how the position of the centre of mass of the sphere varies with time.
 
Last edited:
We only need to find the effective "l" for the pendulum which is the distance of the centre of mass from the point of attaching of the string.
 
Hence... the problem. This sounds like some sort of twisted related rates problem. From the sort of question you posted, I imagine you know how to calculate your moments of inertia (and maybe are familiar with the tensor forms even)... sounds like you just need to express the rate of change of mass for the volume of water within the sphere. Neglecting the mass of the sphere itself, as Hottenanny already mentioned.
 
i am graduated student in physics and i knonw how to calculate its moments of inertia but and how to simply ignore it as a point mass but i can't reach its answer that is first its period becomes large and then becomes smaler and without ignoreing the inertial momentom of the sphere solving becomes very complicated and did not look a meaning ful answer
 
amirghaderi said:
i am graduated student in physics and i knonw how to calculate its moments of inertia but and how to simply ignore it as a point mass but i can't reach its answer that is first its period becomes large and then becomes smaler and without ignoreing the inertial momentom of the sphere solving becomes very complicated and did not look a meaning ful answer
Consider where the centre of mass of the hollow sphere is located at the following points:

(a) When it is full of water
(b) When it is half full of water
(c) When it is empty
 
Last edited:
thanx mr Hootenanny for your hint .
i solve it .
the period first increses and then retern to the orginal value
 
  • #10
amirghaderi said:
thanx mr Hootenanny for your hint .
i solve it .
the period first increses and then retern to the orginal value
It's a pleasure :smile:
 

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