# Can liquids be treated as rigid bodies for moment of inertia

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1. Mar 8, 2015

### Philip17

Consider a partially filled bottle of water. When it's tipped over and rotating about its pivot point where its edge touches the ground, it has an angular acceleration. Assuming the liquid is not viscous at all, so flows perfectly, at a given instant can it be considered to have a moment of inertia equal to that of a rigid body of the same shape? Could its angular acceleration thus be calculated using this moment of inertia and its instantaneous center of mass? This would mean that its moment of inertia changes as it falls.

Thank you :)

2. Mar 8, 2015

Staff Emeritus
It will. The instantaneous moment of inertia is the moment of inertia of the mass distribution at that instant.

3. Mar 8, 2015

### Philip17

As if it were a rigid body at that instant, right?

4. Mar 8, 2015

### mrspeedybob

No
If you can apply a torque to the bottle, and accelerate the bottle without accelerating the liquid inside the the liquid should not be included in the moment of inertia.

Take the simpler example of a round tub containing a hypothetical non viscous, friction-less fluid. If you apply a torque to the tub, the tub will accelerate, but the fluid will remain stationary. Since you did not accelerate the fluid you have not added any angular momentum to it and so it is not included in the tubs moment of inertia.

5. Mar 8, 2015

### Philip17

But in this case the water would have to accelerate. Wouldn't it?
Does that mean the moment of inertia for the water should be ignored when the bottle is falling over?

6. Mar 8, 2015

Staff Emeritus
Rigidity is not something that enters into the moment of inertia. Only the mass distribution matters.

7. Mar 8, 2015

### Philip17

So, just to clarify:
In order to determine the instantaneous angular acceleration of a half-filled bottle of water tipping over - i.e. rotating about the point where it touches the table - I use the moment of inertia for a rigid body with the same instantaneous shape as the water (which will change) (1)
or
Do i simply assume the moment of inertia for the whole system is that of the bottle and ignore the water? (2)

(1) or (2)? Thank you very much by that way :)

8. Mar 8, 2015

### mrspeedybob

Yes, but the mass distribution of what? When the different parts of an object accelerate at different rates then it no longer makes sense to model it as a single simple object. In my example you would clearly only use the moment of the tub because the liquid is not part of the object that is being accelerated. If the liquid were very viscous then treating it as part of the tub would be logical. Philip17's example is between those 2 extremes and is going to be quite difficult to model.

9. Mar 8, 2015

### mrspeedybob

Neither.
The bottle will be experiencing angular acceleration, therefore you should use the moment of inertia for the bottle as a separate object from the water.
The water will be experiencing some angular acceleration due to it's friction with the bottle, some linear acceleration toward the down-tipping end of the bottle, some deformation as the fluid changes shape, some internal friction resisting the change in shape, etc. modeling the water is going to be a rather complex fluid dynamics problem.