Tipping a container of water

[kbannister]

Homework Statement

A weightless cylindrical tank of diameter D and height H rests on a nonslip surface on a flatcar going around a circular track of radius R. The tank contains water to depth h with a free surface, and we assume h < H such that water won't slop over the tank's edge. The flatcar's speed is gradually increased to V. Two possibilities exist: 1. Only part of the water occupies the "ungula" (the heel-shaped volume under the sloping free surface, and 2. All the water ends up in the Ungula, exposing some of the tank's floor. At what value of V will the tank begin to tip over?

Relevant Equations

Radial acceleration, a = V^2/R;
Angle of free surface, alpha = arctan(a/g), where g = acceleration of gravity

Not clear how to proceed. Does the cylindrical surface of the ungula need to be considered?

 

[haruspex]

Does the cylindrical surface of the ungula need to be considered?

Yes.
What attribute of the water volume do you need to determine in order to decide whether it will tip?

 

[kbannister]

Yes.
What attribute of the water volume do you need to determine in order to decide whether it will tip?

Assume the water has a weight density of gamma.

 

[kbannister]

The question concerns determining the speed, V, around the circular track of radius R. A simpler version of the problem is the situation when the container is accelerated in a straight line at acceleration a.

 

[haruspex]

Assume the water has a weight density of gamma.

No, that's an attribute of the water itself. I asked for an attribute of the volume (shape) the water occupies.

 

[kbannister]

Yes.
What attribute of the water volume do you need to determine in order to decide whether it will tip?

The tank is cylindrical, diameter D, height H, water depth h where h < H so that the water won’t spill over.

 

[DaveC426913]

Will a diagram help? Does it need anything added?

 

[haruspex]

The tank is cylindrical, diameter D, height H, water depth h where h < H so that the water won’t spill over.

For any object of given weight standing (not slipping) on a horizontal surface and subject to a sideways force of given magnitude, what determines whether it will tip over?

 

[Lnewqban]

A simpler version of the problem is the situation when the container is accelerated in a straight line at acceleration a.

What (and how much of it) would make the container tip over in that simpler version?