Rigid body solid: center of mass and tendency to topple

[ttja]

Homework Statement

A whimsical piece of furniture has the base shaped like a star.
Formulate the condition in terms of the location of the center of mass of
the object that the piece would not topple over. A sketch would be
helpful.

The Attempt at a Solution

Can someone help me first what the problem is asking for? I understand conceptually that the center of mass has to fall within the area support of the base, which is wherever in the area of the star base. But am i just to find the R vector to the center of mass and deem that its coordinate to fall onto the area of the plane where the base is located?

 

[Delphi51]

I think you would say that the center of mass must be within some cylinder centered on the vertical axis up from the center of the star base. Hmm, if it happens to be right above one of the points on the star, it can be a little further out. Maybe the volume where the c of m can be is not a cylinder but something more complicated.

 

[kerimek]

I can help clarify the problem and provide a solution. The problem is asking for the condition in which the whimsical piece of furniture, with a star-shaped base, will not topple over. This means that the center of mass of the object must fall within the area of support of the base in order for it to remain stable.

To determine this condition, we can use the concept of torque. Torque is a measure of the force that can cause an object to rotate. In this case, we need to ensure that the torque acting on the object is balanced and will not cause it to topple over.

To visualize this, imagine a line connecting the center of mass to the point where the object is in contact with the ground. This line is called the line of action. In order for the object to remain stable, the line of action must fall within the area of support of the base, which is the area where the object is in contact with the ground.

In the case of a star-shaped base, the area of support will be the area enclosed by the points where the star points touch the ground. This can be seen in the sketch below:


As you mentioned, the R vector to the center of mass must fall onto the plane where the base is located. This means that the center of mass must be located within the area of support, as shown in the sketch.

In summary, the condition for the whimsical piece of furniture to not topple over is that the center of mass must fall within the area of support of the base. This can be determined by ensuring that the line of action falls within the area enclosed by the points where the star points touch the ground.