Center of gravity and stability of a system

[stellarator]

Homework Statement

Determine the critical stability angle for a system of two shapes

Relevant Equations

centroid = (V1*y1+V2*y2)/(V1+V2)

Suppose I have an object consisting of a hemisphere of radius r and a cone of radius r and height h. The shapes are glued to each other on their faces and the object is set standing on its hemisphere side. Depending on the value of h, the center of gravity for the system will change.

I have calculated that for h values below r*sqrt(3), the centroid will be in the hemisphere region and for h values above r*sqrt(3), the centroid will be in the cone region. Now I am tasked with finding the critical angle for which the system will topple when pushed.

My reasoning is the following:
when h < r*sqrt(3), the object will never topple and always right itself when pushed.
when h > r*sqrt(3), the object will always topple.

There doesn't seem to be a critical angle for the object losing its balance. It just either topples or does not depending on the h value. Is my reasoning correct?

 

[Baluncore]

Welcome to PF.
I am sorry for the delay.

The potential energy of the system is an important parameter.
As the hemisphere is tilted from the vertical the restoring force increases as a function of angle.
As the cone moves away from vertical the overturning force increases.
1. You must balance those two forces to find the critical height of the cone.
2. Then compare the derivatives of the two forces to identify if a stable position might exist at some angle away from the vertical.

I suspect you have done the first, but not looked closely enough at the derivatives.

 

[Albir17]

Yes, i think your reasoning is correct
An alternative to prove this is by considering the geometry of the system

Suppose that the system is tilted by angle of α, and let h_cg as the vertical coordinate of the center of mass of the system when at rest.

This will shift the horizontal position of the contact point between the system and the ground by Rcos(α)
From this, we know that if h_cgcos(α) < R cos(α), the torque produced by the weight of the system will produce negative angular acceleration. Hence, the system will never topple as long as coordinate of the center mass is in the region of the hemisphere (h_cg < R)