Solution
Step 1: Condition for Stable Equilibrium
For a body with a spherical base resting on a horizontal floor, the stability depends on the position of the Center of Mass (CM) of the composite body relative to the geometric center (center of curvature) of the curved surface in contact with the floor.
Let $O$ be the common center of the two hemispheres. Since hemisphere B is in contact with the floor, $O$ is the center of curvature.
For a body with a spherical base resting on a horizontal floor, the stability depends on the position of the Center of Mass (CM) of the composite body relative to the geometric center (center of curvature) of the curved surface in contact with the floor.
Let $O$ be the common center of the two hemispheres. Since hemisphere B is in contact with the floor, $O$ is the center of curvature.
- Stable Equilibrium: The composite CM must lie below the center of curvature $O$.
- Unstable Equilibrium: The composite CM must lie above $O$.
- Neutral Equilibrium: The composite CM must coincide with $O$.
Step 2: Locate Center of Mass
We define the vertical position $y$ relative to the center $O$.
We define the vertical position $y$ relative to the center $O$.
- Center of mass of upper hemisphere A ($m_A, r_A$) is at $y_A = +\frac{3r_A}{8}$.
- Center of mass of lower hemisphere B ($m_B, r_B$) is at $y_B = -\frac{3r_B}{8}$.
Step 3: Apply Stability Conditions
For stable equilibrium, we require $Y_{CM} < 0$ (below O). $$ \frac{3}{8(m_A + m_B)} (m_A r_A - m_B r_B) < 0 $$ Since masses and radii are positive, this implies: $$ m_A r_A < m_B r_B $$
For stable equilibrium, we require $Y_{CM} < 0$ (below O). $$ \frac{3}{8(m_A + m_B)} (m_A r_A - m_B r_B) < 0 $$ Since masses and radii are positive, this implies: $$ m_A r_A < m_B r_B $$
Conclusion:
- Stable: $m_A r_A < m_B r_B$
- Unstable: $m_A r_A > m_B r_B$
- Neutral: $m_A r_A = m_B r_B$