For the toy to be self-righting, its Center of Mass (COM) must be lower than the center of curvature of the base. The base is the lower sphere (radius 9 cm), so the critical height is y = 9 cm.

y_COM < 9

Positions (height y): Load M at 0, Lower mass m₂ at 9, Upper mass m₁ at 24 (9+9+6).

[ M(0) + m₂(9) + m₁(24) ] / [ M + m₂ + m₁ ] < 9
9m₂ + 24m₁ < 9(250) + 9m₂ + 9m₁

Notice that 9m₂ cancels out on both sides.

15m₁ < 2250
m₁ < 150 g

This implies stability depends only on the upper mass being light enough. The lower mass m₂ can be anything. (Validates statement d)

A “sluggish” response means a low frequency of oscillation. This happens when the Moment of Inertia (I) is high or the restoring torque is low.

Increasing the lower mass m₂ adds mass at y=9. This increases the total inertia of the system. Additionally, mathematical analysis shows it raises the COM slightly towards the neutral point, reducing stability stiffness. Both factors slow down the oscillation.

Therefore, more mass in the lower ball makes the toy more sluggish. (Validates statement b)