Problem source

A child's toy consists of a solid cone of height $\lambda a$ and a solid hemisphere of radius $a$, made out of the same uniform material and fastened together so that their plane faces coincide. (Thus the diameter of the hemisphere is equal to that of the base of the cone.)
Show that if $\lambda < \sqrt{3}$ the toy will always move to an upright position if placed with the surface of the hemisphere on a horizontal table, but that if $\lambda > \sqrt{3}$ the toy may overbalance.
Show, however, that if the toy is placed with the surface of the cone touching the table it will remain there whatever the value of $\lambda$.
[The centre of gravity of a uniform  solid cone of height $h$ is a height $h/4$ above its base. The centre of gravity of a uniform solid hemisphere of radius $a$ is at distance $3a/8$ from the centre of its base.]

Solution source

\begin{center}
\begin{tikzpicture}

% Define parameters
\def\r{1.2}      % radius
\def\h{2.1}      % height of cone

% Draw the hemisphere bottom curve (solid line)
\draw (-\r,0) arc (180:360:\r);

% Draw the dashed line for the opening/junction
% \draw[dashed] (-\r,0) -- (\r,0);

% Draw the elliptical curve to indicate 3D shape on the hemisphere
\draw[dashed] plot[domain=0:180,samples=50] ({\r*cos(\x)},{0.3*\r*sin(\x)});
\draw plot[domain=180:360,samples=50] ({\r*cos(\x)},{0.3*\r*sin(\x)});

% Draw the cone
\draw (-\r,0) -- (0,\h) -- (\r,0);

% Mark the center point
% \node at (0,0) {$O$};
\fill (0,0) circle (1.5pt);

% Add the radius label with an arrow
\draw[->] (0,0) -- (\r,0) node[midway, below] {$a$};

% Add the height label with an arrow
\draw[->] (0,0) -- (0,\h) node[midway, right] {$\lambda a$};

\end{tikzpicture}
\end{center}

By symmetry the centre of mass will lie on the main axis. Taking the plane faces as $x = 0$ we have the following centers of mass:

\begin{align*}
    && \text{COM} && \text{Mass} \\
    \text{Hemisphere} && -\frac{3a}{8} && \frac{2\pi a^3}{3} \\
    \text{Cone} && \frac{\lambda a}{4} && \frac{\lambda \pi a^3}{3} \\
    \text{Toy} && \bar{x} && \frac{(\lambda + 2)\pi a^3}{3} \\
\end{align*}

Therefore,

\begin{align*}
    && \frac{(\lambda + 2)\pi a^3}{3} \cdot \bar{x} &= -\frac{3a}{8} \cdot \frac{2\pi a^3}{3} + \frac{\lambda a}{4} \cdot \frac{\lambda \pi a^3}{3} \\
    \Rightarrow && (\lambda + 2) \bar{x} &= \frac{(\lambda^2 -3)a}{4}  
\end{align*}

Therefore the centre of mass will be inside the hemisphere (and it will always move to an upright position) iff $\bar{x} < 0 \Leftrightarrow \lambda < \sqrt{3}$.

\begin{center}
\begin{tikzpicture}

    \def\r{1.2}      % radius
    \def\h{2.1}      % height of cone

    
    \draw (0, 0) -- (\h, 0) -- (\h/2, {sqrt(3)*\h/2}) -- cycle;

    \coordinate (base) at ({\h/2*cos(150) + \h/4},{\h/2*sin(150) + sqrt(3)*\h/4}); 
    \coordinate (centr) at (\h/4, {sqrt(3)*\h/4});
    \coordinate (top) at (\h,0);
    \coordinate (O) at (0,0);
    \draw plot[domain=60:240,samples=50] ({\h/2*cos(\x) + \h/4},{\h/2*sin(\x) + sqrt(3)*\h/4});

    \draw[dashed] (\h, 0) -- (base);
    \draw[dashed] (0, \h) -- (O);

    \fill ($(base)!0.8!(centr)$) circle (1.5pt);

    \pic [draw, angle radius=0.8cm, "$\theta$"] {angle = base--top--O};
    
\end{tikzpicture}
\end{center}

For the toy to topple from this position, $\bar{x}$ must be longer than it would need to be to form a right-angled triangle with the vertical at the plane face. The angle at this point will be $\theta$, so we need:

$\bar{x} > a\tan \theta = a \frac{a}{\lambda a} = \frac{a}{\lambda}$

\begin{align*}
    && \bar{x} &> \frac{a}{\lambda} \\
    \Leftrightarrow && \frac{(3-\lambda^2)a}{4(\lambda + 2)} &> \frac{a}{\lambda} \\
    \Leftrightarrow && {(3-\lambda^2)\lambda} &> {4(\lambda + 2)} \\
    \Leftrightarrow && -\lambda^3-\lambda -8 &> 0 \\
\end{align*}

Contradiction! Therefore it can never topple when laid on its side.