A $1.5 \mathrm{~mm}$ thick-walled, $6.6 \mathrm{~cm}$ inner diameter, $10 \mathrm{~cm}$ inner height cylindrical glass has a base thickness of $3 \mathrm{~mm}$. The density of the glass is twice that of water.

We slowly pour water into the glass until it is full. Determine and graphically represent the position of the center of mass of the system as a function of the height of the water in the glass!
Let's use the notations from Figure 1! We will decompose the cup into the cylindrical side wall and the cylindrical base. Due to the homogeneous material distribution of the cup (and the water), the symmetry centers $(A, B)$ are also the centers of mass. Based on geometric considerations, we have:
$$
h_{A}=h_{t}+h / 2=5.3 \mathrm{~cm}, \quad h_{B}=h_{t} / 2=0.15 \mathrm{~cm}
$$
and
$$
h_{\text{water}}=h_{t}+h_{v} / 2
$$

![](https://cdn.mathpix.com/cropped/2024_05_02_f9d684784733b717d934g-1.jpg?height=368&width=420&top_left_y=323&top_left_x=836)

Figure 1

The mass of the side wall (if $\varrho$ is the density of the water):

$$
m_{\text{cylinder}}=2 h \pi\left(R^{2}-r^{2}\right) \varrho
$$

The mass of the base:

$$
m_{\text{base}}=2 R^{2} \pi h_{t} \varrho
$$

The mass of the water column of height $h_{v}$ in the cup:

$$
m_{\text{water}}=r^{2} \pi h_{v} \varrho
$$

With the above information, the height $y$ of the center of mass of the system (relative to the base) can be expressed as a function of the water height:

$$
y=\frac{h_{\text{water}} m_{\text{water}}+h_{A} m_{\text{cylinder}}+h_{B} m_{\text{base}}}{m_{\text{cylinder}}+m_{\text{base}}+m_{\text{water}}}
$$

Substituting the given numerical values, the following relationship is obtained:

$$
y=\frac{17 \mathrm{~cm}^{-1} h_{v}^{2}+340.5 \mathrm{~cm}}{34.21 \mathrm{~cm}^{-1} h_{v}+86.05}
$$

![](https://cdn.mathpix.com/cropped/2024_05_02_f9d684784733b717d934g-1.jpg?height=512&width=464&top_left_y=1690&top_left_x=814)

The graph of the function $y(h_{v})$ is shown in Figure 2. The function has a minimum value of $y_{\min }=2.75 \mathrm{~cm}$.

Based on the solution by Péter Költl (Győr, Révai Miklós Gymnasium, 2nd year).