This web page contains solved examples from statics, dynamics, linear and nonlinear vibrations, and acoustics. All examples are solved using the Python programming language.
Each material body can be divided into many smaller elementary parts of volume \(\Delta V_1, \Delta V_2, ..., \Delta V_n\) that have mass \(\Delta m_1, \Delta m_2, ..., \Delta m_n\).
The density of a body \(\rho\), ie its mass per unit volume, is defined by the expression:
\begin{eqnarray}
\rho &=& \lim_{\Delta v \rightarrow 0 } \frac{\Delta m}{\Delta V} = \frac{\mathrm{dm}}{\mathrm{dV}}.
\end{eqnarray}
The unit of density is \(\left[\frac{\mathrm{kg}}{\mathrm{m}^3}\right]\). If the density of the body is equal at each point the body is homogeneous. A body whose one dimension, i.e., thickness, is very small in comparison with the other two, i.e., its length and width, is called the material surface. If its density is equal at all points, then it is a homogeneous material surface. Examples are material surfaces, such as various sheets.
Figure 8.1 - Determination of the center of gravity of a homogeneous material surface.
A body whose two transverse dimensions are very small compared to its length is called a material curve. If the density is equal at all points of that body, it is a homogeneous material curve.
Figure 8.2 - Determination of the center of gravity of a homogeneous material curve.
Homogeneous gravitational field.
The Earth's gravitational field is determined by the acceleration vector of gravity, which is approximately directed toward the center of the Earth, so it is not constant in direction. The magnitude of the acceleration g depends on the geographical position and varies on the Earth's surface in the range of 9.79 to 9.83 m / s ^ 2. It changes even more with height because it is inversely proportional to the square of the distance from the center of the earth. However, even for the largest technical structures such as bridges, ships, skyscrapers, etc., the gravitational field can be considered homogeneous, ie the acceleration of gravity at all points of technical structures can be considered constant in direction and magnitude.
Definition of center of gravity
Each material body can be divided into a series of elementary parts \(\Delta V_1, \Delta V_2, ..., \Delta V_n\) in which the elemental weights act \(\Delta \vec{G}_1,\Delta \vec{G}_2,...,\Delta \vec{G}_n\) as shown in Figure 8.3.
Figure 8.3 The definition of the center of the gravity
The total body weight is G and the total volume is V, so:
\begin{eqnarray}
\vec{G} &=& \sum \Delta \vec{G}_i,\\
V &=& \sum \Delta V_i.
\end{eqnarray}
The center of gravity T is the grip of the resultant force \(\Delta \vec{G}_i\) ie. weight grip \(\vec{G}\). Turning the body in space changes the position of the forces \(\Delta \vec{G}_i\), relative to the body, but the position of the center of gravity T does not change. The position of the grip of the resultant G can be determined from the equality of the moment of the resultant and the sum of the moments of its components around the origin, ie:
\begin{eqnarray}
\vec{r}_T\times \vec{G} &=& \sum \vec{r}_i \times \Delta \vec{G}_i.
\end{eqnarray}
Since \(\vec{G}=-G\vec{k}\) and \(\Delta \vec{G}_i = -\Delta G_i \vec{k}\), the previous expression can be written as:
\begin{eqnarray}
-\left(\vec{r}_TG - \sum \vec{r}_i \times \Delta G_i\right)\times \vec{k} &=& 0,
\end{eqnarray}
or:
\begin{eqnarray}
\vec{r}_T &=& \frac{1}{G}\sum\vec{r}_i\Delta G_i.
\end{eqnarray}
If the body is divided into infinitely many infinitesimal parts, the above expression turns into:
\begin{eqnarray}
\vec{r}_T &=& \frac{1}{G}\int_G\vec{r}\mathrm{dG} = \frac{1}{mg}\int_m\vec{r}g\mathrm{dg}.
\end{eqnarray}
\begin{eqnarray}
\vec{r}_T &=& \vec{r}_s = \frac{1}{m}\int_m \vec{r}\mathrm{dm}.
\end{eqnarray}
\begin{eqnarray}
\vec{r}_T &=& \frac{1}{V}\int_V \vec{r}\mathrm{dV},\\
\vec{r}_T &=& \frac{\sum \vec{r}_i \Delta V_i}{\sum \Delta V_i}.
\end{eqnarray}
\begin{eqnarray}
x_T &=& \frac{1}{V}\int x\mathrm{dV}, \quad y_T =\frac{1}{V}\int y \mathrm{dV}, \quad z_T = \frac{1}{V}\int z \mathrm{dV}\\
x_T &=& \frac{\sum x_i \Delta V_i}{\sum \Delta V_i} \quad y_T = \frac{\sum y_i \Delta V_i}{\sum \Delta V_i} \quad z_T = \frac{\sum z_i \Delta V_i}{\sum \Delta V_i}
\end{eqnarray}
Center of gravity of homogeneous material surfaces. If a homogeneous material is a surface of constant thickness \(h\).
\begin{eqnarray}
x_T &=& \frac{1}{S}\int x \mathrm{dS}, y_T = \frac{1}{S}\int y \mathrm{dS}, z_T = \frac{1}{S}\int z \mathrm{dS},\\
x_T &=& \frac{\sum x_i \Delta S_i}{\sum \Delta S_i}, y_T = \frac{\sum y_i}{\sum \Delta S_i}, z_T = \frac{\sum z_i \Delta S_i}{\sum \Delta S_i}
\end{eqnarray}
where \(\Delta S_i\) are the surface area element and \(S\) is the total surface area.
Center of gravity of homogeneous material curves. A material line (curve) is an elongated body whose transverse dimensions are small compared to its length. If the density and cross-sectional area are constant, it is a homogeneous material curve. Then \(V = As, \mathrm{dV} = A \mathrm{ds}\) it follows:
\begin{eqnarray}
x_T &=& \frac{1}{s}\int_s x \mathrm{ds}, y_T = \frac{1}{s}\int_s y\mathrm{ds}, z_T = \frac{1}{s}\int_s z \mathrm{ds}\\
x_T &=& \frac{\sum x_i \Delta s_i}{\sum \Delta s_i}, y_T = \frac{y_i \Delta s_i}{\sum \Delta s_i}, z_T = \frac{\sum z_i \Delta s_i}{\sum \Delta s_i}.
\end{eqnarray}
Center of gravity of flat geometric figures.
The static moment of the surface of a flat figure shown in the Figure 8.4 are defined by the expressions:
\begin{eqnarray}
S_y &=& \int_A z \mathrm{dA} = z_T A,\\
S_z &=& \int_A y \mathrm{dA} = y_T A,
\end{eqnarray}
where \(S_y\) and \(S_z\) are the static moments of the surface about the y and z axes, respectively.
Figure 8.4 - Graphical representation of static moment of the surface.
A is the area of the figure, \(y_T\) and \(z_T\) are the coordinates of the center of gravity. The coordinates of the center of gravity, therefore, are determined by the expressions:
\begin{eqnarray}
y_T &=& \frac{1}{A}\int_A y \mathrm{dA}, \\
z_T &=& \frac{1}{A}\int_A z \mathrm{dA}
\end{eqnarray}
The center of gravity of a complex geometric figure is determined by the expression:
\begin{eqnarray}
y_T &=& \frac{\sum y_i \Delta A_i}{\sum \Delta A_i},\\
z_T &=& \frac{\sum z_i \Delta A_i}{\sum \Delta A_i}.
\end{eqnarray}
Pappus-Guildin rules. If a plane curve rotates about an axis in its plane, it describes a rotating body having a surface:
\begin{eqnarray}
O &=& 2\pi x_T l,
\end{eqnarray}
where \(x_T\) is the distance of the center of gravity of the curve from the axis of rotation, and \(l\) is the length of the curve. If a plane figure rotates about an axis in its plane, a rotating body with volume \(V\) is formed, ie:
\begin{eqnarray}
V &=& 2\pi x_T A
\end{eqnarray}
where \(A\) is the area of the figure and \(x_T\) is the distance of the center of gravity of the Figure 8.4 from the axis of rotation as shown in the figure.
Figure 8.5The cross-section of the body. Example for Pappus-Guldins theorem.