• Home
  • Textbooks
  • Solving Practical Engineering Mechanics Problems Fluid Mechanics.
  • Topic FM-5: Determination of Hydrostatic Force on a Surface and Buoyancy

Solving Practical Engineering Mechanics Problems Fluid Mechanics.

Sayavur I. Bakhtiyarov

Chapter 5

Topic FM-5: Determination of Hydrostatic Force on a Surface and Buoyancy - all with Video Answers

Educators

Section 1

Problems

Problem 1

A fluid flow is defined by $u=f_1(x, y)$ and $v=f_2(\mathrm{x}, y)$ (Table 5,1). Determine the equation of the streamline passing through point $M\left(x_M y_M\right)$. Also, find the acceleration components of the particle at point $M\left(x_{M_1} y_M\right)$ and sketch the acceleration on the streamline.

$$
\begin{aligned}
&\begin{array}{|c|c|c|c|}
\hline u=f_1\left(x_i y\right), \frac{m}{s} & y=f_2\left(x_i y\right) / \frac{\mathrm{m}}{\mathrm{s}} & x_m \cdot m & y_m m \\

\hline 3 y^2 & 5 x-1 & 1 & 2 \\

\hline
\end{array}\\
\end{aligned}
$$

Check back soon!

Problem 2

A fluid flow is defined by $u=f_1(x, y)$ and $v=f_2(\mathrm{x}, y)$ (Table 5,1). Determine the equation of the streamline passing through point $M\left(x_M y_M\right)$. Also, find the acceleration components of the particle at point $M\left(x_{M_1} y_M\right)$ and sketch the acceleration on the streamline.

$$
\begin{aligned}
&\begin{array}{|c|c|c|c|}
\hline u=f_1\left(x_i y\right), \frac{m}{s} & y=f_2\left(x_i y\right) / \frac{\mathrm{m}}{\mathrm{s}} & x_m \cdot m & y_m m \\

\hline 2 x-3 & 6 y+4 x & 1 & 1 \\

\hline
\end{array}\\
\end{aligned}
$$

Check back soon!

Problem 3

A fluid flow is defined by $u=f_1(x, y)$ and $v=f_2(\mathrm{x}, y)$ (Table 5,1). Determine the equation of the streamline passing through point $M\left(x_M y_M\right)$. Also, find the acceleration components of the particle at point $M\left(x_{M_1} y_M\right)$ and sketch the acceleration on the streamline.

$$
\begin{aligned}
&\begin{array}{|c|c|c|c|}
\hline u=f_1\left(x_i y\right), \frac{m}{s} & y=f_2\left(x_i y\right) / \frac{\mathrm{m}}{\mathrm{s}} & x_m \cdot m & y_m m \\

\hline 3 x^2-2 y^2 & 8 x y & 2 & 2 \\

\hline
\end{array}\\
\end{aligned}
$$

Check back soon!

Problem 4

A fluid flow is defined by $u=f_1(x, y)$ and $v=f_2(\mathrm{x}, y)$ (Table 5,1). Determine the equation of the streamline passing through point $M\left(x_M y_M\right)$. Also, find the acceleration components of the particle at point $M\left(x_{M_1} y_M\right)$ and sketch the acceleration on the streamline.

$$
\begin{aligned}
&\begin{array}{|c|c|c|c|}
\hline u=f_1\left(x_i y\right), \frac{m}{s} & y=f_2\left(x_i y\right) / \frac{\mathrm{m}}{\mathrm{s}} & x_m \cdot m & y_m m \\

\hline 4 x+6 y & 8 y & 1 & 1 \\

\hline
\end{array}\\
\end{aligned}
$$

Check back soon!

Problem 5

A fluid flow is defined by $u=f_1(x, y)$ and $v=f_2(\mathrm{x}, y)$ (Table 5,1). Determine the equation of the streamline passing through point $M\left(x_M y_M\right)$. Also, find the acceleration components of the particle at point $M\left(x_{M_1} y_M\right)$ and sketch the acceleration on the streamline.

$$
\begin{aligned}
&\begin{array}{|c|c|c|c|}
\hline u=f_1\left(x_i y\right), \frac{m}{s} & y=f_2\left(x_i y\right) / \frac{\mathrm{m}}{\mathrm{s}} & x_m \cdot m & y_m m \\

\hline 9 x y & -3 x & 3 & 2 \\

\hline
\end{array}\\
\end{aligned}
$$

Check back soon!

Problem 6

A fluid flow is defined by $u=f_1(x, y)$ and $v=f_2(\mathrm{x}, y)$ (Table 5,1). Determine the equation of the streamline passing through point $M\left(x_M y_M\right)$. Also, find the acceleration components of the particle at point $M\left(x_{M_1} y_M\right)$ and sketch the acceleration on the streamline.

$$
\begin{aligned}
&\begin{array}{|c|c|c|c|}
\hline u=f_1\left(x_i y\right), \frac{m}{s} & y=f_2\left(x_i y\right) / \frac{\mathrm{m}}{\mathrm{s}} & x_m \cdot m & y_m m \\

\hline 3 y^3-1 & 2 x^2+6 & 2 & 2 \\

\hline
\end{array}\\
\end{aligned}
$$

Check back soon!

Problem 7

A fluid flow is defined by $u=f_1(x, y)$ and $v=f_2(\mathrm{x}, y)$ (Table 5,1). Determine the equation of the streamline passing through point $M\left(x_M y_M\right)$. Also, find the acceleration components of the particle at point $M\left(x_{M_1} y_M\right)$ and sketch the acceleration on the streamline.

$$
\begin{aligned}
&\begin{array}{|c|c|c|c|}
\hline u=f_1\left(x_i y\right), \frac{m}{s} & y=f_2\left(x_i y\right) / \frac{\mathrm{m}}{\mathrm{s}} & x_m \cdot m & y_m m \\

\hline 2 x-3 y & 5 x^2-1 & 3 & 3 \\

\hline
\end{array}\\
\end{aligned}
$$

Check back soon!

Problem 8

A fluid flow is defined by $u=f_1(x, y)$ and $v=f_2(\mathrm{x}, y)$ (Table 5,1). Determine the equation of the streamline passing through point $M\left(x_M y_M\right)$. Also, find the acceleration components of the particle at point $M\left(x_{M_1} y_M\right)$ and sketch the acceleration on the streamline.

$$
\begin{aligned}
&\begin{array}{|c|c|c|c|}
\hline u=f_1\left(x_i y\right), \frac{m}{s} & y=f_2\left(x_i y\right) / \frac{\mathrm{m}}{\mathrm{s}} & x_m \cdot m & y_m m \\

\hline 3 x-2 y^2 & 6 y+x & 1 & 3 \\

\hline
\end{array}\\
\end{aligned}
$$

Check back soon!

Problem 9

A fluid flow is defined by $u=f_1(x, y)$ and $v=f_2(\mathrm{x}, y)$ (Table 5,1). Determine the equation of the streamline passing through point $M\left(x_M y_M\right)$. Also, find the acceleration components of the particle at point $M\left(x_{M_1} y_M\right)$ and sketch the acceleration on the streamline.

$$
\begin{aligned}
&\begin{array}{|c|c|c|c|}
\hline u=f_1\left(x_i y\right), \frac{m}{s} & y=f_2\left(x_i y\right) / \frac{\mathrm{m}}{\mathrm{s}} & x_m \cdot m & y_m m \\

\hline 2 x+12 y & 5 x-y & 1 & 1 \\
\hline
\end{array}\\
\end{aligned}
$$

Check back soon!

Problem 9

A fluid flow is defined by $u=f_1(x, y)$ and $v=f_2(\mathrm{x}, y)$ (Table 5,1). Determine the equation of the streamline passing through point $M\left(x_M y_M\right)$. Also, find the acceleration components of the particle at point $M\left(x_{M_1} y_M\right)$ and sketch the acceleration on the streamline.

$$
\begin{aligned}
&\begin{array}{|c|c|c|c|}
\hline u=f_1\left(x_i y\right), \frac{m}{s} & y=f_2\left(x_i y\right) / \frac{\mathrm{m}}{\mathrm{s}} & x_m \cdot m & y_m m \\

\hline 4 x+6 y^2 & 8 x y+2 & 2 & 3 \\

\hline
\end{array}\\
\end{aligned}
$$

Check back soon!

Problem 10

A fluid flow is defined by $u=f_1(x, y)$ and $v=f_2(\mathrm{x}, y)$ (Table 5,1). Determine the equation of the streamline passing through point $M\left(x_M y_M\right)$. Also, find the acceleration components of the particle at point $M\left(x_{M_1} y_M\right)$ and sketch the acceleration on the streamline.

$$
\begin{aligned}
&\begin{array}{|c|c|c|c|}
\hline u=f_1\left(x_i y\right), \frac{m}{s} & y=f_2\left(x_i y\right) / \frac{\mathrm{m}}{\mathrm{s}} & x_m \cdot m & y_m m \\

\hline 6 x y+1 & 3-8 y & 3 & 1 \\

\hline
\end{array}\\
\end{aligned}
$$

Check back soon!

Problem 10

A fluid flow is defined by $u=f_1(x, y)$ and $v=f_2(\mathrm{x}, y)$ (Table 5,1). Determine the equation of the streamline passing through point $M\left(x_M y_M\right)$. Also, find the acceleration components of the particle at point $M\left(x_{M_1} y_M\right)$ and sketch the acceleration on the streamline.

$$
\begin{aligned}
&\begin{array}{|c|c|c|c|}
\hline u=f_1\left(x_i y\right), \frac{m}{s} & y=f_2\left(x_i y\right) / \frac{\mathrm{m}}{\mathrm{s}} & x_m \cdot m & y_m m \\

\hline 9 x y-8 & y+4 x & 3 & 2 \\
\hline
\end{array}\\
\end{aligned}
$$

Check back soon!

Problem 11

A fluid flow is defined by $u=f_1(x, y)$ and $v=f_2(\mathrm{x}, y)$ (Table 5,1). Determine the equation of the streamline passing through point $M\left(x_M y_M\right)$. Also, find the acceleration components of the particle at point $M\left(x_{M_1} y_M\right)$ and sketch the acceleration on the streamline.

$$
\begin{aligned}
&\begin{array}{|c|c|c|c|}
\hline u=f_1\left(x_i y\right), \frac{m}{s} & y=f_2\left(x_i y\right) / \frac{\mathrm{m}}{\mathrm{s}} & x_m \cdot m & y_m m \\

\hline 3 y^2-4 & -9 x^2 & 2 & 1 \\

\hline
\end{array}\\
\end{aligned}
$$

Check back soon!

Problem 12

A fluid flow is defined by $u=f_1(x, y)$ and $v=f_2(\mathrm{x}, y)$ (Table 5,1). Determine the equation of the streamline passing through point $M\left(x_M y_M\right)$. Also, find the acceleration components of the particle at point $M\left(x_{M_1} y_M\right)$ and sketch the acceleration on the streamline.

$$
\begin{aligned}
&\begin{array}{|c|c|c|c|}
\hline u=f_1\left(x_i y\right), \frac{m}{s} & y=f_2\left(x_i y\right) / \frac{\mathrm{m}}{\mathrm{s}} & x_m \cdot m & y_m m \\

\hline 2 x-3 y & 2 x^2 & 1 & 2 \\

\hline
\end{array}\\
\end{aligned}
$$

Check back soon!

Problem 13

A fluid flow is defined by $u=f_1(x, y)$ and $v=f_2(\mathrm{x}, y)$ (Table 5,1). Determine the equation of the streamline passing through point $M\left(x_M y_M\right)$. Also, find the acceleration components of the particle at point $M\left(x_{M_1} y_M\right)$ and sketch the acceleration on the streamline.

$$
\begin{aligned}
&\begin{array}{|c|c|c|c|}
\hline u=f_1\left(x_i y\right), \frac{m}{s} & y=f_2\left(x_i y\right) / \frac{\mathrm{m}}{\mathrm{s}} & x_m \cdot m & y_m m \\

\hline 3 x^2-2 y & 5 x-y & 2 & 2 \\
\hline
\end{array}\\
\end{aligned}
$$

Check back soon!

Problem 14

A fluid flow is defined by $u=f_1(x, y)$ and $v=f_2(\mathrm{x}, y)$ (Table 5,1). Determine the equation of the streamline passing through point $M\left(x_M y_M\right)$. Also, find the acceleration components of the particle at point $M\left(x_{M_1} y_M\right)$ and sketch the acceleration on the streamline.

$$
\begin{aligned}
&\begin{array}{|c|c|c|c|}
\hline u=f_1\left(x_i y\right), \frac{m}{s} & y=f_2\left(x_i y\right) / \frac{\mathrm{m}}{\mathrm{s}} & x_m \cdot m & y_m m \\

\hline x+2 y & y+x & 1 & 1 \\
\hline
\end{array}\\
\end{aligned}
$$

Check back soon!

Problem 15

A fluid flow is defined by $u=f_1(x, y)$ and $v=f_2(\mathrm{x}, y)$ (Table 5,1). Determine the equation of the streamline passing through point $M\left(x_M y_M\right)$. Also, find the acceleration components of the particle at point $M\left(x_{M_1} y_M\right)$ and sketch the acceleration on the streamline.

$$
\begin{aligned}
&\begin{array}{|c|c|c|c|}
\hline u=f_1\left(x_i y\right), \frac{m}{s} & y=f_2\left(x_i y\right) / \frac{\mathrm{m}}{\mathrm{s}} & x_m \cdot m & y_m m \\

\hline 9 x y-6 & 8-x^2 & 1 & 4 \\
\hline
\end{array}\\
\end{aligned}
$$

Check back soon!

Problem 16

A fluid flow is defined by $u=f_1(x, y)$ and $v=f_2(\mathrm{x}, y)$ (Table 5,1). Determine the equation of the streamline passing through point $M\left(x_M y_M\right)$. Also, find the acceleration components of the particle at point $M\left(x_{M_1} y_M\right)$ and sketch the acceleration on the streamline.

$$
\begin{aligned}
&\begin{array}{|c|c|c|c|}
\hline u=f_1\left(x_i y\right), \frac{m}{s} & y=f_2\left(x_i y\right) / \frac{\mathrm{m}}{\mathrm{s}} & x_m \cdot m & y_m m \\

\hline 3 y^2+2 & 8 . x^2 & 1 & 2 \\
\hline
\end{array}\\
\end{aligned}
$$

Check back soon!

Problem 17

A fluid flow is defined by $u=f_1(x, y)$ and $v=f_2(\mathrm{x}, y)$ (Table 5,1). Determine the equation of the streamline passing through point $M\left(x_M y_M\right)$. Also, find the acceleration components of the particle at point $M\left(x_{M_1} y_M\right)$ and sketch the acceleration on the streamline.

$$
\begin{aligned}
&\begin{array}{|c|c|c|c|}
\hline u=f_1\left(x_i y\right), \frac{m}{s} & y=f_2\left(x_i y\right) / \frac{\mathrm{m}}{\mathrm{s}} & x_m \cdot m & y_m m \\

\hline 8 x-y^2 & -9 x & 1 & 1 \\
\hline
\end{array}\\
\end{aligned}
$$

Check back soon!

Problem 18

A fluid flow is defined by $u=f_1(x, y)$ and $v=f_2(\mathrm{x}, y)$ (Table 5,1). Determine the equation of the streamline passing through point $M\left(x_M y_M\right)$. Also, find the acceleration components of the particle at point $M\left(x_{M_1} y_M\right)$ and sketch the acceleration on the streamline.

$$
\begin{aligned}
&\begin{array}{|c|c|c|c|}
\hline u=f_1\left(x_i y\right), \frac{m}{s} & y=f_2\left(x_i y\right) / \frac{\mathrm{m}}{\mathrm{s}} & x_m \cdot m & y_m m \\

\hline x^2-2 y & 2 x^2+2 & 2 & 2 \\
\hline
\end{array}\\
\end{aligned}
$$

Check back soon!

Problem 21

A fluid flow is defined by $u=f_1(x, y)$ and $v=f_2(\mathrm{x}, y)$ (Table 5,1). Determine the equation of the streamline passing through point $M\left(x_M y_M\right)$. Also, find the acceleration components of the particle at point $M\left(x_{M_1} y_M\right)$ and sketch the acceleration on the streamline.

$$
\begin{aligned}
&\begin{array}{|c|c|c|c|}
\hline u=f_1\left(x_i y\right), \frac{m}{s} & y=f_2\left(x_i y\right) / \frac{\mathrm{m}}{\mathrm{s}} & x_m \cdot m & y_m m \\

\hline 3 y^2+5 & 8 x & 2 & 2 \\

\hline
\end{array}\\
\end{aligned}
$$

Check back soon!

Problem 22

A fluid flow is defined by $u=f_1(x, y)$ and $v=f_2(\mathrm{x}, y)$ (Table 5,1). Determine the equation of the streamline passing through point $M\left(x_M y_M\right)$. Also, find the acceleration components of the particle at point $M\left(x_{M_1} y_M\right)$ and sketch the acceleration on the streamline.

$$
\begin{aligned}
&\begin{array}{|c|c|c|c|}
\hline u=f_1\left(x_i y\right), \frac{m}{s} & y=f_2\left(x_i y\right) / \frac{\mathrm{m}}{\mathrm{s}} & x_m \cdot m & y_m m \\

\hline 2 x-3 y^3 & y^2 & 3 & 3 \\

\hline
\end{array}\\
\end{aligned}
$$

Check back soon!

Problem 23

A fluid flow is defined by $u=f_1(x, y)$ and $v=f_2(\mathrm{x}, y)$ (Table 5,1). Determine the equation of the streamline passing through point $M\left(x_M y_M\right)$. Also, find the acceleration components of the particle at point $M\left(x_{M_1} y_M\right)$ and sketch the acceleration on the streamline.

$$
\begin{aligned}
&\begin{array}{|c|c|c|c|}
\hline u=f_1\left(x_i y\right), \frac{m}{s} & y=f_2\left(x_i y\right) / \frac{\mathrm{m}}{\mathrm{s}} & x_m \cdot m & y_m m \\

\hline 7 x^2-y^2 & -x & 1 & 3 \\

\hline
\end{array}\\
\end{aligned}
$$

Check back soon!

Problem 23

A fluid flow is defined by $u=f_1(x, y)$ and $v=f_2(\mathrm{x}, y)$ (Table 5,1). Determine the equation of the streamline passing through point $M\left(x_M y_M\right)$. Also, find the acceleration components of the particle at point $M\left(x_{M_1} y_M\right)$ and sketch the acceleration on the streamline.

$$
\begin{aligned}
&\begin{array}{|c|c|c|c|}
\hline u=f_1\left(x_i y\right), \frac{m}{s} & y=f_2\left(x_i y\right) / \frac{\mathrm{m}}{\mathrm{s}} & x_m \cdot m & y_m m \\

\hline 4 x^2+6 y & 2 x^2+3 & 2 & 3 \\

\hline
\end{array}\\
\end{aligned}
$$

Check back soon!

Problem 24

A fluid flow is defined by $u=f_1(x, y)$ and $v=f_2(\mathrm{x}, y)$ (Table 5,1). Determine the equation of the streamline passing through point $M\left(x_M y_M\right)$. Also, find the acceleration components of the particle at point $M\left(x_{M_1} y_M\right)$ and sketch the acceleration on the streamline.

$$
\begin{aligned}
&\begin{array}{|c|c|c|c|}
\hline u=f_1\left(x_i y\right), \frac{m}{s} & y=f_2\left(x_i y\right) / \frac{\mathrm{m}}{\mathrm{s}} & x_m \cdot m & y_m m \\

\hline 4 x^2+6 y & 2 x^2+3 & 2 & 3 \\

\hline
\end{array}\\
\end{aligned}
$$

Check back soon!

Problem 25

A fluid flow is defined by $u=f_1(x, y)$ and $v=f_2(\mathrm{x}, y)$ (Table 5,1). Determine the equation of the streamline passing through point $M\left(x_M y_M\right)$. Also, find the acceleration components of the particle at point $M\left(x_{M_1} y_M\right)$ and sketch the acceleration on the streamline.

$$
\begin{aligned}
&\begin{array}{|c|c|c|c|}
\hline u=f_1\left(x_i y\right), \frac{m}{s} & y=f_2\left(x_i y\right) / \frac{\mathrm{m}}{\mathrm{s}} & x_m \cdot m & y_m m \\

\hline 2-x y & 5 x+1 & 3 & 1 \\

\hline
\end{array}\\
\end{aligned}
$$

Check back soon!

Problem 26

A fluid flow is defined by $u=f_1(x, y)$ and $v=f_2(\mathrm{x}, y)$ (Table 5,1). Determine the equation of the streamline passing through point $M\left(x_M y_M\right)$. Also, find the acceleration components of the particle at point $M\left(x_{M_1} y_M\right)$ and sketch the acceleration on the streamline.

$$
\begin{aligned}
&\begin{array}{|c|c|c|c|}
\hline u=f_1\left(x_i y\right), \frac{m}{s} & y=f_2\left(x_i y\right) / \frac{\mathrm{m}}{\mathrm{s}} & x_m \cdot m & y_m m \\

\hline 3 y^2-x & 6 y+x & 2 & 1 \\

\hline
\end{array}\\
\end{aligned}
$$

Check back soon!

Problem 27

A fluid flow is defined by $u=f_1(x, y)$ and $v=f_2(\mathrm{x}, y)$ (Table 5,1). Determine the equation of the streamline passing through point $M\left(x_M y_M\right)$. Also, find the acceleration components of the particle at point $M\left(x_{M_1} y_M\right)$ and sketch the acceleration on the streamline.

$$
\begin{aligned}
&\begin{array}{|c|c|c|c|}
\hline u=f_1\left(x_i y\right), \frac{m}{s} & y=f_2\left(x_i y\right) / \frac{\mathrm{m}}{\mathrm{s}} & x_m \cdot m & y_m m \\

\hline 2 x-5 & 12 x y & 1 & 2 \\

\hline
\end{array}\\
\end{aligned}
$$

Check back soon!

Problem 28

A fluid flow is defined by $u=f_1(x, y)$ and $v=f_2(\mathrm{x}, y)$ (Table 5,1). Determine the equation of the streamline passing through point $M\left(x_M y_M\right)$. Also, find the acceleration components of the particle at point $M\left(x_{M_1} y_M\right)$ and sketch the acceleration on the streamline.

$$
\begin{aligned}
&\begin{array}{|c|c|c|c|}
\hline u=f_1\left(x_i y\right), \frac{m}{s} & y=f_2\left(x_i y\right) / \frac{\mathrm{m}}{\mathrm{s}} & x_m \cdot m & y_m m \\

\hline 6 x^2-2 y & 8+y & 2 & 2 \\

\hline
\end{array}\\
\end{aligned}
$$

Check back soon!

Problem 29

A fluid flow is defined by $u=f_1(x, y)$ and $v=f_2(\mathrm{x}, y)$ (Table 5,1). Determine the equation of the streamline passing through point $M\left(x_M y_M\right)$. Also, find the acceleration components of the particle at point $M\left(x_{M_1} y_M\right)$ and sketch the acceleration on the streamline.

$$
\begin{aligned}
&\begin{array}{|c|c|c|c|}
\hline u=f_1\left(x_i y\right), \frac{m}{s} & y=f_2\left(x_i y\right) / \frac{\mathrm{m}}{\mathrm{s}} & x_m \cdot m & y_m m \\

\hline 3 x^2+y & 9-x & 1 & 1 \\

\hline
\end{array}\\
\end{aligned}
$$

Check back soon!

Problem 30

A fluid flow is defined by $u=f_1(x, y)$ and $v=f_2(\mathrm{x}, y)$ (Table 5,1). Determine the equation of the streamline passing through point $M\left(x_M y_M\right)$. Also, find the acceleration components of the particle at point $M\left(x_{M_1} y_M\right)$ and sketch the acceleration on the streamline.

$$
\begin{aligned}
&\begin{array}{|c|c|c|c|}
\hline u=f_1\left(x_i y\right), \frac{m}{s} & y=f_2\left(x_i y\right) / \frac{\mathrm{m}}{\mathrm{s}} & x_m \cdot m & y_m m \\

\hline 6-9 x y & 2 x+6 & 1 & 4 \\

\hline
\end{array}\\
\end{aligned}
$$

Check back soon!