91影视

• The water is flowing at a rate of$\frac{dV}{dt}=7200c{m}^3/s$.
• The absolute pressure of water is$P_1=2.40脳{10}^{5}Pa$.
• The radius at one point is$r_1=4cm$.
• The radius at second point is$r_2=2cm$.

In this problem, Bernoulli鈥檚 equation will be applied to evaluate the absolute pressure. In this case, the pipe is horizontal, so the vertical components will be equal to zero.

The relation of speed can be written as:

$\begin{array}{r}{V}_1=\frac{\left(\frac{dV}{dt}\right)}{A}\\ V_1=\frac{\left(\frac{dV}{dt}\right)}{蟺r_1^2}\end{array}$

Substitute$7200c{m}^3/s$ for$\frac{dV}{dt}$ and$4cm$for $r_1$in the above relation.

$\begin{array}{r}{v}_1=\frac{\left(7200{\mathrm{cm}}^3/\mathrm{s}脳\frac{1{\mathrm{m}}^3/\mathrm{s}}{{10}^6{\mathrm{cm}}^3/\mathrm{s}}\right)}{蟺{\left(4\mathrm{cm}脳\frac{1\mathrm{m}}{100\mathrm{cm}}\right)}^2}\\ v_1=1.43\mathrm{m}/\mathrm{s}\end{array}$

The relation of speed can be written as:

$\begin{array}{r}{A}_1{V}_1=A_2{v}_2\\ V_2=\left(\frac{\left(蟺脳r_1^2\right)v_1}{\left(蟺脳r_2^2\right)}\right)\end{array}$

Substitute 1.43 m/s for$v_1$ , 4 cm for$r_1$ and$2cm$ for$r_2$ in the above relation.

$\begin{array}{r}{v}_2=\left(\frac{\left(蟺脳{\left(4\mathrm{cm}脳\frac{1\mathrm{m}}{100\mathrm{cm}}\right)}^2\right)(1.43\mathrm{m}/\mathrm{s})}{\left(蟺脳{\left(2\mathrm{cm}脳\frac{1\mathrm{m}}{100\mathrm{cm}}\right)}^2\right)}\right)\\ v_2=5.72\mathrm{m}/\mathrm{s}\end{array}$

The relation of absolute pressure of water can be written as:

$\begin{array}{r}{P}_1+\frac{1}{2}蚁{v_1}^2=P_2+\frac{1}{2}蚁{v_2}^2\\ P_2=P_1+\frac{1}{2}蚁\left(v_1^2鈭�{v_2}^2\right)\end{array}$

Here,$蚁$is the density of water and$P_2$is the required pressure.

Substitute $2.40脳脳{10}^{5}Pa$for $P_1$, $1000kg/m^3$for $蚁$, 1.43 m/s for $v_1$and$5.72m/s$ for in the above relation.

$\begin{array}{r}{P}_2=2.40脳{10}^5\mathrm{Pa}+\frac{1}{2}\left(1000\mathrm{kg}/{\mathrm{m}}^3\right)\left((1.43\mathrm{m}/\mathrm{s}{)}^2鈭�(5.72\mathrm{m}/\mathrm{s}{)}^2\right)\\ P_2=2.24脳{10}^5\mathrm{Pa}\end{array}$

Thus, the absolute pressure is$2.24脳{10}^{5}Pa$.