Exercises: Bernoulli's Equation

Write Bernoulli's equation for an ideal fluid that flows between two points 1 and 2. Identify which term is related to pressure, which to kinetic energy per unit volume, and which to gravitational potential energy per unit volume.

Water flows through a horizontal pipe that narrows. Compared with the wide section, is the pressure in the narrow section greater, smaller, or the same? Explain in one or two sentences using Bernoulli's equation.

Water flows through a horizontal pipe. At point 1 the pressure is $1.50\times10^{5}\ \text{Pa}$ and the speed is $2.0\ \text{m/s}$. At point 2 the speed is $4.0\ \text{m/s}$. Find the pressure at point 2.

Water flows through a pipe that rises. At the lower point the gauge pressure is $2.0\times10^{5}\ \text{Pa}$, the speed is $1.0\ \text{m/s}$, and the height is $0$. At the upper point the height is $5.0\ \text{m}$ and the speed is still $1.0\ \text{m/s}$. Find the gauge pressure at the upper point.

A large, open water tank in Houston has a small hole $3.0\ \text{m}$ below the water surface. The tank is open to the atmosphere, and the surface is large enough that its downward speed is negligible.

(a) Use Bernoulli's equation to find the speed of water leaving the hole.

(b) Explain why the answer does not depend on the size of the hole, as long as the hole is small.

Water flows through a horizontal pipe. A wide section has a cross-sectional area of $0.030\ \text{m}^2$ and a speed of $1.0\ \text{m/s}$. The pipe narrows to an area of $0.010\ \text{m}^2$.

(a) Find the speed in the narrow section.

(b) Find the pressure difference between the wide and narrow sections.

A horizontal pipe carries water. In the wide section the cross-sectional area is $4.0\times10^{-3}\ \text{m}^2$ and the absolute pressure is $1.80\times10^{5}\ \text{Pa}$. In the narrow section the area is $1.0\times10^{-3}\ \text{m}^2$. The volume flow rate is $8.0\times10^{-3}\ \text{m}^3/\text{s}$.

(a) Find the water speed in each section.

(b) Find the absolute pressure in the narrow section.

(c) Describe what would happen to the pressure in the narrow section if the volume flow rate were doubled, and state whether the pressure could physically drop below zero in a real pipe. Justify your answer.

Water flows from a large open tank through a pipe that first runs horizontally at the level of the tank bottom, and then rises. The tank's water surface is $6.0\ \text{m}$ above the horizontal pipe. The tank is open to the atmosphere, and its surface speed is negligible. The water leaves the end of the pipe, which is open to the atmosphere, at a point $2.0\ \text{m}$ above the horizontal section.

(a) Find the speed of water leaving the end of the pipe.

(b) The pipe has the same cross-sectional area everywhere. Find the gauge pressure in the horizontal section.

(c) Explain why the gauge pressure at the top of the rising pipe is lower than in the horizontal section, even though the speed is the same.