Use r_water = 1,000 kg/m³ in the following problems.

I. Warm-up Exercises

1. Water is flowing out the end of a pipe with a speed of 0.35 m/s.  What volume of water flows out the end of the pipe in a time of 55 s if the cross sectional area of the pipe is 0.0032 m²?

2. Water flows out the end of a pipe with a mass flow rate of 0.39 kg/s.  What is the volume flow rate?

3. What is the cross-sectional area of a pipe if the speed of water flowing through it is 1.2 m/s and the mass flow rate is 3.5 kg/s?

4. Water is flowing through a pipe with a speed of 35 cm/s in a region where the pipe has a radius of 2.7 cm.  What is the speed of  the water flow if the radius of the pipe decreases to 1.0 cm?  (Recall: For a circle, A = pr².)

5. Explain as clearly and completely as possible how Bernoulli’s equation can help explain how planes fly.

6. A large semi-truck is carrying a cargo which has a canvas tarpaulin (a large sheet of canvas tied over the cargo to keep it from blowing away as  the truck moves) pulled over its cargo.  When the truck is moving quickly down the interstate, the tarpaulin is seen to bulge upwards and outwards.  Explain why this is so.

II. Some Standards

7. A horizontal section of pipe contains water flowing at 30 cm/s with a pressure of 2.3 x 10⁵ Pa.  The cross-sectional  area of the pipe in this region is 5.3 cm².  The pipe then contains a section which moves vertically downward through a vertical distance of 2.3 m, at which point it connects with  another horizontal section of pipe which has a cross-sectional area of 2.5 cm².  (a) What is the speed of water flowing in the lower section of the pipe?  (b) What is the pressure of the water in the lower section? (Hint: Let y = 0 at the level of the lower section of pipe.)

8. An airplane travels horizontally with a constant velocity in a region where the density of air is 1.0 kg/m³.  Each of the  two wings of the plane has an area of 35 m².  If the speed of the air under the lower wing surface is 50 m/s, and that over the upper wing surface is 65 m/s, what is the weight  of the plane (in MKS units)?

III. So, you think you’re pretty good...?

9. You are at home and find yourself in an awful situation of having a terrible wind storm suddenly blow up with tornado-like conditions.  (a) What is the pressure difference Dp = p_inside – p_outside between the inside of your house and just above the roof outside your house if a 66 mph (30 m/s) wind is blowing over the roof? Use r_air = 1.3 kg/m³.  (Hint: Assume that y_inside = y_outside in Bernoulli’s equation.  What is the speed of air inside your house?)  (b) What force trying to lift the roof off of your house results from this wind if your roof has an area of about 200 m²?

10. A Venturi meter is a device used to determine the speed of fluid flow (of density r_pipe = 1350 kg/m³) in a pipe without disturbing the flow which we are attempting to measure.  The meter (shown in blue in the diagram)  is connected to a pipe of area A₁ at its two ends.  The speed of flow in the pipe is v.

The meter works by taking advantage of the difference in pressure between the points 1 and 2 due to the different cross sectional areas A₁ and A₂ of the meter. The dashed line shows the level of liquid (density r_tube = 982 kg/m³) in the U-tube when the fluid in the pipe is not  moving.  When it is moving, the liquid on the right side of the U-tube rises a distance Dy , while the liquid on the left of course lowers by the same amount.  Find the speed of fluid flow v in the original pipe (point 1) if the pipe radii are r₁ = 4.5 cm, r₂ = 2.2 cm, and Dy = 2.4 cm.  (Hint: The pressure at point a must be the same as the pressure at point b since they  are at the same height in the same fluid.  Do you remember the equation for the variation of pressure with depth from the last lecture?)