Exercises
1. At one point in a pipeline, the speed of water is \( 3 \mathrm{~m} / \mathrm{s} \) and the gauge pressure is \( 5 \times 10^{4} \mathrm{~Pa} \). Calculate the gauge pressure at a second point in the pipeline, \( 11 \mathrm{~m} \) lower than the first, if the pipe diameter at the second point is twice that at the first. (Density of water is \( 1000 \mathrm{~kg} / \mathrm{m}^{3} \) ).
2. A horizontal pipe \( 10.0 \mathrm{~cm} \) in diameter has a smooth reduction to a pipe \( 5.00 \mathrm{~cm} \) in diameter. If the pressure of the water in the larger pipe is \( 8.00 \times 10^{4} \mathrm{~Pa} \) and the pressure in the smaller pipe is \( 6.00 \times 10^{4} \mathrm{~Pa} \), at what rate does water flow through the pipe?
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3. A large water tank is completely filled with water. The tank is \( 3.0 \mathrm{~m} \) high and its surface is open. If a small hole is made on one side of the tank at a height of \( 0.8 \mathrm{~m} \), calculate the speed with which the water comes out of the pipe.
4. A typical capillary is \( 1 \mathrm{~mm} \) long and has radius of \( 2 \mu \mathrm{m} \). (a) If the pressure difference between its end is \( 2.66 \times 10^{3} \mathrm{~N} / \mathrm{m}^{2} \), find the average speed of the blood that flows through such a caplliary. (b) If it pumps \( 80 \mathrm{~cm}^{3} \) of blood through a person's body per second, how many capillaries does the body have? (Viscosity of blood \( =1.2 \times 10^{-3} \mathrm{PI} \) ).
5. The capillaries in a tree are typically \( 0.02 \mathrm{~mm} \) in radius. If the contact angle is \( 0^{\circ} \), find the maximum height to which water can rise in a tree under the influence of surface tension alone. (Density of water is \( 1000 \mathrm{~kg} / \mathrm{m}^{3} \) and surface tension of water at \( 20^{\circ} \mathrm{C} \) is \( 0.073 \mathrm{~N} / \mathrm{m} \) ).
6. A glass capillary whose radius is \( 0.5 \mathrm{~mm} \) is dipped in a dish of mercury, whose surface tension is \( 0.44 \mathrm{~N} / \mathrm{m} \). The contact angle is \( 140{ }^{\circ} \mathrm{C} \). Find the height of the mercury in the capillary relative to the mercury surface in the dish. (Density of mercury is \( 13.6 \times 10^{3} \mathrm{~kg} / \mathrm{m}^{3} \) ).
