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If a vertical cylinder of water (or any other liquid)
Chapter 8, Problem 66CP(choose chapter or problem)
If a vertical cylinder of water (or any other liquid) rotates about its axis, as shown in FIGURE CP8.66, the surface forms a smooth curve. Assuming that the water rotates as a unit (i.e., all the water rotates with the same angular velocity), show that the shape of the surface is a parabola described by the equation \(z=\left(\omega^{2} / 2 g\right) r^{2}\).
Hint: Each particle of water on the surface is subject to only two forces: gravity and the normal force due to the water underneath it. The normal force, as always, acts perpendicular to the surface.
Questions & Answers
QUESTION:
If a vertical cylinder of water (or any other liquid) rotates about its axis, as shown in FIGURE CP8.66, the surface forms a smooth curve. Assuming that the water rotates as a unit (i.e., all the water rotates with the same angular velocity), show that the shape of the surface is a parabola described by the equation \(z=\left(\omega^{2} / 2 g\right) r^{2}\).
Hint: Each particle of water on the surface is subject to only two forces: gravity and the normal force due to the water underneath it. The normal force, as always, acts perpendicular to the surface.
ANSWER:
Step 1 of 4
In this problem, we have to find the expression in the z-direction. The free body diagram of this problem is given below.