# As shown in following figure (a), a rightcircular cylinder partially filled with fluid is rotated with a constant angular velocity Ï about a vertical y-axis through its center. The rotating fluid forms a surface of revolution S. To identify S, we first establish a coordinate system consisting of a vertical plane determined by the y-axis and an x-axis drawn perpendicular

Chapter 1, Exerise Questions #87
As shown in following figure (a), a rightcircular cylinder partially filled with fluid is rotated with a constant angular velocity Ï‰ about a vertical y-axis through its center. The rotating fluid forms a surface of revolution S. To identify S, we first establish a coordinate system consisting of a vertical plane determined by the y-axis and an x-axis drawn perpendicular to the y-axis such that the point of intersection of the axes (the origin) is located at the lowest point on the surface S. We then seek a function y = f (x) that represents the curve C of intersection of the surface S and the vertical coordinate plane. Let the point P(x, y) denote the position of a particle of the rotating fluid of mass m in the coordinate plane. See the following figure (b).

(a) At P there is a reaction force of magnitude F due to the other particles of the fluid which is normal to the surface S. By Newton€™s second law the magnitude of the net force acting on the particle is mÏ‰2x. What is this force? Use the following figure (b) to discuss the nature and origin of the equations

FcosÎ¸ = mg, FsinÎ¸ = mÏ‰2x

(b) Use part (a) to find a first-order differential equation that defines the function y = f (x).