A cylindrical container of an incompressible liquid with density p rotates with constant angular speed w about its axis of symmetry, which we take to be the y-axis (Fig. 14.41).
(a) Show that the pressure at a given height within the fluid increases in the radial direction (outward from the axis of rotation) according to ∂p/∂r = pw2r.
(b) Integrate this partial differential equation to find the pressure as a function of distance from the axis of rotation along a horizontal line at y = O.
(c) Combine the result of part (b) with Eq. (14.5) to show that the surface of the rotating liquid has a parabolic shape, that is, the height of the liquid is given by h( r) = w2r2/2g. (This technique is used for matching parabolic telescope mirrors; liquid glass is rotated and allowed to solidify while rotating.)