Derive the equation for the shape of the surface, Mechanical Engineering

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(Rigid body rotation) Derive the equation for the shape of the surface zsurf(rsurf) for a rigid-body rotation of water at angular speed (rad/sec). Follow this process:

1) Start with pressure gradient in cylindrical coordinates:

2) Write out force balance in r and z directions

3) Integrate using the method we did in class: F = ma in r-direction gives: P(r,z) = P(r) + f1(z) F = ma in z-direction gives: P(r,z) = P(z) + f2(r)

4) Solve for z(r) at surface, where P = Patm = constant

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