Forces due to Curved Surfaces

A plane surface exerts forces in the plane of the surface, as shown in Fig. 1,10. A curved surface can also exert a net force at right angles to itself, as shown in Fig. 17.8. 

Here we see a small piece of liquid surface perpendicular to the z axis, whose projection on the xy plane is the small rectangle Ajt by Ay. The forces exerted by the surface along its four edges are F = a Ay for the two edges perpendicular to the x axis and F = t Ax for the two edges perpendicular to the y axis. Assuming that this piece of surface is symmetric about the x and y axes, we can see that the x and y components of the force on the surface are zero, because the plus and minus parts cancel. 

To find the z component of this force, we sum the z components of the four surface forces on the edges to find 

Example 17.1. Evaluate the force per unit area due to surfaces in the shape of (n) a sphere, (b) a cylinder, and (c) a plane. j 

In the calculus it is proved that the. radius of curvature of any curve on the xy plane is given by 

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