Supporting a Sphere

A metal bowl is in the shape of a partial unit sphere centered on the origin. The bowl's mass is $1 \, \text{kg}$ , and gravity is $10 \, \text{m/s}^2$ in the $-z$ direction.

$\large{x = \cos \theta \, \sin \phi \\ y = \sin \theta \, \sin \phi \\ z = \cos \phi \\ 0 \leq \theta \leq 2 \pi \\ \frac{\pi}{6} \leq \phi \leq \pi}$

A force is applied normally at all points on the outside surface, such that each unit of surface area experiences an infinitesimal force of the same magnitude:

$\large\frac{d \, |F|}{d S} = \alpha{}$

In the above equation, the units on $\alpha$ are $\text{N/m}^2$ (it is therefore a pressure value). If the applied force keeps the bowl in stasis, what is the value of $\alpha$ ?