In about 1657, Otto von Guericke, inventor of the air pump,...

In about $1657,$ Otto von Guericke, inventor of the air pump, cvacuated a sphere made of two brass hemi- spheres. Two teams of eight horses each could pull the hemispheres apart only on some trials and then "with greatest difficulty," with the resulting sound likened to a cannon firing (Fig. Pl4.60). (a) Show that the force $F$ required to pull the thin-walled evacuated hemispheres apart is $\pi R^{2}\left(P_{0}-P\right),$ where $R$ is the radius of the hemispheres and $P$ is the pressure inside the hemispheres, which is much less than $P_{0}$ . (b) Determine the force for $P=0.100 P_{0}$ and $R=0.300 \mathrm{m} .$

In about $1657,$ Otto von Guericke, inventor of the air pump, cvacuated a sphere made of two brass hemi-
spheres. Two teams of eight horses each could pull the hemispheres apart only on some trials and then "with greatest difficulty," with the resulting sound likened to a cannon firing (Fig. Pl4.60). (a) Show that the force $F$ required to pull the thin-walled evacuated hemispheres apart is $\pi R^{2}\left(P_{0}-P\right),$ where $R$ is the radius of the hemispheres and $P$ is the pressure inside the hemispheres, which is much less than $P_{0}$ . (b) Determine the force for $P=0.100 P_{0}$ and $R=0.300 \mathrm{m} .$

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Key Concepts

Integration Over a Curved Surface

Calculating the net force on a curved surface involves integrating the pressure over the surface. However, by taking advantage of symmetry about an axis, the integration can be simplified by considering the projected area of the curved surface, which in many cases leads to the use of simple geometric formulas such as the area of a circle (?R²). This technique is central in problems involving pressure forces on surfaces with curved geometries.

Atmospheric Pressure and Vacuum Conditions

The concept of atmospheric pressure, or external pressure, and its contrast with lower internal or vacuum pressures is fundamental in explaining phenomena where force is exerted due to pressure imbalance. In scenarios involving evacuated spaces, understanding the relative magnitudes of ambient and internal pressures allows one to predict and calculate the resultant forces acting on structural components.

Pressure Differential

This concept refers to the difference in pressure between two regions, which is the driving factor for net force in many physical situations. When there is a discrepancy between the internal and external pressures acting on a surface, the resulting force is determined by multiplying the pressure difference by the area over which it acts.

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