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2. Consider a hemispherical shell of radius R pictured below. Water of density p is fills the shell. To prevent the shell from lifting off the surface and allowing the water to escape, a series of bolts around a peripheral flange attach the shell to the ground. A small hole at the top of the shell maintains equal pressure between the water at the top of the shell and the air that surrounds the shell. (Assume that $P_{air} \ll P_{water}$.) (a) What downward force must the bolts apply to prevent the water from lifting the shell off the surface and leaking? (Assume the shell has negligible weight.) (b) Now imagine a flanged cylindrical shell of the same radius R and volume as the hemisphere in part (a). Like the hemisphere, the cylinder has a small hole at its top to equalize pressure. What force must the flange bolts provide to prevent the water from leaking? (c) Explain the physical reason why your answers to parts (a) and (b) are equal or unequal. The following integrals may be helpful: $\int \sin^2\theta \cos\theta \,d\theta = \frac{\sin^3\theta}{3}$ and $\int \sin\theta \cos^2\theta \,d\theta = -\frac{\cos^3\theta}{3}$

          2. Consider a hemispherical shell of radius R pictured below. Water of density p is fills the shell.
To prevent the shell from lifting off the surface and allowing the water to escape, a series of
bolts around a peripheral flange attach the shell to the ground. A small hole at the top of the
shell maintains equal pressure between the water at the top of the shell and the air that
surrounds the shell. (Assume that $P_{air} \ll P_{water}$.)
(a) What downward force must the bolts apply to prevent the water from lifting the shell off
the surface and leaking? (Assume the shell has negligible weight.)
(b) Now imagine a flanged cylindrical shell of the same radius R and volume as the
hemisphere in part (a). Like the hemisphere, the cylinder has a small hole at its top to
equalize pressure. What force must the flange bolts provide to prevent the water from
leaking?
(c) Explain the physical reason why your answers to parts (a) and (b) are equal or unequal.
The following integrals may be helpful:
$\int \sin^2\theta \cos\theta \,d\theta = \frac{\sin^3\theta}{3}$ and $\int \sin\theta \cos^2\theta \,d\theta = -\frac{\cos^3\theta}{3}$

2. Consider a hemispherical shell of radius R pictured below. Water of density p is fills the shell.
To prevent the shell from lifting off the surface and allowing the water to escape, a series of
bolts around a peripheral flange attach the shell to the ground. A small hole at the top of the
shell maintains equal pressure between the water at the top of the shell and the air that
surrounds the shell. (Assume that Pair≪ Pwater.)
(a) What downward force must the bolts apply to prevent the water from lifting the shell off
the surface and leaking? (Assume the shell has negligible weight.)
(b) Now imagine a flanged cylindrical shell of the same radius R and volume as the
hemisphere in part (a). Like the hemisphere, the cylinder has a small hole at its top to
equalize pressure. What force must the flange bolts provide to prevent the water from
leaking?
(c) Explain the physical reason why your answers to parts (a) and (b) are equal or unequal.
The following integrals may be helpful:
∫sin^2θcosθ dθ = (sin^3θ)/(3) and ∫sinθcos^2θ dθ = -(cos^3θ)/(3)

Added by Jason W.

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