Spherical Shell

Solid enclosed between two concentric spheres is called a spherical shell. For a spherical shell if R and r are outer and inner radii respectively, then the volume of the shell is

 = \frac{4}{3}\pi \left( {{R^3} - {r^3}} \right)


 = \frac{\pi }{6}\left( {{D^3} - {d^3}} \right)


A sphere of radius 5cm is dropped into a cylindrical vessel partly filled with water. The diameter of the vessel is 10cm. If the sphere is completely submerged, by how much will the surface of water be raised?


The sphere will displace a volume of water equal to that of itself and this amount of water will go up.
\therefore   Volume of sphere  = \frac{4}{3}\pi {r^3} = \frac{4}{3}\pi {\left( {\frac{5}{2}} \right)^3} = \frac{{125}}{6}\pi cu.cm

The radius of the cylinder vessel  = \frac{{10}}{2} = 5cm

The volume height occupied by \frac{{125}}{6}\pi cu.cm. of water in a cylindrical vessel of 5cm radius  = \frac{{125}}{6}\pi + \pi {\left( 5 \right)^3}

 = \frac{{125}}{{6 \times 25}} = \frac{5}{6}cm


A solid cylinder of glass, the radius of whose base is 9cm and height 12cm is melted and turned into sphere. Find the radius of the sphere so formed.


Volume of the cylinder  = \pi {r^2}h = \pi \times {\left( 9 \right)^2} \times 12
 = 972\pi cu.cm       --- (1)

Volume of the sphere    = \frac{4}{3}\pi {r^3}                 --- (2)

By the given condition,
(1) = (2)  \Rightarrow \frac{4}{3}\pi {r^3} = 972\pi
 \Rightarrow {r^3} = 729 \Rightarrow r = 9