A solid sphere of mass m and radius R rotating about its diameter. A solid cylinder of the same mass and same radius is also rotating about its geometrical axis with an angular speed twice that of the sphere. The ratio of their kinetic energies of rotation `((E_("sphere"))/(E_("cylinder")))` will be = ..............

Options

A. `1:4`
B. `3:1`
C. `2:3`
D. `1:5`

Answer

D

Explanation

Moment of inertia of sphere `I_(S)=(2)/(5)MR^(2)`
Moment of inertia of cylinder `I_(C )=(MR^(2))/(2)`
Rotational kinetic energy of body `E=(1)/(2)Iomega^(2)`
Here rotational kinetic energy of sphere
`E_(S)=(1)/(2)I_(S)omega_(S)^(2)`
`=(1)/(2)xx(2)/(5)MR^(2)omega_(S)^(2)`
Rotational kinetic energy of cylinder
`E_(C)=(1)/(2)I_(C)omega_(C)^(2)`
`=(1)/(2)xx(MR^(2))/(2)xx(2omega_(S))^(2)`
`=(1)/(2)xx(MR^(2))/(2)xx4omega_(S)^(2)`
`therefore (E_(S))/(E_(C))=(1)/(5)`
`therefore E_("sphere"):E_("cylinder")=1:5`