If the earth suddenly contracts to one third of its present size without any change in its mass, the ratio of the kinetic energies of the earth after and before contraction will be

Options

A. 9
B. 8
C. 7
D. 3

Answer

A

Explanation

By the principle of conservation of angular momentum,
`I_(1)omega_(1)=I_(2)omega_(2)`
It is given that, `M_(1)=M_(2)=M and R_(2)=(R_(1))/(3)`
For a sphere, `I=(2)/(5)MR^(2)`
`therefore" "(2)(5)MR_(1)^(2)omega_(1)=(2)/(5)M((R_(1))/(3))^(2)omega_(2)=(2)/(5)(MR_(1)^(2))/(9)omega_(2)`
`therefore" "omega_(1)=(omega_(2))/(9)" or "(omega_(2))/(omega_(1))=9" ...(1)"`
`therefore" "("RotationalK.E. after contraction")/("Rotational K.E. before contraction")=((1)/(2)I_(2)omega_(2)^(2))/((1)/(2)I_(1)omega_(1)^(2))`
`therefore" "(K_(2))/(K_(1))=((2)/(5)(MR_(1)^(2))/(9)omega_(2)^(2))/((2)/(5)MR_(1)^(2)omega_(1)^(2))=(omega_(2)^(2))/(9omega_(1)^(2))=((omega_(2))/(omega_(1)))^(2)xx(1)/(9)`
`therefore" "(K_(2))/(K_(1))=(9)^(2)xx(1)/(9)=9" "therefore K_(2)=9K_(1)`