Pertaining to two planets, the ratio of escape velocities from respective surfaces is `1:2` , the ratio of the time period of the same simple pendulum at their respective surfaces is `2:1` (in same order). Then the ratio of their average densities is

Options

A. `1:1`
B. `1:2`
C. `1:4`
D. `8:1`

Answer

C

Explanation

`T=2pisqrt(1)/(g)` or `T prop (1)/sqrt(g) rarr (T_(1))/(T(2))=sqrt(g_(2))/(g_(1))=2/1`
`therefore (g_(2))/(g_(1))=4 because v_(e )=sqrt(2gR)` or `v_(e ) prop sqrt(gR)`
`(v_(e))/(v_(e ))=sqrt(g_(1))/(g_(2)).(R_(1))/(R_(2))=1/2 rarr sqrt(1/4xx(R_(1))/(R(2))=1/2 therefore (R_(1))/(R_(2))=1`
`therefore (g_(1))/(g_(2))=(R_(1))/(R_(2)).(P_(1))/(P_(2))` or `1/4=1(P_(1))/(P_(2)) rarr (P_(1))/(P_(2))=1/4`