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

Time period of simple pendulum,
`T=2pi sqrt((l)/(g))or T prop (1)/(sqrt(g))`
`(T_(1))/(T_(2))=sqrt((g_(2))/(g_(1)))=(2)/(1)rArr (g_(2))/(g_(1))=4`....(i)
`v_(e)=sqrt(2gR)or v_(e)prop sqrt(gR)rArr (v_(e1))/(v_(e2))=sqrt((g_(1))/(g_(2)).(R_(1))/(R_(2)))=1:2`
`:. sqrt((1)/(4)xx(R_(1))/(R_(2)))=(1)/(2)" "(because "from Eq. (i)" (g_(2))/(g_(1))=4)`
`:. (R_(1))/(R_(2))=1`
Further, `g=(Gm)/(R^(2))=(G((4)/(3)piR^(3)rho))/(R^(2))or g prop R rho`
`:. (g_(1))/(g_(2))=(R_(1))/(R_(2)).(rho_(1))/(rho_(2))or((1)/(4))=(1)(rho_(1))/(rho_(2))or (rho_(1))/(rho_(2))=(1)/(4)`.