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Mars and earth have masses in the ratio ...

Mars and earth have masses in the ratio `1:11` and radii in the ratio `42:79`. Compare
a. their densities, assuming them to be spheres of uniform density,
b. gravitational field strengths at their surfaces:
c. escape velocities from their surfaces,
d. periods of their satellites near their surfaces.

Text Solution

Verified by Experts

a. `rho=M/(4/3piR^(3)`
`(rho_(1))/(rho_(2))=(M_(1))/(M_(2))((R_(2))/(R_(1)))^(2)=0.605`
b.` fg=-(GM)/(R^(2))`
`(f_(1))/(f_(2))=(M_(1))/(M_(2))((R_(2))/(R_(1)))^(2)=1/11xx(79/42)^(2)`
c. `v_(e)=sqrt((2GM)/R)`
`(v_(1))/(v_(2))=sqrt((M_(1))/ M_(2) (R_(2))/(R_(1)))=sqrt(1/11xx79/42)=0.4135`
d. Period near surface of planet is
`T=2pisqrt(R^(3)/(GM))`
`(T_(1))/(T_(2))=sqrt(((R_(1))/(R_(2)))^(3)(M_(2))/(M_(1)))=sqrt((42/79)^(3)xx11/1)=1.286`
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