Gravitational acceleration on the surface of a planet is (√6/11)g, where g is the gravitational acceleration on the surface - Physics - Gravitation

Question

Gravitational acceleration on the surface of a planet is
(√6/11)g, where g is the gravitational acceleration on the
surface of the earth The average mass density of the planet is 2/3
times that of the earth if the escape speed on the surface of the
earth is taken to be 11km/s , the escape speed on the surface of
the palnett is km/s will be

Ujjval Chauhan · Accepted Answer

Dear student

We have vesc=2gR where g is the acceleration due to gravity and R is the radius of the planet or earth
 Therefore vesc∝gR 
 If gp and ge are the accelerations due to gravity on the surface of the planet and the earth, Rp and Re are their radii and vp and ve the escape velocities respectively, we havevpve =gpRpgeRe  Since the gravitational acceleration on the surface is given by g=GMR2 where G is the gravitational constant and M is the mass of the planet or earth, we can write g=G43πR3ρR2 ,where ρ is the average mass density of the planet or earth
 Therefore g∝Rρ and gpge=ρpRpρeRe
RpRe=gpgeρpρegpge=611   and    ρpρe=23so   vpve =611×611×32vpve =611×11×32vp=ve×311vp=3 km/s