I\'m really confused with how to solve this quesiton, beenstuck for hours =/ Wha

Question

I'm really confused with how to solve this quesiton, beenstuck for hours =/
What i know is that for solid spheres,
if the mass lies outside hte sphere, r >R and if the mass lies inside the sphere than r<R
Equations g=GM/r^2 Q. A Uniform solid sphere of radius R produces a gravitationalacceleration of ag on its surface. At what distance from thesphere's center are there points (a) inside and (b) outside thesphere where the gravitational acceleration is ag/3 I'm really confused with how to solve this quesiton, beenstuck for hours =/ What i know is that for solid spheres, if the mass lies outside hte sphere, r > R and if the mass lies inside the sphere than r

Explanation / Answer

a = GM/r^2 = ag on surface so GM =ag*R^2 We now have: a = ag*R^2/r^2. This isfor r >=R a. For a = ag/3 = ag*R^2/r^2 we have r^2 = 3ag*R^2 or r =R*sqrt(3ag) where r is outside of sphere b. When r is inside of sphere, the amount of massunder r is lessened affecting gravitation force. Note that ahollow sphere does not provide a gravitational force if you areinside it. So, we can disregard the mass of the sphere thatis above r. Density of sphere = M/volume = M/(4/3 pi R^3) Mr = mass underneath at radius r = [M/(4/3 piR^3)]*(4/3 pi r^3) = M*(r/R)^3 Gravitation formula a = G*M*(r/R)^3/r^2 = ag*R^2*r/R^3 = ag* r/R So, a = ag/3 =  ag* r/R or r =R/3

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