Gravitational acceleration magnitude - confused

[help6]

Homework Statement

Let g be the magnitude of the gravitational acceleration at the surface of a perfectly spherical planet with mass M, radius R, and uniform density. What is the magnitude of gravitational acceleration at a distance R/4 from the center of the planet.

Relevant Equations

I figured the relevant equation would be g=(GM)/r^2

The given answer is g/4. But when I substituted R/4 into the radius, I get 16GM. Am I just using the wrong equation altogether? He also said that you also got g/4 if the distance was 2R.

 

[Ibix]

Google "shell theorem".

 

[help6]

Google "shell theorem".

I'm sorry, I understand shell theorem, but I still don't get why r/4 and 2r (which is also from the center of the planet) have the same gravitational acceleration. Can you please expound?

 

[Ibix]

If you understand the shell theorem you know how to calculate the gravitational acceleration at $R/4$ and you know how to calculate it at $2R$ (note that if you are using $r$ for a radial coordinate it is unwise to also use it for the fixed radius of your sphere - so I am using $R$ for the latter).

Why are they equal? If gravitational acceleration is a maximum at $r=R$ and goes smoothly to zero at $r=0$ and $r\rightarrow\infty$ then there has to be somewhere outside the sphere where the acceleration is equal to any given point inside.

 

[Orodruin]

Put differently, the M in your relevant equation is the total mass inside of the radius R. Outside of the planet this is always the total mass of the planet but not so inside.