Gravitational Force on Mass in Cavity of Planet

[anshuman3105]

A large spherical planet of radius R made of a material of density d, has a spherical cavity of radius R/2, with center of cavity a distance R/2 from the centre of the planet. Find the gravitational Force on a small mass m at the centre of the cavity

 

[haruspex]

A large spherical planet of radius R made of a material of density d, has a spherical cavity of radius R/2, with center of cavity a distance R/2 from the centre of the planet. Find the gravitational Force on a small mass m at the centre of the cavity

As per forum rules, you should quote any relevant standard equations of which you are aware and must show some attempt at a solution. If totally stuck, you should at least be able to provide some thoughts.

 

[anshuman3105]

using the formula F = Gm1m2/r^2, i am getting 16Gpidrm/3 but the solution is 2Gpidrm/2

 

[haruspex]

using the formula F = Gm1m2/r^2, i am getting 16Gpidrm/3 but the solution is 2Gpidrm/2

I get $\frac 23 G\pi d r m$ (I'm guessing the "/2" in what you posted is a typo).
Please post your working.
(There is a very quick method here.)

 

[anshuman3105]

Yes your answer is right...but how did you do it?

 

[anshuman3105]

I used Gm1m2/r^2
So F = (G(d*4/3pir^3)m)/(r/2)^2

 

[haruspex]

I used Gm1m2/r^2

That formula is essentially for point masses. It also works if one mass is a uniform spherical shell (or assembly of concentric uniform spherical shells) and the other (point) mass is outside all the shells.
The trick when dealing with cavities is to treat the cavity as filled in (i.e. no cavity) then add a 'negative mass' at the cavity. so in this case we have a complete sphere (S₁) radius R minus a complete sphere (S₂) radius R/2.
What do you know about the gravitational field inside a uniform spherical shell?
What is the force on m due to S₁?
What is the force on m due to S₂?

 

[anshuman3105]

Can you show it to me the solved part..?

 

[haruspex]

Can you show it to me the solved part..?

http://hyperphysics.phy-astr.gsu.edu/hbase/mechanics/sphshell2.html