Net gravitational force inside a shell

In summary, the theorem states that the net force inside a shell is always zero. The equation for this force is dependent on the shape of the ring.
  • #1
jaydnul
558
15
When I was in introductory physics I remember being told that the net force (gravitational for example) inside a shell is always 0. I always felt that, intuitively, this would only be true at the center of the ring. Not sure what made me think of it today, by I decided to sit down and do the calculation.

The vector formulation for a force of gravity

[itex]F=\frac{Gm_1m_2}{|r|^2}\frac{r}{|r|}[/itex]

m1 will be the mass of the particle inside the shell, m2 will be the mass of the shell. Inside a shell this becomes

[itex]F=\frac{Gm_1m_2}{|r|^2}<cos(θ),sin(θ)>[/itex]

[itex]dF=\frac{Gm_1dm_2}{|r|^2}<cos(θ),sin(θ)>[/itex]

where

[itex]dm_2=\frac{m_2}{2π}dθ[/itex]

If r is changing as a function of θ, we can write it in this form

[itex]dF=\frac{Gm_1m_2}{2π}<\frac{cos(θ)}{r(θ)^2},\frac{sin(θ)}{r(θ)^2}>dθ[/itex]

For the r(θ), I made up a simple function that satisfied my question. [itex]r(θ)=2+sin(θ-π/2)[/itex] will put my particle somewhere in the right section (as long as it's not centered) of my oval shaped ring. (the biggest radius from my particle would be 3, the smallest would be 1). Finally we get

[itex]F=\frac{Gm_1m_2}{2π}∫<\frac{cos(θ)}{(2+sin(θ-π/2))^2},\frac{sin(θ)}{(2+sin(θ-π/2))^2}>dθ[/itex]

The integral is from 0 to 2π (not sure how to do a definite). After doing the integrals in my calculator I get roughly

[itex]F=\frac{Gm_1m_2}{2π}<1.21,0>[/itex]

There would be a pull to the right. So unless the particle was exactly centered, it would never just float, there would always be a force on it. Anyways, just thought I'd share. Why are we told to assume the net force at every point inside the shell is 0?
 
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  • #2
  • #3
It's the same in principle, isn't it? I just found it easier to make up an equation where r is a function of θ. If you were to take a point inside a spherical shell that wasn't at the center, wouldn't the result for net force be non-zero?
 
  • #4
It would be zero in a spherically symmetric shell. Try it out, and then check the wikipedia I posted which has the derivation.

Intuitively you could think that in any direction you will be closer to a small amount of mass and further from a large amount of mass such that the force from each is equal in magnitude and opposite in direction (they cancel out). If the shell isn't spherically symmetric then this balance won't necessarily happen.
 
  • #5
Ahh I see. I'm not quite sure how to develop that equation for r, but assume it has something to do with a coordinate transformation. I will go work on that, thanks ModusPwnd.
 

What is net gravitational force inside a shell?

The net gravitational force inside a shell refers to the combined gravitational forces exerted by all the individual masses inside a spherical shell on an object located at any point inside the shell.

How is the net gravitational force inside a shell calculated?

The net gravitational force inside a shell is calculated using the formula F = GMm/r^2, where G is the gravitational constant, M is the total mass of the shell, m is the mass of the object, and r is the distance between the object and the center of the shell.

Does the net gravitational force inside a shell always equal zero?

No, the net gravitational force inside a shell is not always zero. It depends on the distribution of mass inside the shell. If the mass is evenly distributed, the net gravitational force will be zero. However, if the mass is not evenly distributed, there will be a non-zero net gravitational force.

How does the net gravitational force inside a shell change with distance from the center?

The net gravitational force inside a shell decreases as the distance from the center of the shell increases. This is because the further away an object is from the center, the less gravitational force it experiences from the masses on the opposite side of the shell.

What is the significance of the net gravitational force inside a shell?

The net gravitational force inside a shell helps us understand how gravity works in celestial bodies, such as planets and stars. It also has practical applications in fields such as astronomy and astrophysics.

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