Which method to use? Gravitational problem with a cut-out cylinder

[21385]

Homework Statement

A small cylinder with radius R has been cut out of a larger cylinder with radius 2R. (With part of the circumference of the small cylinder and the large cylinder touching each other, so it kinda looks like a crescent) The mass of the larger is M. What is its gravitational pull from a distance of 4R from the center of the large cylinder.

Homework Equations

a=(GM/(d^2)) (first equation)
x(center of mass)=(m1x1+m2x2)/(m1+m2) (second equation)

The Attempt at a Solution

I have two methods of solving this problem in mind. One is to find the new center of mass using the second equation above and then just using the first equation to solve the rest. Another is to use the first equation to find the gravitational pull (acceleration) of the small cylinder and subtract it from the large cylinder (which includes the small cylinder). These two methods do not yield the same result...I don't know which one to use? Can someone help?

Thanks

 

[Alabran]

I don't know if you see this or not, but integration is a very important part of this problem. I am not very strong in this matter, but from what you expressed, neither equation you are using would do the trick.

Edit: Ok, I just noticed the "kinda like a crescent" part. I imagine I would use calculus to solve for the gravitational attraction at each of the said points and then subtract gravitational attraction of the smaller one. Based on the difficulty of this problem, I assume you do know that calculus will be required to solve this problem. Unfortunately, as I mentioned above, I am not at all strong in this matter and can't help you with it mathmatically. I believe you can solve it through first finding the gravitational attraction of a disc-crescent of a neglegable depth dz. You find this through intigating rings of mass through the radius of the disk. Then you can integrate that down the cylinder, make sure you account for the change in the distance between 4R away from the center and that which is away from the center. Also, the direction from the center of the cylinder is to be taken into account

 

[Doc Al]

I assume you are able to find the gravitational field from a cylinder of mass? Hint: View the object as the superposition of a cylinder (radius R) of negative mass and a solid cylinder (radius 2R).

 

[21385]

i am unsure but wouldn't the find a new center of mass method also work? in theory at least? because you are believing that all the mass is concentrated in the new center of mass.

 

[Doc Al]

I don't understand how finding the center of mass would help. The mass distribution is not symmetric.

 

[21385]

What do you mean when you say the mass distribution is not symmetric? Sorry but I still don't understand.

 

[becca4]

I'd think that you'd use F = (G m1 m2)/r^2 for this... Can you assume that the point at 4r that you're using has the mass of a point mass?

 

[Doc Al]

What do you mean when you say the mass distribution is not symmetric? Sorry but I still don't understand.

If you had a spherically symmetric mass distribution, then you can treat the mass as being concentrated at the center of mass for the purposes of calculating the gravitational field. But that's not the case here. For a cylinder with a cut out, you don't even have cylindrical symmetry. The center of mass is irrelevant. Use the principle of superposition, as I described earlier.