Statics 2D Equilibrium Problem

[J0sh8830]

Homework Statement

This is a three step problem (I am not fond of the multi-step problems as I can usually do better without multiple steps. Here is the problem:

A uniform ring of mass m = 10 kg and radius r = 195 mm carries an eccentric mass m0 = 18 kg at a radius b = 160 mm and is in an equilibrium position on the incline, which makes an angle α = 19° with the horizontal. If the contacting surfaces are rough enough to prevent slipping, solve for the angle θ which defines the equilibrium position.

Part 1.) The free-body diagram of the body is shown. Identify the weight W (of the entire structure).

This part was not that difficult, finding the weight of the entire structure is just m(of the entire structure)g or
(28kg)(9.81m/s²). The next part is the step where I am having trouble.

Part 2.) Point G represents the center of mass of the object. Find the distance d between point O and point G.

Homework Equations

[∑F_x=0]
[∑F_y=0]
[∑M_z=0][/B]

The Attempt at a Solution

[/B]
1.) I drew the free body diagram first, just like was shown in the picture from my homework.
2.) I chose an axes system with G as the origin (my reasoning for this was that I believed this was the only way to solve for d, by summing the moments about point O.) However, this is where I am having some trouble. I think the angle is confusing me and I am not sure how to set up the moment equation. I want to try to work through the problem on my own, but if anyone could tell me if I am on the right track or perhaps point me in the right direction, it would be greatly appreciated. Thank you!

Josh

 

[Chestermiller]

Hi Josh. Welcome to Physics Forums.

Here are some hints:

1. The location of the center of mass is independent of the specific geometry for this problem
2. Choose a convenient angular orientation of the system so that determining the location of the center of mass is much easier
3. Make use of the symmetry of the geometry
4. The line Ob bisects the mass $m_0$
5. If you split the outer rim along a center line through O, where would the center of mass of each half be located?

 

[J0sh8830]

Hi Josh. Welcome to Physics Forums.

Here are some hints:

1. The location of the center of mass is independent of the specific geometry for this problem
2. Choose a convenient angular orientation of the system so that determining the location of the center of mass is much easier
3. Make use of the symmetry of the geometry
4. The line Ob bisects the mass $m_0$
5. If you split the outer rim along a center line through O, where would the center of mass of each half be located?

Mr. Miller,
Thank you so much for your reply. I was able to solve the problem with your hints. It was a lot easier than I first imagined. I was trying to use too much information when the center of mass of the system and its distance from O were much easier to solve for. Thank you!

Josh