Calculate Center of Mass for a Figure of Uniform Density

[draotic]

Homework Statement

Find the center of mass of figure , of uniform density

Homework Equations

X = m1 x1 + m2 x2 + ... / m1 + m2 ...

The Attempt at a Solution

i broke the figure in 4 rectangles and got individual center of masses
my answers comes out to be
(13b/8 , 5b/2)...
someone told me that's not correct , can someone guide me please ?

 

[grzz]

More details are required.

 

[Doc Al]

Show how you got your answer. What did you get for the center of mass of each rectangle? What is the mass of each?

 

[draotic]

More details are required.

thats what is given

 

[draotic]

Show how you got your answer. What did you get for the center of mass of each rectangle? What is the mass of each?

i got 4 coordinates as
1) 2b , b/2
2) 2b , 2.5b
3) 2b , 4.5b
4) 0.5b , 2.5b

 

[Doc Al]

i got 4 coordinates as
1) 2b , b/2
2) 2b , 2.5b
3) 2b , 4.5b
4) 0.5b , 2.5b

Which is which?

And what's the height and width of this object?

 

[draotic]

Which is which?

And what's the height and width of this object?

here are the measurements

 

[draotic]

total height of object is 5b

 

[Doc Al]

OK, looks good. So what did you use for the mass of each piece?

 

[draotic]

OK, looks good. So what did you use for the mass of each piece?

thats where i am going wrong...
dont know how to proceed , grateful if you could take it from here

 

[Doc Al]

thats where i am going wrong...
dont know how to proceed , grateful if you could take it from here

I'll start you off. Since it's uniform, the mass is proportional to the area. We don't care about the actual mass, only the relative mass of each piece.

What's the area of each piece? Let that represent the mass of each piece.

For example, piece #1 has an area of 2b*b = 2b².

 

[draotic]

all right , 3 pieces have area 2b^2 and the larger one has 5b^2
what now?

 

[Doc Al]

all right , 3 pieces have area 2b^2 and the larger one has 5b^2
what now?

Now start cranking out the center of mass using the formula.

 

[draotic]

Now start cranking out the center of mass using the formula.

could you please do it for 1 rectangle , i'll do rest

 

[Xyius]

A trick here is to start your coordinate axis off at the bottom left corner of each rectangle. (Or other corner, just make sure it is all consistent.) What you will find is that each of the smaller rectangles will all have the same center of mass. You could then find the center of mass of the bigger one. Be careful though, they are all (referring to the 3 little ones) only the same in their own respective axes. You would then need to find their coordinates on a different more convenient axis. (Such as the lower left hand corner of the entire E.) From there you can find the center of mass of the entire object.

Sounds kinda tricky but it isn't so bad. This is how I just did it, there are definitely other ways.

 

[Doc Al]

The OP has already figured out the center of mass of each rectangle measured from the same point. All that's left to do is to apply the formula that was posted in the very first post.

 

[Xyius]

Oh sorry ha! Kind of got a head of my self there. Probably should have read through the post more thoroughly. :p