Formally finding the center of mass

[guitarman]

Question:
Determine the location of the center of mass of a "L" whose thin vertical and horizontal members have the same length L and the same mass M. Use the formal definition to find the x and y coordinates, and check your result by doing the calculation with respect to two different origins, one in the lower left corner at the intersection of the horizontal and vertical members and one at the right end of the horizontal member.
(a) origin at the lower left
xcm =
ycm =

(b) origin at the right end of the horizontal member
xcm =
ycm =

Relevant Equations:
The center of mass can be found by (m1r1+m2r2)/2

Attempt at solution:
Since I have not gotten part A, I have not been able to attempt part B.
Since I am not given a mass, I have simply tried doing (r1+r2)/2, or (L+L)/2 in this case, but it is not the right answer. Can somebody please tell me what I'm doing wrong?

 

[Galileo]

First draw a picture of the situation, indicating r1,r2, L, M etc.
You are give the lines both have the same mass M.

Since this is a two dimensional problem, you need to use different components. Not sure you realized that from what you wrote. r1=(x_1,y_1), r2=(x_2,y_2)

 

[Doc Al]

Relevant Equations:
The center of mass can be found by (m1r1+m2r2)/2

That should be (m1r1+m2r2)/(m1+m2). (Of course, for your example you can simplify it since the masses are equal.)

Since I am not given a mass, I have simply tried doing (r1+r2)/2, or (L+L)/2 in this case, but it is not the right answer. Can somebody please tell me what I'm doing wrong?

When using this equation, r1 and r2 represent the coordinates of the two point masses, m1 and m2. You'll apply this twice: once for the x-coordinate; once for the y.

Since you don't have two point masses, but have two "rods" of length L, what coordinates do you think you should use?

 

[guitarman]

Oh okay, so if I have r1 = <x1,y1> and r2 = <x2,y2> I will find the center of mass by

(M1<x1,y1> + M2<x2,y2>) / (M1 + M2)

which gives would ultimately give me <(M1x1+M2x2) / (M1+M2), (M1y1 + M2y2)/(M1+M2)>

Can somebody please confirm this for me?

 

[guitarman]

So I tried using (M1y1 + M2y2)/(M1+M2) for the y component with the origin at the lower left, but this was incorrect. Now I am thinking that if I am at the lower left region, my y1 will be zero, and so wouldn't the y component simply be M2y2/(M1+M2)?

 

[Doc Al]

You are using the wrong coordinates for r1 and r2. Answer the question I asked at the end of my last post.

 

[guitarman]

Why would my coordinates be in something other than x and y? The only other thing that jumps to my mind would be radians, although I am unsure if radians can be classified as a coodinate system.

 

[Doc Al]

A rod is not a point, it has length. So when you give the coordinates of the rod, what part of the rod are you referring to?

 

[guitarman]

You refer to the rod in a length of meters, in reference to its distance.

 

[Doc Al]

Coordinates x1, y1 refer to a point on the rod. What point?

 

[guitarguy1]

I'm working on this same problem and I can't make heads or tails of what's being asked. The explanation doesn't seem very clear, so if it makes sense to somebody please explain what they're asking for.

 

[guitarguy1]

Coordinates x1, y1 refer to a point on the rod. What point?

You're talking about the centerpoints of the rod? if so they would just be L/2, but how would you put that into a useful form to fit the equation for both axes?

 

[guitarman]

Ahh okay so after rereading the problem I understand it better. Our teacher did an example in class with finding the center of mass of a letter "T", and now he wants us to do the same for the letter "L".
So I suppose that although it is not literally made out of rods, for simplicity's sake we can assume that it is.
Now, I believe that the first question is asking to find the center of mass of the bottom,leftmost corner of the letter "L", and the second one is asking for the center of mass near the right bottom end of the column.
Now what is confusing me is what I need to use for x1,y1, x2,y2, etc. I do not have any numerical values, so clearly I will end up with my answer in variables.
Now if it asks me for the center of mass with respect to the origin at the left bottom corner, does that mean that I arbitrarily set that value to zero?

 

[Doc Al]

You're talking about the centerpoints of the rod?

Yes! You want the coordinates of the centers of each "rod" (arm of the L shape).

if so they would just be L/2, but how would you put that into a useful form to fit the equation for both axes?

The distance from the end of the rod to the center is L/2, of course. What you need is the coordinates of those centers, which depend on your origin.

Now, I believe that the first question is asking to find the center of mass of the bottom,leftmost corner of the letter "L", and the second one is asking for the center of mass near the right bottom end of the column.

The difference is just the origin used to reference the coordinates. The physical location of the center of mass doesn't change, but its coordinates will.

Now what is confusing me is what I need to use for x1,y1, x2,y2, etc. I do not have any numerical values, so clearly I will end up with my answer in variables.

Yes, your answer will be in terms of the length "L".

Now if it asks me for the center of mass with respect to the origin at the left bottom corner, does that mean that I arbitrarily set that value to zero?

What it means is that you set the coordinates of that point (left bottom corner) equal to 0,0. Then you use that as your reference to find the coordinates of everything else.