Why Is the Center of Gravity Calculation Resulting in Negative Coordinates?

[webren]

Hello,
I with this problem, I am getting the correct answer, but I do not know why the book's answer is negative:

"Consider the following mass distribution: 5.00 kg at (0,0) m, 3.00 kg at (0, 4.00) m, and 4.00 kg at (3.00, 0) m. Where should a fourth object of mass 8.00 kg be placed so that the center of gravity of the four-object arrangement will be at (0,0)?"

I understand that the center of gravity is the sum of the masses multiplied by the x and y coordinates divided by the sum of the masses of the objects.

The answer I come to are (1.5, 1.5). The book says both of those coordinates are negative. What did I do wrong?

Thank you.

 

[Cyrus]

You need to show me your work so I can see where you have made a mistake. Keep in mind, you will get a negative number if the mass is to the left of your origin

 

[Office_Shredder]

Remember, if the center of mass is on the origin, then the sum of the x components multiplied by the masses, and the sum of the y components multiplied by the masses, is zero

 

[webren]

Here is my work for the x coordinate of the center of gravity:

The sum of the coordinates multiplied by their respective masses are:
(5kg)(0) + (3kg)(0) + (4kg)(3m) + (8kg)(x) = 0.
I know I need to find x, so the equation becomes: (8)(x) + 12 = 0

The sum of all the masses are (5 + 3 + 4 + 8) = 20.
If you divide the two sums, you will get (12 + 8x) divided by 20. Solving for x yields positive 1.5 The book claims that there is a negative as the answer.

When my professor worked this problem out in class, at the end of the calculations (when he was trying to solve for x), he randomly placed a negative sign onto the 12, which of course yields the -1.5. I do not understand why this negative keeps popping up. I do understand if the mass is going to be at the left of the origin, it will be negative, but my calculations say it should be to the right, which should make the coordinates positive.

Thank you.

 

[Doc Al]

Here is my work for the x coordinate of the center of gravity:

The sum of the coordinates multiplied by their respective masses are:
(5kg)(0) + (3kg)(0) + (4kg)(3m) + (8kg)(x) = 0.
I know I need to find x, so the equation becomes: (8)(x) + 12 = 0

Actually, it's the coordinates of the CM that equal zero, not just that sum. To find those coordinates you must divide that sum by the total mass. (Since the coordinates of the CM are zero, you'll end up with the same equation.)

The sum of all the masses are (5 + 3 + 4 + 8) = 20.
If you divide the two sums, you will get (12 + 8x) divided by 20. Solving for x yields positive 1.5 The book claims that there is a negative as the answer.

To get the x-coordinate of the CM, you must divide (8x + 12) by 20 and set that equal to 0. So:
(8x + 12)/20 = 0

Since the right hand side is 0, you can just multiply both sides by 20 and end up with your earlier equation:
8x + 12 = 0

Now just solve that equation for x; x is what you are trying to find.

(Edited to be clearer and more accurate.)​

 

[webren]

Ah, I think I see. So is dividing not necessary at all for this kind of problem?

 

[Doc Al]

Ah, I think I see. So is dividing not necessary at all for this kind of problem?

No, dividing by the total mass to find the CM is the right thing to do. But for the special case where the CM is given as zero, it won't make a difference. (I'm going to update my earlier response, since it might be misleading.)

 

[webren]

I understand a lot better. Thank you.

There is another similar problem where it shows an L-shaped ruler and each side's length. There is no mass given, so I assumed the center of gravity of the scenario is the sum of the coordinates multiplied by the area, divided by the total area of the shapes (which is two rectangles). It seems like you initially have to divide each side by 2 and figure it out from there. Why is it mandatory to divide by two?

 

[Doc Al]

It seems like you initially have to divide each side by 2 and figure it out from there. Why is it mandatory to divide by two?

To find the coordinates of the center of mass of each piece--which is right in the middle--you would divide the length by two.