Calculating y-Coord of Center of Mass in Unique Shape

[phsyics_197]

Homework Statement

____
l _ l
l_l l_l

Find the y-coordinate of the center of mass. Each side l / _ is length a.
I got an answer of 1.25a, but that isn't right.

Homework Equations

The Attempt at a Solution

Area of each section * x-coordinate of center of mass. (the area missing is negative)
Then divided by the total area

 

[tiny-tim]

Welcome to PF!

Hi phsyics_197! Welcome to PF!

____
l _ l
l_l l_l

Find the y-coordinate of the center of mass. Each side l / _ is length a.
I got an answer of 1.25a, but that isn't right.

(hmm … I can see from the "QUOTE" box that that's supposed to be two squares on level 0, and three squares on level one)

Show us how you got 1.25, and then we can see what went wrong, and we'll know how to help!

 

[kerimek]

.

First, it's important to clarify what shape we are dealing with. From the given information, it seems like we are dealing with a shape that consists of two rectangles connected at the corners, creating a "V" shape. Please correct me if I am wrong.

Assuming this is the case, we can use the formula for the center of mass of a composite object, which is:

x_cm = (m1x1 + m2x2 + ... mnxn) / (m1 + m2 + ... + mn)

where m is the mass of each component and x is its respective x-coordinate of the center of mass.

Since we are dealing with a 2D shape, we can use the formula for the center of mass in the y-direction, which is:

y_cm = (m1y1 + m2y2 + ... + mny) / (m1 + m2 + ... + mn)

In this case, we can divide the shape into two rectangles with equal width (a) and length (l). The first rectangle has a mass of (l/a) * a = l, and its center of mass in the y-direction is at y = l/2. The second rectangle also has a mass of l and its center of mass in the y-direction is at y = 3l/2. Therefore, the total mass of the shape is 2l.

Plugging in the values into the formula, we get:

y_cm = (l * l/2 + l * 3l/2) / (2l) = (2l^2/2) / (2l) = l/2

Therefore, the y-coordinate of the center of mass for this unique shape is y = l/2.

It's important to note that this solution only applies if my assumption about the shape is correct. If the shape is different, the solution may be different as well. In general, to find the center of mass of a unique shape, we need to divide it into smaller, simpler shapes and use the formulas for the center of mass for each individual shape. Then, we can use the composite object formula to find the overall center of mass.