Simple center of mass problem but i cant figure it out

[huk]

I feel silly for ask you for help on this simple center of mass problem but i got wrong answers...

Where is the center of mass of a uniform, L-shaped iron rod of sides X = 0.8 m and Y = 0.5 m, respectively? Take the corner to be at (x,y) = (0,0), with the X and Y sides along those axes, respectively. (Assume that the rod is so narrow that the dimensions of the outer bend are the same as those of the inner bend of the L.)

i thought the answer is x = .2667m and y = .1667m perhaps my answers are wrong?

 

[arildno]

Now, the C.O.M of the horizontal leg is (0.4,0), for the vertical leg, (0,0.25)
(Agreed?)
Let $$m_{h}=0.8\rho$$ be the mass of the horizontal leg, and $$m_{v}=0.5\rho$$ the mass of the vertical leg ($$\rho$$ being the density).
Hence, the coordinates of the C.M of the L-shaped object fulfill:
$$(x_{C.M},y_{C.M})=\frac{m_{h}(0.4,0)+m_{v}(0,0.25)}{m_{h}+m_{v}}$$

 

[thepatientmental]

Don't feel silly for asking for help on a simple center of mass problem. Sometimes, even the simplest problems can be tricky to solve. It's always better to ask for help and get the correct answer, rather than struggling on your own and potentially getting the wrong answer.

Regarding your specific problem, your answers for the center of mass coordinates seem to be incorrect. To find the center of mass of a uniform object, we can use the formula:

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

Where x_cm is the x-coordinate of the center of mass, m is the mass of each part of the object, and x is the x-coordinate of each part.

For this L-shaped iron rod, we can divide it into two parts - the longer side with length 0.8 m and the shorter side with length 0.5 m. The longer side has a mass of 0.8 m, and its x-coordinate is 0.4 m (half of its length). The shorter side has a mass of 0.5 m, and its x-coordinate is also 0.4 m (half of its length).

Plugging these values into the formula, we get:

x_cm = (0.8*0.4 + 0.5*0.4) / (0.8 + 0.5) = 0.36 m

This means that the center of mass of the L-shaped rod is located at x = 0.36 m. Similarly, we can find the y-coordinate of the center of mass using the same formula.

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

For this L-shaped rod, the y-coordinate of the longer side is 0.4 m (half of its length), and the y-coordinate of the shorter side is 0 m (since it is along the x-axis). Plugging these values into the formula, we get:

y_cm = (0.8*0.4 + 0.5*0) / (0.8 + 0.5) = 0.256 m

This means that the center of mass of the L-shaped rod is located at y = 0.256 m.

In