Center of Mass: xcom, ycom Calculations

[vysero]

What are (a) the x coordinate and (b) the y coordinate of the center of mass for the uniform plate? Since the plate is uniform, we can split it up into three rectangular pieces, with the mass of each piece being proportional to its area and its center of mass being at its geometric center. We’ll refer to the large 35 cm × 10 cm piece (shown to the left of the y-axis in Fig. 9-38) as section 1; it has 63.6% of the total area and its center of mass is at (x1 ,y1) = (−5.0 cm, −2.5 cm). The top 20 cm × 5 cm piece (section 2, in the first quadrant) has 18.2% of the total area; its center of mass is at (x2,y2) = (10 cm, 12.5 cm). The bottom 10 cm x 10 cm piece (section 3) also has 18.2% of the total area; its center of mass is at (x3,y3) = (5 cm, −15 cm). Answers:
(a) xcom = (0.636)x1 + (0.182)x2 + (0.182)x3 = – 0.45 cm
(b)ycom = (0.636)y1 + (0.182)y2 + (0.182)y3 = – 2.0 cm

Correct me if I am wrong and please explain the right answer to me. xcom = m(x1) + m(x2) + m(x3) right? So what I am not understanding is how .636 is = m which stands for mass right?

 

[frogjg2003]

You aren't given the mass for any of the pieces, but you were given the percentage of total area. You were told each that it was a uniform plate, so the the mass and area are proportional. Part of finding the center of mass is dividing by the total mass, so you are just given the fraction to begin with and you won't have to divide.

 

[vysero]

You aren't given the mass for any of the pieces, but you were given the percentage of total area. You were told each that it was a uniform plate, so the the mass and area are proportional. Part of finding the center of mass is dividing by the total mass, so you are just given the fraction to begin with and you won't have to divide.

The fraction being the 63.4% or the other %'s of the total area respectively?

 

[frogjg2003]

Yes.

 

[Nickie]

Yes, you are correct. In this case, m represents the mass of each rectangular piece, which is proportional to its area. So, for example, the first rectangular piece (section 1) has a mass of 0.636 times the total mass of the plate since it makes up 63.6% of the total area. This is why we multiply each x coordinate by its corresponding mass (0.636 for section 1, 0.182 for sections 2 and 3) in the equation for xcom. The same applies for the y coordinates. This method allows us to find the center of mass for a complex shape by breaking it down into simpler shapes.