Resultant of || forces: Eng mechncs

[Edwardo_Elric]

Homework Statement

The shaded area in the figure represents a steel plate of uniform thickness. A hole of 4in diameter has been cut in the plate. Locate the center of gravity of the plate.

hint: The weight of the plate is equivalent to the weight of the original plate minus the weight of the material cut away. Represent the original weight of the plate by a downward force acting at the center(7X5in.) of the 10X14in. rectangle. Represent the weight of the material cut away by an upward force acting at the center of the circle. Locate the position of the resultant of these two forces with respect to the left edge and bottom of the plate.

Answer: Center of gravity is 6.80in. from left edge and 4.90in above bottom edge

Homework Equations

M = R * d

The Attempt at a Solution

This problem may involve moment which is M = Fd.
W = weight
m = material
p = plate

W(p) = W(m+p) - W(m)

Drawing with respect to the left edge

^ That is W(m+p)
R*x = W(p)x = 7*W(m+p) - 9*W(m)

(W(m+p) - W(m))x = 7*W(m+p) - 9*W(m)


Drawing with respect to the bottom edge

R * y = W(p)y = 5*W(m+p) - 6*W(m)
(W(m+p) - W(m))y = 5*W(m+p) - 6*W(m)

Now I am stuck with these equations
did i do the right thing?

 

[Shooting Star]

I have not gone through your calc in detail, but I’m sure you have done correctly. However, the whole thing can be done just by applying the formula for the CM of a body.

Let M be the whole mass, Mp be the mass of the circular plate and Mr be the mass of the rest of it. (The suffix p and r denote quantities for the plate and the rest of the plate respectively.)

Choosing the origin at the bottom left corner, and letting the Xs denote the x co-ords of the CMs of the pieces,

M*X = Mr*Xr + Mp*Xp, where X and Xp are known. So,
X= (Mr/M)Xr + (Mp/M)Xp

Xr and similarly Yr may be found, because Mr/M and Mp/M are known.