A.: Finding Center of Mass for a Rectangular Surface with a Hole

[Jalo]

Homework Statement

Given a homogenous rectangular surface, sides of length a and b=4*a, with a circular hole in x=a and y=a/2, find the center of mass.

Homework Equations

R=1/M*Ʃm_i*r_i , M= total mass, r= position vector

area of a circle = pi*r² , r being the radius

The Attempt at a Solution

I determined the center of mass of the surface and circle and tried to subtract one from another, however the result did not agree with the solutions.

A_circle=pi*r² = pi*a²/4
A_rectangle= a*4a = 4a²
Center of mass of the rectangle alone:
R_r=(2a,a/2)
Center of mass of the circle alone:
R_c=(a,a/2)

R=[R_r*A_rectangle - R_c*A_circle] / Total area ⇔
⇔R=a*(32-pi)/(16+pi)

The correct center of mass is (2,05,a/2)

Thanks!

D.

 

[vela]

You need to subtract the areas in the denominator because you're considering the circle to have negative mass. Right now, your calculation is for the solid slab plus a positive-mass circle located at (-a, -a/2).

Also, to get the answer you cited, I think you need a different radius for the circle. Is the radius of the circle really equal to a/2? If so, you won't get that answer.