The Center of Gravity — Question 3

Question 3

Removing off-center material shifts the balance point away from the hole

A uniform plate initially has mass $5\,\mathrm{kg}$ and center of gravity at the origin. A hole removes $1\,\mathrm{kg}$ whose original center was at $(2,1)\,\mathrm m$. Gravity is uniform.

1. Find the center of gravity of the remaining plate.

2. Where should a $0.50\,\mathrm{kg}$ point ballast be placed to restore the original center?

Question 3 - Solution

Removing off-center material shifts the balance point away from the hole

(a) Derivation and calculation

Subtract the removed mass and its first moment: $$M_r=5-1=4\ \mathrm{kg},\qquad M_r\vec r_G=5\vec0-1(2\hat x+\hat y).$$ Thus $$\boxed{\vec r_G=(-.50\hat x-.25\hat y)\ \mathrm m}.$$

(b) Result and interpretation

To restore the center to the origin with $.5$ kg of ballast, $$4\vec r_G+.5\vec r_b=\vec0.$$ Therefore $$\vec r_b=-\frac4{.5}(-.5\hat x-.25\hat y) =\boxed{(4\hat x+2\hat y)\ \mathrm m}.$$ The final mass is $4.5$ kg; its first moment about the origin is zero.

Physical check

Removing mass moves the center away from that mass; replacement ballast must lie on the opposite side of the new center.