The Center of Gravity — Question 2

Question 2

The center of gravity of an L-shaped plate lies in empty space

A uniform thin plate is the union of rectangles $0\le x\le4$, $0\le y\le1$ and $0\le x\le1$, $1\le y\le4$, with coordinates in metres. Gravity is uniform.

1. Locate its center of gravity.

2. Is a material point required at that location? Explain how a rigid extension could balance the plate there.

Question 2 - Solution

The center of gravity of an L-shaped plate lies in empty space

(a) Derivation and calculation

For the two nonoverlapping rectangles, use area weights: $$A_1=4,\quad(x_1,y_1)=(2,.5),\qquad A_2=3,\quad(x_2,y_2)=(.5,2.5).$$ $$x_G=\frac{4(2)+3(.5)}7=\boxed{\frac{19}{14}\ \mathrm m},$$ $$y_G=\frac{4(.5)+3(2.5)}7=\boxed{\frac{19}{14}\ \mathrm m}.$$

(b) Result and interpretation

Both coordinates exceed $1$ m, so $G$ lies in the missing upper-right region. A massless rigid extension can transmit the supporting force through this point: $$\vec R=Mg\hat y,\qquad \vec\tau_G=\vec0.$$ The center of gravity is a weighted position and need not lie in material.

Physical check

The plate is symmetric under interchange of $x$ and $y$, consistent with $x_G=y_G$.