The Center of Gravity — Question 4

Question 4

Two suspension tests locate a center in body coordinates

A thin rigid plate is suspended freely from two different holes in separate trials. The downward plumb lines, drawn on the plate in its own coordinates, are $x+y=3$ and $2x-y=0$, with coordinates in centimetres. Each trial settles without other contact.

1. Find its center of gravity.

2. Why does one suspension line not usually determine the center by itself?

Question 4 - Solution

Two suspension tests locate a center in body coordinates

(a) Derivation and calculation

Each suspension orientation places the center on the vertical line through its support. In body coordinates, $$x+y=3,\qquad 2x-y=0.$$ Adding gives $$3x=3\quad\Longrightarrow\quad\boxed{(x_G,y_G)=(1,2)\ \mathrm{cm}}.$$

(b) Result and interpretation

One suspension gives only one equation for two coordinates. Two nonparallel lines give a unique intersection. For line directions separated by angle $\phi$, intersection error scales approximately as $$\Delta r\sim\frac{\Delta d}{|\sin\phi|}.$$ Nearly parallel lines therefore amplify measurement errors.

Physical check

The reported point satisfies both measured line equations exactly.