Centre of Gravity

The centre of gravity of a body is defined as the single point at which the ENTIRE WEIGHT of the

body -- the total downward gravitational force acting on every one of its constituent particles -- may be

taken to act, for the purpose of computing the net torque due to gravity on the body as a whole.

To locate it, imagine the body broken into a very large number of small mass elements $m_1, m_2, \ldots$,

each experiencing its own small weight $m_1 g_1, m_2 g_2, \ldots$ pulling straight down, where $g_1, g_2, \ldots$ is the local acceleration due to gravity at each element's own location. The centre of gravity is

the point about which the net torque due to all these individual weights, taken together, exactly equals

the torque that a single force equal to the body's total weight $Mg$ would produce if it acted at that one

point alone.

When the centre of gravity coincides with the centre of mass. If the body is small enough, or is

located in a region where the acceleration due to gravity $g$ can be treated as exactly uniform in both

magnitude and direction across the whole body (an excellent approximation for any laboratory-sized or even

building-sized object on the Earth's surface), then every mass element's own weight is simply

$m_i g$ with the SAME value of $g$ throughout, and the mass-weighted centre of these weights works out to

be located at EXACTLY the same point as the ordinary, purely mass-weighted centre of mass defined in

Sections 5.2-5.3. Under this uniform-gravity condition -- which holds for essentially every rigid body

problem met in this chapter and in ordinary laboratory or engineering practice -- centre of mass and

centre of gravity are simply two different names for the very same point, and the two terms are often used

interchangeably.

The two points genuinely separate only for a body so large, or so situated, that $g$ measurably varies in

strength or direction from one part of the body to another -- for instance, an extremely tall structure or …