10 STATICS 10.3 Equilibrium of a laminar object in a gravitational field
Xi=1,N..
we calculate the net torque at will—this choice is usually made so as to simplify
the calculation.
Another question which needs clarification is as follows. At which point shouldwe assume that the weight of the system acts in order to calculate the contribu-
tion of the weight to the net torque acting about a given point? Actually, in
Sect. 8.11, we effectively answered this question by assuming that the weight
acts at the centre of mass of the system. Let us now justify this assumption. The
external force acting on the ith component of the system due to its weight is
Fi = mi g, (10.11)where g is the acceleration due to gravity (which is assumed to be uniform
throughout the system). Hence, the net gravitational torque acting on the system
about the origin of our coordinate scheme is
τ =
i=1,Nri × mi g =X
mi ri × g = rcm × M g, (10.12)where M = (^) i=1,N mi is the total mass of the system, and rcm = i=1,N mi ri/M
is the position vector of its centre of mass. It follows, from the above equation,
that the net gravitational torque acting on the system about a given point can be
calculated by assuming that the total mass of the system is concentrated at its
centre of mass.
10.3 Equilibrium of a laminar object in a gravitational field
Consider a general laminar object which is free to pivot about a fixed perpendic-
ular axis. Assuming that the object is placed in a uniform gravitational field (such
as that on the surface of the Earth), what is the object’s equilibrium configuration
in this field?
Let O represent the pivot point, and let C be the centre of mass of the ob-
ject. See Fig. 90. Suppose that r represents the distance between points O and
C, whereas θ is the angle subtended between the line OC and the downward