5.6 Static properties of matter 149
where is the total force acting on the system of masses. The position Xof the
centre of mass is then also called the centre of gravity, and the quantity
is called the moment of forceor torqueof the system of forces about the point O. If
the masses are attached to a uniform rigid rod (itself of negligible mass) then the
torque is a measure of the tendency of the forces to rotate the system around Oas
pivot and, by equation (5.41), it is equal to the torque produced by the total force F
concentrated at the centre of mass. If the point Ois atthe centre of mass (gravity) then
X 1 = 10 and
(5.42)
so that the total torque about the centre of mass is zero.
The second moment of mass, (iii), is the moment of inertiaof the system of masses
with respect to the point O. It is the property of the mass distribution that is most
important in the description of the dynamics of rotating bodies.
EXAMPLE 5.13A system of two masses
Figure 5.17 shows two bodies, massesm
1andm
2, joined by a rigid rod (of negligible
mass). The forcesF
1andF
2are the weights of the bodies, andFis a counter force
acting at the pivot point O. If the pivot is at the centre of mass then, puttingx
11 = 1 −r
1andx
21 = 1 r
2in equation (5.42),
F
1r
11 = 1 F
2r
2(law of levers)
and the body is at equilibrium with respect to rotation about the pivot O. The total
vertical force acting on the body is F
11 + 1 F
21 − 1 F, so that the body is at equilibrium with
respect to vertical motion ifF 1 = 1 F
11 + 1 F
2.
The moment of inertia of the two masses is I 1 = 1 m
1r
211 + 1 m
2r
22.When Ois at the
centre of mass thenX 1 = 10 in (5.40) so thatm
1r
11 = 1 m
2r
2, and the distancesr
1andr
2can be written as
r
m
mm
Rr
m
mm
R
12122112=
,=
TFx
iNii==
=∑
10
TFx
ii=
∑
FF
i=
∑
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Figure 5.17