Geoscience Reference
In-Depth Information
c
d
a
b
O 2
O 2
r 2 x F 2
F
A
F
r 2
A
r 2 x F 2
r 2 x F 2
r 1 x F 1
r 1 x F 1
r 2
A
A
CMI
r 1
r 1
r 1 x F 1
O
O
O 1
O 1
Fig. 9.1 Torque and reaction torque. ( a ) Reaction torque in opposite direction to the torque, ( b )a
specific case, r 1 D
r 2 ,( c )and( d ) reaction torque in the same direction with the torque
on a body should be mathematically equal to the moment of inertia times angular
acceleration of the body. For Newton's third law, however, a reaction torque is not
defined in physics to lead to a complete correspondence.
Considering Fig. 9.1 , there should be a reaction torque by exactly the same reason
as the reaction force in Newton's third law which is simply a special case that the
radius vectors of torque and reaction torque are equal in magnitude and direction.
Let us suppose that the rod is rigid and its mass is negligible. As a person in Fig. 9.1a
actsasatorque, r 2 F 2 , on an object via the rod to angularly accelerate the object
with respect to the pivot O , the person will feel a reaction torque, r 1 F 1 , acted
by the object, where the torque and reaction torque have the same magnitudes and
opposite directions.
Newton's third law is a specific case of r 1 D r 2 , as shown in Fig. 9.1b .Sincea
torque is a product of two vectors, as shown in Fig. 9.1c, d , torque and reaction
torque may have the same directions depending on the positions of pivots. If all
the bodies are in an isolated system and in a stationary state initially, the person
in Fig. 9.1a rotates in opposite direction to the object to conserve the total angular
momentum before and after the person acts the torque, J D J 1 C J 2 , J D 0, J 1 D J 2 .
On the other hand, since the torque and reaction torque are in the same directions
in Fig. 9.1c, d , there should be two different kinds of rotations, spin and orbital
angular momenta, to conserve the total angular momentum. The person rotates with
a spin angular momentum S 1 with respect to the pivot O 1 , and the object rotates
with S 2 with respect to the pivot O 2 . To conserve the total angular momentum,
J D 0, the whole system rotates with orbital angular momentum L D ( S 1 C S 2 ),
J D L C S 1 C S 2 , in the direction indicated by arrows in O 1 and O 2 . The pivot of the
rotation will be a point, center of moment of inertia (CMI), about which moment of
Search WWH ::




Custom Search