644 12. Collision and Rigid Body Dynamics
mass .) This is illustrated in Figure 12.25. The torque N caused by a force F ap-
plied at a location r is
(12.7)
Equation (12.7) implies that torque increases as the force is applied farther
from the center of mass. This explains why a lever can help us to move a heavy
object. It also explains why a force applied directly through the center of mass
produces no torque and no rotation—the magnitude of the vector r is zero in
this case.
When two or more forces are applied to a rigid body, the torque vectors
produced by each one can be summed, just as we can sum forces. So in general
we are interested in the net torque, Nnet.
In two dimensions, the vectors r and F must both lie in the xy-plane, so
N will always be directed along the positive or negative z-axis. As such, we’ll
denote a two-dimensional torque via the scalar Nz , which is just the z-compo-
nent of the vector N.
Torque is related to angular acceleration and moment of inertia in much
the same way that force is related to linear acceleration and mass:
Angular:
N It
I t It
z=
==
α
θ
()
ω() ();
Linear:
Fa
vr
=
==
mt
m t mt
()
() ().
(12.8)
12.4.5.5. Solving the Angular Equations of Motion in Two Dimensions
For the two-dimensional case, we can solve the angular equations of motion
using exactly the same numerical integration techniques we applied to the lin-
ear dynamics problem. The pair of ODEs that we wish to solve is as follows:
Angular:
N t It
t
net^
() ();
() ();
=
=
ω
ωθt
Linear:
Fv
vr
net()^ ();
() (),
t mt
t
=
=
t
and their approximate explicit Euler solutions are
F r
r sin θ
Figure 12.25. Torque is calculated by taking the cross product between a force’s point of
application in body space (i.e., relative to the center of mass) and the force vector. The
vectors are shown here in two dimensions for ease of illustration; if it could be drawn, the
torque vector would be directed into the page.
NrF=×.