Physically Based Animation - Computer Animation: Algorithms and Techniques - page 216

F ¼ ma (7.31)

a ¼ F =m (7.32)

( t )( Eq. 7.33 ).
Given the mass of the point, the acceleration due to the total external force can be calculated and then
used to modify the velocity of the point. This can be done at each time step. If the point is assumed to be
part of a rigid object, then the point's location on the object must be taken into consideration, and the
effect of the force on the point has an impact on the object as a whole. The rotational equivalent of linear
force is torque , t ( t ). The torque that arises from the application of forces acting on a point of an object is
given by Equation 7.34 .

The various forces acting on a point can be summed to form the total external force, F

¼ P f i ðÞ F (7.33)

t i ¼ ð q i ðÞ x ðÞ Þ f i ðÞ

t ¼ P t i ðÞ

(7.34)

Momentum
As with force, the momentum (mass

velocity) of an object is decomposed into a linear component and
an angular component. The object's local coordinate system is assumed to be located at its center of mass.
The linear component acts on this center of mass, and the angular component is with respect to this center.
Linear momentum and angular momentum need to be updated for interacting objects because these values
are conserved in a closed system. Saying that the linear momentum is conserved in a closed system, for
example, means that the sum of the linear momentum does not change if there are no outside influences
on the system. The case is similar for angular momentum. That they are conserved means they can be
used to solve for unknown values in the system, such as linear velocity and angular velocity.
Linear momentum is equal to velocity times mass ( Eq. 7.35 ). The total linear momentum ( t ) of a rigid

body is the sum of the linear momentums of each particle (Eq. 36). For a coordinate system whose origin
coincides with the center of mass, Equation 7.36 simplifies to the mass of the object times its velocity
( Eq. 7.37 ). Further, since the mass of the object remains constant, taking the derivative of momentum
with respect to time establishes a relationship between linear momentum and linear force ( Eq. 7.38 ). This
states that the force acting on a body is equal to the change in momentum. Interactions composed of equal
but opposite forces result in no change in momentum (i.e., momentum is conserved).

P

P ¼ m v (7.35)

PðÞ¼Sm i q_ i ðÞ (7.36)

P ðÞ¼M v ðÞ (7.37)

P ðÞ¼M_

v ðÞ¼ F ðÞ (7.38)

Angular momentum , L

, is a measure of the rotating mass weighted by the distance of the mass from
the axis of rotation. For a mass point in an object, the angular momentum is computed by taking the
cross-product of a vector to the mass and a velocity vector of that mass point

the mass of the point.
These vectors are relative to the center of mass of the object. The total angular momentum of a rigid
body is computed by integrating this equation over the entire object. For the purposes of computer