Background Information and Techniques - Computer Animation: Algorithms and Techniques - page 471

is said to be constant. Angular velocity is referred to here as o ( t )( Eq. B.97 ). As for constant-velocity
linear motion, in which the distance equals speed multiplied by time, for constant angular velocity the
angle equals angular velocity multiplied by time ( Eq. B.98 ) . If y ( t ) is measured in radians and time in
seconds, then o ( t ) is measured in radians per second. To simplify the following equations, the func-
tional dependence on time will often be omitted when the dependence is obvious from the context.

pðÞ¼ r cos y ðÞ ð ð Þ Þi þ r sin y ðÞ ð ð Þ Þj (B.96)

d
dt yðtÞ¼o ðÞ

(B.97)

y ðÞ¼ot (B.98)

Taking the derivative of EquationB.96 with respect to time gives the instantaneous velocity ( Eq. B.99 ) .
Notice that the velocity vector, v ( t ), is perpendicular to the position vector, p ( t ). This can be demonstrated
by taking the dot product of the two vectors v ( t )and p ( t ) and showing that it is identically zero.

dp
dt ¼ ððrÞo sin

vðtÞ¼ ðotÞÞi þðro cos ðotÞÞj (B.99)

¼ r o and, therefore, that the velocity is independent of t
(i.e., constant). Notice, however, that a constant circular motion still gives rise to an acceleration.
Taking the derivative of Equation B.99 produces Equation B.100 , which is called the centripetal accel-
eration. The centripetal acceleration resulting from uniform circular motion is directed radially inward
and has constant magnitude. With the equation for the length of v ( t ) from earlier, the magnitude of the
acceleration can be written using Equation B.101 .

Computing the length of v ( t ) shows that | v |

dv
dt ¼ðo

2

aðtÞ¼ Þððr cos ðotÞÞi þðr sin ðotÞÞjÞ

(B.100)

2

¼ðo ÞpðtÞ

2

a ¼ v =r (B.101)

For any particle in a rigid mass undergoing a rotation, that particle is undergoing the same rotation
about its own center. In addition, if the particle is displaced from the center of rotation, then it is also
undergoing an instantaneous positional translation as a result of its circular motion ( Figure B.52 ).

B.7.3 Newton's laws of motion
It is useful to review Newton's laws of motion. The first law is the principle of inertia. The second law
relates force to the acceleration of a mass ( Eq. B.102 ) . In another form, this law relates force to change
in momentum ( Eq. B.103 ), where momentum is mass times velocity ( m ￿ v ). The third law states that
when an object pushes with a force on another object, the second object pushes back with an equal but
opposite force. It is important to note that the force F used here is considered to be the sum of all exter-
nal forces acting on an object. Force is a vector quantity, and these equations really represent three sets
of equations, one for each coordinate ( Eq. B.104 ) . Newton's laws of motion are stated as follows:

First Law: If no force is acting on an object, then it will not accelerate. It will maintain a constant
velocity.