Background Information and Techniques - Computer Animation: Algorithms and Techniques - page 476

v ¼ mð = ð 6 pRn Þ (B.119)

m dv

dt ¼ mg

6 pRnv (B.120)

B.7.11 Centrifugal force
Consider an object (a frame of reference) that rotates in uniform circular motion with respect to a post
(an inertial frame) because it is held at a constant distance by a rope. Relative to the inertial frame, each
point in the uniformly rotating frame has centripetal acceleration expressed by Equation B.121 ; r is the
distance of the point from the axis of rotation, R is the unit vector from the inertial frame to the rotating
frame, and v is the speed of the point. The tension in the rope supplies the force necessary to produce the
centripetal acceleration. Relative to the rotating frame (not an inertial frame), the frame itself does not
move and therefore the centrifugal force necessary to counteract the force supplied by the rope is cal-
culated by using Equation B.122 .

2
r R

v

a ¼ (B.121)

2

mv R

F c ¼ðmaÞ¼

(B.122)

r

B.7.12 Work and potential energy
For a constant force of magnitude F moving an object a distance h parallel to the force, the work W
performed by the force is shown in Equation B.123 . If a mass m is lifted up so that it does not accelerate,
then the lifting force is equal to the weight (mass gravitational acceleration) of the object. Since the

weight is constant, the work done to raise the object up to a height h is presented in Equation B.124 .
Energy that a body has by virtue of its location is called potential energy . The work in this case is con-
verted into potential energy.

W ¼ Fh (B.123)

W ¼ mgh (B.124)

B.7.13 Kinetic energy
Energy of motion is called kinetic energy and is shown in Equation B.125 . The velocity of a falling
body that started at a height h is calculated by Equation B.126 . Its kinetic energy is therefore calculated
by Equation B.127 .

1
2 mv

2

K ¼ (B.125)

2

v ¼

2 gh

(B.126)

1
2 mv

2

K ¼ ¼ mgh (B.127)