Background Information and Techniques - Computer Animation: Algorithms and Techniques - page 474

gravitational constant of Equation B.108 can all be combined to produce the object's acceleration
( Eq. B.109 ) . This acceleration is usually denoted as g.
F ¼ G m 1 m 2
r

(B.108)

2

F
m object ¼ G

m earth
radius earth

s 2 s 2

a ¼

2 ¼

9 : 8m = ¼ 32f = ¼ g (B.109)

ð Þ

B.7.9 Centripetal force
Centripetal force is any force that is directed inward, toward the center of an object's motion (centrip-
etal means center seeking ). In the case of a body in orbit, gravity is the centripetal force that holds the
body in the orbit. Consider a body, such as the moon, with mass M m and a distance r away from the
earth. The unit vector from the earth to the moon is R. The earth has a mass M e . Equation B.110 cal-
culates the force vector that results from gravity. The acceleration induced by a circular orbit always
points toward the center (centripetal), as in Equation B.111 . As previously shown, the acceleration due
to circular motion has magnitude v

2 / r ( Eq. B.100 ) . This can be used to solve for velocity ( Eq. B.112 ).

F ¼G M m M e
r

R (B.110)

2

a ¼ F=M m
a ¼ ððGM e Þ=r

2

ÞR (B.111)

2

a ¼ðv =rÞR

2

a ¼ðGM e =r ÞR

p

GM e v ¼ ð Þ=r (B.112)

B.7.10 Contact forces
Gravity is one of the four fundamental forces (gravity, electromagnetic, strong, and weak) that act at a
distance. Another important category is contact forces . The tension in a wire or rope is an example of a
contact force, as is the compression in a rigid rod. These forces arise from the complex interactions of
electric forces that tend to keep atoms a certain distance apart. The empirical law that governs such
forces, and that is familiar when dealing with springs, is Hooke's law ( Eq. B.113 ). Variable x is the
change from the equilibrium length of the spring, k is the spring constant, and F is the restoring force
of the spring to return to rest length. The constant k is a measure of the stiffness of the spring; the larger
k is, the more sensitive the spring is to motion away from the rest position.

F ¼kx (B.113)

Another example of contact force, known as the normal force , is the result of repulsion between any
two objects pressed against each other. It arises from the repulsion of the atoms of the two objects. It is
always perpendicular to the surfaces, and its magnitude is proportional to how hard the two objects are
pressed against each other. When objects in motion come in contact, an impulse force due to collision is