9781118230725.pdf

stage (Fig. 9-6). These actions shift her center of mass upward through her body.

Although the shifting center of mass faithfully follows a parabolic path across the

stage, its movement relative to the body decreases the height that is attained by her

head and torso, relative to that of a normal jump. The result is that the head and torso

follow a nearly horizontal path, giving an illusion that the dancer is floating.

Proof of Equation 9-14

Now let us prove this important equation. From Eq. 9-8 we have, for a system of n

particles,

(9-16)

in which Mis the system’s total mass and is the vector locating the position of

the system’s center of mass.

Differentiating Eq. 9-16 with respect to time gives

(9-17)

Here is the velocity of the ith particle, and is the

velocity of the center of mass.

Differentiating Eq. 9-17 with respect to time leads to

(9-18)

Here is the acceleration of the ith particle, and is

the acceleration of the center of mass. Although the center of mass is just a geo-

metrical point, it has a position, a velocity, and an acceleration, as if it were a particle.

From Newton’s second law, is equal to the resultant force that acts on

theith particle. Thus, we can rewrite Eq. 9-18 as

(9-19)

Among the forces that contribute to the right side of Eq. 9-19 will be forces that

the particles of the system exert on each other (internal forces) and forces

exerted on the particles from outside the system (external forces). By Newton’s

third law, the internal forces form third-law force pairs and cancel out in the sum

that appears on the right side of Eq. 9-19. What remains is the vector sum of

all the externalforces that act on the system. Equation 9-19 then reduces to

Eq. 9-14, the relation that we set out to prove.

Ma:comF 1

:

F 2

:

F 3

:





Fn

:

.

Fi

:

mia:i

a:i (dv:i/dt) :acom (dv:com/dt)

Ma:comm 1 a: 1 m 2 a: 2 m 3 a: 3 



mna:n.

:vi :vcom (dr:com/dt)

(dr:i/dt)

Mv:comm 1 :v 1 m 2 v: 2 m 3 v: 3 





mnv:n.

r:com

Mr:comm 1 r: 1 m 2 r: 2 m 3 :r 3 



mnr:n,

222 CHAPTER 9 CENTER OF MASS AND LINEAR MOMENTUM

Path of head

Path of center of mass

Figure 9-6A grand jeté. (Based on The Physics of Dance,by Kenneth Laws, Schirmer

Books, 1984.)