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10.3 The Motion of the Center of Mass 325

If the net external force vanishes, then the acceleration of the center of mass
vanishes; hence the center of mass remains at rest or it moves with uniform velocity.

During a “space walk,” an astronaut floats in space
EXAMPLE 9
8.0 m from his spacecraft orbiting the Earth. He is teth-
ered to the spacecraft by a long umbilical cord (see Fig. 10.22); to return,
he pulls himself in by this cord. How far does the spacecraft move toward
him? The mass of the spacecraft is 3500 kg, and the mass of the astronaut,
including his space suit, is 110 kg.
SOLUTION: In the reference frame of the orbiting (freely falling) astronaut
and spacecraft, each is effectively weightless; that is, the external force on
the system is effectively zero. The only forces in the system are the forces
exerted when the astronaut pulls on the cord; these forces are internal.The
forces exerted by the cord on the spacecraft and on the astronaut during the
pulling in are of equal magnitudes and opposite directions; the astronaut
is pulled toward the spacecraft, and the spacecraft is pulled toward the astronaut. FIGURE 10.22 Astronaut on a “space
In the absence of external forces, the center of mass of the astronaut–spacecraft walk” during the Gemini 4 mission.
system remains at rest. Thus, the spacecraft and the astronaut both move toward
the center of mass, and there they meet.
With the x axis as in Fig. 10.23, the x coordinate of the center of mass is

m x m x
1 1
2 2
x (10.41)
CM
m m
1 2
where m 3500 kg is the mass of the spacecraft and m 110 kg is the mass of
1 2
the astronaut. Strictly, the coordinates x and x of the spacecraft and of the astro-
1 2
naut should correspond to the centers of mass of these bodies, but, for the sake of
simplicity, we neglect their size and treat both as particles.The initial values of the
coordinates are x 0 and x 8.0 m; hence
1 2
0 110 kg 8.0 m
x 0.24 m
CM
3500 kg 110 kg
During the pulling in, the spacecraft will move from x 0 to x 0.24 m;
1 1
simultaneously, the astronaut will move from x 8.0 m to x 0.24 m.
2 2
(a) (b)
Distances to the center of Position of the
mass are in inverse proportion center of mass
y y
to the masses. remains fixed.

x x
O O
CM
CM
x = 8.0 m

FIGURE 10.23 (a) Initial position of the astronaut and the spacecraft. The center of mass is between them.
(b) Final position of the astronaut and the spacecraft. They are both at the center of mass.