Simple Nature - Light and Matter

n/An improperly balanced
wheel has a center of mass that
is not at its geometric center.
When you get a new tire, the
mechanic clamps little weights to
the rim to balance the wheel.

o/This toy was intentionally
designed so that the mushroom-
shaped piece of metal on top
would throw off the center of
mass. When you wind it up, the
mushroom spins, but the center
of mass doesn’t want to move,
so the rest of the toy tends to
counter the mushroom’s motion,
causing the whole thing to jump
around.

is at

xcm=

1

n

∑n

j=1

xj,

wherex 1 is the mass of the first particle, and so on. The velocity of

the center of mass is

vcm= dxcm/dt

=

1

n

∑n

j=1

vj

=

1

nm

∑n

j=1

mvj

=

ptotal

mtotal

What about a system containing objects with unequal masses,

or containing more than two objects? The reasoning above can be

generalized to a weighted average:

xcm=

∑n

∑j=1mjxj

n

j=1mj

The solar system’s center of mass example 13

In the discussion of the sun’s gravitational field on page 99, I

mentioned in a footnote that the sun doesn’t really stay in one

place while the planets orbit around it. Actually, motion is relative,

so it’s meaningless to ask whether the sun is absolutely at rest,

but it is meaningful to ask whether it moves in a straight line at

constant velocity. We can now see that since the solar system

is a closed system, its total momentum must be constant, and

pt ot al=mt ot alvcmthen tells us that it’s the solar system’s center

of mass that has constant velocity, not the sun. The sun wob-

bles around this point irregularly due to its interactions with the

planets, Jupiter in particular.

The earth-moon system example 14

The earth-moon system is much simpler than the solar system

because it contains only two objects. Where is the center of mass

of this system? Letx=0 be the earth’s center, so that the moon

lies atx= 3.8× 105 km. Then

xcm=

∑ 2

j=1mjxj

∑ 2

j=1mj

=

m 1 x 1 +m 2 x 2

m 1 +m 2

,

144 Chapter 3 Conservation of Momentum