


Center of Mass
When calculating trajectories and collisions, it’s convenient
to treat extended bodies, such as boxes and balls, as point masses.
That way, we don’t need to worry about the shape of an object, but
can still take into account its mass and trajectory. This is basically
what we do with freebody diagrams. We can treat objects, and even
systems, as point masses, even if they have very strange
shapes or are rotating in complex ways. We can make this simplification because
there is always a point in the object or system that has the same
trajectory as the object or system as a whole would have if all
its mass were concentrated in that point. That point is called the
object’s or system’s center of mass.
Consider the trajectory of a diver jumping into the water.
The diver’s trajectory can be broken down into the translational
movement of his center of mass, and the rotation of the rest of
his body about that center of mass.
A human being’s center of mass is located somewhere around
the pelvic area. We see here that, though the diver’s head and feet
and arms can rotate and move gracefully in space, the center of
mass in his pelvic area follows the inevitable parabolic trajectory
of a body moving under the influence of gravity. If we wanted to
represent the diver as a point mass, this is the point we would
choose.
Our example suggests that Newton’s Second Law can be rewritten
in terms of the motion of the center of mass:
Put in this form, the Second Law states that the net force
acting on a system, , is equal to the
product of the total mass of the system, M,
and the acceleration of the center of mass, . Note that if the net force acting on a
system is zero, then the center of mass does not accelerate.
Similarly, the equation for linear momentum can be written
in terms of the velocity of the center of mass:
You will probably never need to plug numbers into these
formulas for SAT II Physics, but it’s important to understand the
principle: the rules of dynamics and momentum apply to systems as
a whole just as they do to bodies.
Calculating the Center of Mass
The center of mass of an object of uniform density is
the body’s geometric center. Note that the center of mass does not
need to be located within the object itself. For example, the center
of mass of a donut is in the center of its hole.
For a System of Two Particles
For a collection of particles, the center of mass can
be found as follows. Consider two particles of mass and separated by a distance d:
If you choose a coordinate system such that both particles
fall on the xaxis, the center of mass of this
system, , is defined by:
For a System in One Dimension
We can generalize this definition of the center of mass
for a system of n particles on a line. Let
the positions of these particles be , , . . ., . To simplify our notation, let M be
the total mass of all n particles
in the system, meaning . Then, the center of
mass is defined by:
For a System in Two Dimensions
Defining the center of mass for a twodimensional system
is just a matter of reducing each particle in the system to its x and ycomponents.
Consider a system of n particles in
a random arrangement of xcoordinates , , . . . , and ycoordinates , , . . ., . The xcoordinate of the
center of mass is given in the equation above, while the ycoordinate
of the center of mass is:
How Systems Will Be Tested on SAT II Physics
The formulas we give here for systems in one and two dimensions
are general formulas to help you understand the principle by which
the center of mass is determined. Rest assured that for SAT II Physics,
you’ll never have to plug in numbers for mass and position for a system
of several particles. However, your understanding of center of mass
may be tested in less mathematically rigorous ways.
For instance, you may be shown a system of two or three
particles and asked explicitly to determine the center of mass for
the system, either mathematically or graphically. Another example,
which we treat below, is that of a system consisting of two parts,
where one part moves relative to the other. In this cases, it is
important to remember that the center of mass of the system as a
whole doesn’t move.
Example

If you’ve ever tried to walk from one end of
a small boat to the other, you may have noticed that the boat moves
backward as you move forward. That’s because there are no external
forces acting on the system, so the system as a whole experiences
no net force. If we recall the equation , the center of mass of the system cannot
move if there is no net force acting on the system. The fisherman
can move, the boat can move, but the system as a whole must maintain
the same center of mass. Thus, as the fisherman moves forward, the
boat must move backward to compensate for his movement.
Because the boat is symmetrical, we know that the center
of mass of the boat is at its geometrical center, at x =
^{L}/_{2} .
Bearing this in mind, we can calculate the center of mass of the
system containing the fisherman and the boat:
Now let’s calculate where the center of mass
of the fishermanboat system is relative to the boat after the fisherman
has moved to the front. We know that the center of mass of the fishermanboat
system hasn’t moved relative to the water, so its displacement with
respect to the boat represents how much the boat has been displaced
with respect to the water.
In the figure below, the center of mass of the boat is
marked by a dot, while the center of mass of the fishermanboat
system is marked by an x.
At the front end of the boat, the fisherman is now at
position L, so the center of mass
of the fishermanboat system relative to the boat is
The center of mass of the system is now ^{3}
/_{5} from the back
of the boat. But we know the center of mass hasn’t moved, which
means the boat has moved backward a distance of
^{1}/_{5 } L,
so that the point ^{3}/
_{5 } L is
now located where the point ^{2}
/_{5} L was
before the fisherman began to move.
