Center of Mass

Center of Mass
Newtons 2nd Law and
1. Forces cause changes in an object’s momentum. That
is, forces cause the object’s velocity to change over
2. We can determine the change in an object’s velocity
over time using this expression in the form:
p p f  pi
 F  t  t  t
A special point…
•If the net external force on a system of
particles is zero, then (even if the velocity
of individual objects changes), there is a
point associated with the distribution of
objects that moves with zero
acceleration (constant velocity).
•This point is called the “center of mass” of
the system. It is the balancing point for the
mass distribution.
Location of Center of Mass
The CM could be located:
• within the object (human
standing straight)
• outside the object (high jumper
as she goes over the bar)
Center of Mass is outside the
Center of Mass
• System of Particles
• Extended Object
Center of Mass of two Objects in
one Dimension
• mi is the mass of each object
• xi is the position of each objects center
of mass with respect to the origin
Example 1: Center of Mass
in one Dimension
• Find the CM of a system of two
spheres that have a mass of 2 kg
each. One is located 3 cm from the
origin on the + x-axis and one is 4
cm from the origin on the – x-axis
• Answer:
-.5 cm
Center of Mass of a Rod
• Find the center of mass of a rod of mass
M and length L.
Ans: xCM = L / 2
•The overall translational motion of a
system of particles can be analyzed
using Newton's laws as if all the mass
were concentrated at the center of
mass and the total external force were
applied at that point.
Internal and External Forces
•There are two classes of forces that act on
and within systems.
•INTERNAL FORCES are forces between an
object within the system and another object
within the system.
•EXTERNAL FORCES are forces between an
object within the system and an object
outside the system.
Only External Forces
Change the Total
Momentum of the System
Recall Newton’s third law. For every force there is an equal and
opposite force. These “paired” forces are called Newton’s third law
Two forces are a Newton’s third law pair if
1. The two different forces are between the same two objects
2. The object exerting the force and the object being acted on switch
roles in the two forces.
Third Law force pairs cancel out. So only forces external to a system
matter. ALL internal forces are third law pairs.
Horse and the Cart
Horse and the Cart
If there are no external forces this is a true statement. Not only will
the center of mass not be able to start moving, but if it is, there can
be no way for it to stop if it is moving.
The only way for the center of mass of the system can possibly
change its momentum is that there must be a net EXTERNAL force
acting on it.
It is still possible for individual parts to change their momentum (by
preforming an explosion), but the center of mass cannot no change