Until now, we've been looking at how individual particles behave.
Only "translational" motion was explored -
Now we look at ROTATIONAL motion
(More complicated vibrational motion we'll save for other courses)
First, consider a system of particles:
think of examples...
Describe the system in terms of
what the center of mass is doing, and
what the rest is doing relative to the center of mass
Example: think of a swarm of bees...
the center of the swarm of bees can travel
and the bees also move around the center of the swarm
Example: meter stick twirling as I walk around
the meterstick is made up of many many particles - but is actually
considered a rigid body
Center of Mass for a system of particles
Define the center of mass - a point that is characteristic of the entire
system
concept:
a point that has as much mass on either side (in any direction)
a balancing point (especially if the object is one-dimensional)
mathematically:
a weighted average of all the positions of bits of mass
add up all the positions
weighted by how much mass is at each position
divide by the total mass of all
we could consider the class as a system of particles...
simple average of all positions is the "centroid"
use mass as a weighting in a weighted average, get the COM
can do each component separately
Center of Mass for a continuous body
Same idea for a continuous body like the meterstick, but
instead of a sum over discrete bits of mass,
use an integral over infinitesimal bits of mass
Example:
Meterstick (expect it to be in the center)
Motion in two parts: of the COM, and about the COM
All of our laws for particle motion also work for systems of particles:
the COM has position, velocity, and acceleration
the momentum of the system is equivalent to the momentum of a single
particle
with a mass = total mass of the system
located at the center of mass
moving with the velocity of the COM
the net external force on the system accelerates the COM by Newton's
second law
What about energy?
The kinetic energy of a system of particles is
the kinetic energy of the center of mass (using it's velocity and
the total system mass)
the kinetic energy relative to the center of mass
find the velocity of each particle relative to the COM
find kinetic energy of each particle using that velocity
add up KE from all particles
As above, the motion is thought of as
motion of center of mass
motion relative to center of mass
What causes motion relative to COM? We'll get into that next time...