Download Topic 2_1_Ext N__Center of mass 1

Topic 2.1 Extended
N – Center of mass 1
Up to this point we've considered only "point masses"
which can be treated as if they have no "extent."
The reason we do this is so that we can treat all of
the particles in the mass as one - all having the same
position, speed and acceleration.
Even if the mass is extended, as in this wrench, we
can treat it as a point mass as long as it is not
rotating.
If the wrench is rotating as it slides, we can no
longer treat it as a point mass:
Note that of all of the points on the extended body,
ONLY THE WHITE DOT FOLLOWS A STRAIGHT LINE.
We call the position of the white dot the center of
mass. The center of mass (CM) is that point about
which all other points of the extended mass rotate (if
rotation even occurs).
This section will teach you how to find the center of
mass of an extended body. We'll need this skill later
when we look at rotation, and balancing structures.
Topic 2.1 Extended
N – Center of mass 1
It turns out that if you have an
extended mass, Newton's 2nd law
applies to the center of mass (cm)
of the extended mass:
We write
Fnet,
external
= MAcm
Newton's 2nd
Law for a
System of
Particles
where M is the total mass of the
extended mass.
Note that only the external force
contributes to the acceleration of
the system.
Since the spreading explosion of
the fireworks was caused by
internal forces, the whole mass
will continue along the original
parabolic trajectory:
Topic 2.1 Extended
N – Center of mass 1
Since Acm =
Vcm
, we can write
t
Fnet, external = MAcm
Vcm
Fnet, external = M
t
MVcm
Fnet, external =
t
Newton's 2nd Law
Fnet, external = P
t
P-form
where P = MVcm is the total momentum of the
system of particles making up the extended mass.
If Fnet, external = 0 then we see that P = MVcm is
zero.
FYI: Thus, if the net external force acting on a system of particles is
ZERO, that cm of that system must move at a constant velocity or be
at rest.
FYI: The cm is where we can think of all the mass of an extended
Topic
2.1
Extended
body as being concentrated
as far
as translational
motion is
concerned.
N – Center of mass 1
CENTER OF MASS
The center of mass is applicable to ANY system of
masses - even those of a gas.
particles we have
Xcm =
For a system of n
m1x1 + m2x2 + ... + mnxn
m1 + m2 + ... + mn
n
Xcm =
 mixi
M
Location
of Xcm
i= 1
Find the center of mass of the following system:
3 kg
2 kg
y
8 kg
1 kg
x
m1x1 + m2x2 + m3x3 + m4x4
m1 + m2 + m3 + m4
3-7 + 2-2 + 80 + 17
=
3 + 2 + 8 + 1
xcm =
= -1.3 m
Topic 2.1 Extended
N – Center of mass 1
CENTER OF MASS
The center of mass doesn't necessarily have to lie
within the mass itself.
n
(xcm,ycm)
Ycm
(xcm,ycm)
n
(xcm,ycm,zcm)
Zcm =
 mizi
M
i= 1
 miyi
i= 1
=
M
FYI: The cg is now known, but a third trial will ensure that our first two
mappings are correct:Topic 2.1 Extended
N –directly
Center
ofpoint
mass
1
FYI: The cg must hang
below the
of suspension
at the tip
of Albert's tail. Albert is playing "dead" for this experiment, and so is
CENTER OFstiff.
GRAVITY
completely
The center of gravity (CG) is where we can think of
the weight of an extended body to be concentrated as
far as translational motion is concerned.
In a region where g is constant, the CG can be found
from
n
Xcg
 m gx
= i= 1 i i
Mg
Location
of Xcg
How would you find the CG of Albert?
Albert the physics cat
Since Albert is asymmetric, you find his center of
gravity by hanging him three different ways: