Tittle is about the mass centre

1. Center of Mass
BY
Usman Sajid

2. • The point of an object at which all the mass
of the object is thought to be concentrated.
• This point is called the “center of mass” of
the system. It is the balancing point for the
mass distribution

3. 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).
.

4. Center of Mass
• The CM will lie along
the central axis (the
axis of symmetry)
• And it is the CM that
faithfully follows the
line of a parabola

5. Center of Mass
• We define the center of mass for these two
particles to be:
d
mm
m
x
21
2
com
+
=

6. Center of Mass
• From this we can see that if m2 = 0, then xcom
= 0
• Similarly, if m1 = 0, then xcom = d

7. Center of Mass
• Finally, if m1 = m2, then xcom = ½d
• So we can see that the center of mass in this
case is constrained to be somewhere
between x = 0 and x = d

8. Center of Mass
• Now lets shift the
origin of the
coordinate system a
little
• We now need a
more general
definition of the
center of mass

9. Center of Mass
• The more general
definition (for two
particles) is:
• Note that if x1 = 0 we
are back to the
previous equation
21
2211
com
mm
xmxm
x
+
+
=

10. Center of Mass
• Now let’s suppose that we have lots of
particles – all lined up nicely for us on the x
axis
• The equation would now be:
where M = m1 + m2 + … + mn
M
xmxmxm
x nn+++
=
...2211
com

11. Center of Mass
• The collection of terms in the numerator
can be rewritten as a sum resulting in:
∑=
=
n
i
ii xm
M
x
1
com
1

12. CENTER OF MASS
∑∑∑ === nncmnncmnncm zm
M
zym
M
yxm
M
x
111
:equationscomponentThree
 M = m1 + m2 + ···
M = Σ mn

13. Center of Mass
• Noticing that xcom and xi, etc. are distances
along the main axis of our coordinate
system, we could just as easily switch to
vector notation
• First recall that the position of mass mi
using vector notion would be:
kzjyixr iiii
ˆˆˆ ++=


14. Center of Mass
• Our center of mass equation using vector
notation would therefore be:
remembering that M is the total mass of the
system
∑=
=
n
i
iirm
M
r
1
com
1 

15. Position, Velocity and Acceleration of
CENTER OF MASS
∑
∑
∑
=
=
=
nncm
nncm
nncm
am
M
a
m
M
xm
M
x
1
1
1
vv