### Center of Mass and Center of Gravity

```Why is the concept of center of mass useful?
Center of mass is a useful concept because it
simplifies how we look at objects. Rather than
trying to do complex calculations, we can just
find the center of mass and act as if all the
mass is there.
Center of Mass and Center of Gravity
Center of Mass: The center of mass of an object is the
average point of the mass of the object.
Center of Gravity: The point on an object where
all of its weight can be
considered to act.
These are the same points for an object in a constant
gravitational field.
The center of mass lies at the
geometric center for a symmetric,
uniform density object.
R
h
h
h
h/2
h/3
h/2
Center of Mass
M
M
L/2
L/2
L
3M
Center of Mass
1/4L
3/4L
L
M
Calculating the Center of Mass
Two particles of mass m1 and m2 that are located on
the x axis at the position x1 and x2. The distance xcm
of the center of mass point from the origin is:
Sample:
A 1.0 m long dumbbell has a 10 kg mass on the
left and a 5.0 kg mass on the right. Find the center
of mass (the position of the center of gravity).
m1 = 10kg
x1 = 0 m
m2 = 5.0kg
xcm
x2 = 1.0m
Fin the center of mass of an object modeled as
two separate masses on the x-axis. The first
mass is 2 kg at an x-coordinate of 2 and the
second mass is 6 kg at an x- coordinate of 8.
We can use the same strategy for finding the
center of mass of a multi-dimensional object.
Sample Problem: Find the coordinates of the
center of mass for the system shown below.
Center of mass
The center of mass of a system will not move if there is no
external forces acting on the system.
EXAMPLE
A fisherman stands at the back of a perfectly
symmetrical boat of length L. The boat is at rest in
the middle of a perfectly still and peaceful lake, and
1
the fisherman has a mass /4 that of the boat. If the
fisherman walks to the front of the boat, by how
much is the boat displaced?
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:
In the figure below, the center of mass of the
boat is marked by a dot, while the center of
mass of the fisherman-boat 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.
Walking along the slab
A child of mass m is standing at the left end of a thin and
uniform slab of wood of length L and mass M. The slab lies
on a horizontal and frictionless icy surface of a lake. Starting
from rest, the child walks towards the right end of the slab.
a) How far and in what direction did the center of mass of the
child-slab system move when the child has reached the right
end of the slab?
b) Where is the left end of the slab with respect
to its initial position when the child has reached
the right end?
```