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Print this file using A4 and write your solutions on it
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All or nothing per set-up
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A 3-ft long rod has a density that varies linearly from one end to the other
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Find the center of mass of the rod
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A

B

In terms of x
A
Center of
Mass

Centroid with
AOR
Y=0
Centroid with
AOR
X=0
B
Center of
Mass
Centroid with
AOR
X=0

End of Exercise 10

In terms of y

Unit 3
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4
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4
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1

Find the center of mass of the three masses: 1, 3,
and 7 kg with the directed distance of 2,  4, and
9, respectively from the origin
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0

L

x

Center of Mass of a Rod
The center of mass of a rod of length L meters
with its left endpoint at the origin and   x  kg
per meter as the linear density at any point x in
the rod is at

 x  x  dx
x
   x  dx
L

0

L

0

Example 3
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1
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Find the center of
mass of the rod
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4
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3
Show that the center of mass of a homogeneous
rod of length L is located at the middle of the rod
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4
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4
The linear density at any point on the rod varies
directly as the third power of the distance of that
point from the left end
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Find the center of mass of the rod
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4
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5
The density at any point on an 8-meter rod is a
linear function of the distance of the point from
the left end
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(set up only)
For   x  :

  x   ax  b

  0  2

  0  b

  8  5

 8  8a  b

For   x  :

  x   ax  b

  0  2

  0  b

  8  5

 8  8a  b

b2

8a  b  5
8a  2  5
8a  3
a  38
3
Therefore,   x   x  2
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Exercises 6
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End of Unit 3
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1

Unit 3
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2
Center of Mass
of a Plane Region

n

x

m x
i 1
n

i i

m
i 1

i

mi xi  moment of mass of the ith particle
with respect to the origin
n

M 0   mi xi  moment of mass of the system
i 1

with respect to the origin
n

M   mi  total mass of the system
i 1

Center of Mass of a Rod
The center of mass of a rod of length L meters
with its left endpoint at the origin and   x  kg
per meter as the linear density at any point x in
the rod is at

 x  x  dx
x
   x  dx
L

0

L

0

Consider a system of n particles with masses
m1, m2 , , mn located at the points  x1, y1  ,

 x2 , y2  , ,  xn , yn  in the

xy  plane
...

n

M x   mi yi  moment of mass of the system
i 1

with respect to the x  axis
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i 1

Consider a flat plate (lamina) of constant density
that occupies the shaded region
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4
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1
Find the centroid of the shaded region (set up
only)

y4
yx

2

 2,4

using
a
...
horizontal strips

a
...
horizontal strips
A

4

0

y4
y

y dy

centroid at  x , y  where
1
M y  0 2 y  y dy
x

4
M
  ydy
4

0

M x  0 y y dy
y

4
M
  y dy
4

0

x y

 2,4

Example 3
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2
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vertical strips

b
...
vertical strips
1 

A    x  x  dx
0
2 

centroid at  x , y  where
4

y x
x

1 

 0 x  x  x  dx
My
2 

x

4
1 
M
 0  x  x  dx
2 

41
1 
1 
 0  x  x  x  x  dx
Mx
2
2 
2 
y

4
1 
M
 0  x  x  dx
2 

4

 4,2
1
y x
2

x  y2
y

b
...
If R
is rotated about l , then the volume of the
resulting solid is the product of the area A
of R and the distance d traveled by the
centroid of R
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Example 3
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2
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Find
the volume of the torus
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Exercises 6
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End of Unit 3
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2

Unit 3
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3
Center of Mass
of a Solid of Revolution

xy  plane:
xz  plane:
yz  plane:

z0
y0
x0

Consider a system of n particles with masses
m1, m2 , , mn located at the points  x1, y1, z1  ,

 x2 , y2 , z2  , ,  xn , yn , zn  in the space
...

n

M xz   mi yi  moment of mass of the system
i 1

with respect to the xz  plane
...

n

M   mi
i 1

 total mass of the system
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Theorem

The center of mass of a solid of revolution is
located at its axis of revolution
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4
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1
Set up the expressions in terms of x that give
the centroid of the solid generated when the
shaded region is revolved about y  1
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4
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2
Set up the expressions in terms of x that give
the centroid of the solid generated when the
shaded region is revolved about x  3
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4
...
3
Set up the expressions in terms of y that give
the centroid of the solid generated when the
shaded region is revolved about y  1
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4
...
4
Set up the expressions in terms of y that give
the centroid of the solid generated when the
shaded region is revolved about x  3
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4

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