Transcript Slide 1

Rotational Inertia of a Continuous Object
r
M
m
n
I   mi ri2
Approximate rotational inertia
i 1
n

2
I  lim   mi ri 
n i 1

I   r 2dm
Rotational Inertia of a
continuous body
Rotational Inertia Derivations
Determine the rotational inertia (moment of inertia) for the following objects.
a. A hoop of mass M and radius R about an axis through the center and
perpendicular to the plane of the hoop.
R
I   r 2 dm
But for all dm,
r = R = constant
I   R2dm
I  R2  dm
I hoop  MR 2
Hoop about central axis
Rotational Inertia Derivations
Determine the rotational inertia (moment of inertia) for the following objects.
b. A thin cylindrical shell of mass M and radius R about its axis.
R
Treat the cylindrical shell as a series of hoops each
with a mass of mi and radius R.
I   I hoops
mi
I   mi R 2
But R2 is constant
I  R 2  mi
I  MR
2
Cylindrical shell
about a central axis
Rotational Inertia Derivations
Determine the rotational inertia (moment of inertia) for the following objects.
c. A thin rod of mass M and length L about an axis perpendicular to the rod and
through its center of mass.
dx

dm
L
2
L
2
I   r 2 dm
M

L
dm

dx
dm   dx
Change r to x since rod is
horizontal
L
I
I   x dm
2
Linear Density
2
2
x
  dx
L
2
Rotational Inertia Derivations
Determine the rotational inertia (moment of inertia) for the following objects.
c. A thin rod of mass M and length L about an axis perpendicular to the rod and
through its center of mass.
  L 
dx

dm
L
2
L
I 
L
2
3

L


I        
3  2   2  
I
2
2
x
 dx
L
I
2
L
1 3 2

I   x 
3   L 2
I
3
  L3   L3 
    
3  8   8 
  L3 
3  4 
L3
12
Rotational Inertia Derivations
Determine the rotational inertia (moment of inertia) for the following objects.
c. A thin rod of mass M and length L about an axis perpendicular to the rod and
through its center of mass.
I
dx

L
2
dm
L
2
But
L3
12
M

L
1 M  3
I   L
12  L 
Thin Rod about axis through
center perpendicular to length
1
I  ML2
12
Rotational Inertia Derivations
Determine the rotational inertia (moment of inertia) for the following objects.
d. A solid disk of mass M, radius R, and uniform density about an axis through its
center.
Calculate I by taking mass elements
dm as shown in the diagram. Each
mass element is a hoop of radius r
and thickness dr.
dm
dr
R
r
I   r 2dm
But the density of the disk is
constant. Hence,
M dm

A dA
Where dA  2rdr
M
dm  2rdr
A
Rotational Inertia Derivations
Determine the rotational inertia (moment of inertia) for the following objects.
d. A solid disk of mass M, radius R, and uniform density about an axis through its
center.
R

2 M
dm
dr
r

I  r  2rdr 
A


0
R
M 3
I  2
r dr
A

0
R
M
I  2
A
R
1 r 4 
 4  0
1 M 4
I   R 0
2 A
Rotational Inertia Derivations
Determine the rotational inertia (moment of inertia) for the following objects.
d. A solid disk of mass M, radius R, and uniform density about an axis through its
center.
1 M 4
I  R
2 A
dm
dr
r
But the area of the entire
disk is
2
A  R
R
1 M 4
I  2R
2 R
Thin solid disk about central axis
1
I  MR 2
2
Rotational Inertia Derivations
Determine the rotational inertia (moment of inertia) for the following objects.
e. A solid cylinder of mass M, radius R, and uniform density about a central axis.
R
Consider the cylinder as a set of disks,
each with mass mi.
I   I disks
1
2
I   mi R
2
mi
But the radius is constant.
Solid cylinder about central axis
1 2
I  R  mi
2
1
I  MR 2
2