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 2rdr
M
dm 2rdr
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 2rdr
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