Rotational dynamics
•  CAPA homework due tomorrow.
•  Chapter 10 (Rotational Motion) today.
1
Relationships to linear acceleration
!
What about acceleration? If speed is v = v = rω then
dv d(rω )
dω
atan =
=
=r
= rα
dt
dt
dt
v2
We know that centripetal (or radial) acceleration is arad = .
r
2
a
=
r
ω
Using v = rω, this can be rewritten as rad
.
Total linear acceleration is composed of both tangential and radial
acceleration which are always perpendicular to each other.
2
2
atotal = arad
+ atan
2
Summary of angular kinematics
Constant angular acceleration equations:
ω z = ω0 z + α zt
1
2
θ = θ 0 + ω0 zt + α zt
2
ω z2 = ω 02z + 2α z (θ − θ 0 )
Relationship between linear and angular variables
(angular variables need to use radians):
s = rθ
v = rω
atan = rα
3
Clicker question 1
Set frequency to BA
A small wheel and a large wheel are connected by a belt. The
small wheel is turned at a constant angular velocity ωs. How
does the magnitude of the angular velocity of the large wheel
ωL compare to that of the small wheel?
ωL
ωS
A.  ω L > ωS
B.  ω L < ωS
C.  ω L = ωS
The speed of the belt is constant: v
Small wheel: v = RSω S
Large wheel: v = RLω L
RSω S = RLω L
Since RS < RL it must
be that ω L < ω S
4
Moment of inertia
Consider an object of mass m at the end
of a massless rod of length R spinning
around an axis with angular velocity ω.
What is the kinetic energy of the mass?
R
m
Remember that v = rω
K = 12 mv2 = 12 mR2ω2 = 12 Iω2
For a particle which is a distance R from the axis
(or pivot) the moment of inertia is I = mR2
As the particle moves further out (R increases), ω doesn’t
change but velocity increases so kinetic energy increases.
This comes from the moment of inertia increasing
5
Moment of inertia
If you have a bunch of particles rotating about an axis,
can find the total moment of inertia by adding up the
moment of inertia of all the particles
I total = I1 + I2 + … = m1r12 + m2r22 + … = ∑ miri2
i
If there is a smooth distribution of matter then the
sum becomes an integral but the idea is the same.
Remember that r is the distance from the axis.
Therefore, the moment of inertia depends on the axis location.
Moment of inertia is also called rotational inertia
6
Moment of inertia (ring & cylindrical shell)
Rings and cylindrical shells (hollow cylinders)
behave like a bunch of little particles all
located at a distance of R from the axis.
I = MR 2
This is only true for the axis shown.
What about a filled in disk or solid cylinder?
Not all of the mass is at a distance R from the axis.
End up with something less than MR2.
1
2 (solid disk or cylinder,
I = MR
including pulleys)
2
7
Moment of inertia
What about a solid sphere spinning about its axis?
Not all of the bits of matter are
at a distance R from the axis.
End up with something less than MR2.
I = 2 MR 2 (solid sphere)
5
A hollow sphere of radius R still doesn’t have all bits of matter
a distance of R from the axis. I = 2 MR 2 (hollow sphere)
3
Moments of inertia for other bodies can be found in Table 10.2
8
Clicker question 2
Set frequency to BA
A pottery wheel has a mass M and a radius of R. A small lump of
clay with mass m is located on the edge of the wheel (a distance
R from the center). What is the moment of inertia of the combined
wheel + clay about the axis through the center of the wheel?
2
A.  ( M + m ) R
I
= 2 MR 2 I = MR 2
B.
1
2
( M + m) R
(
D.  ( M + m ) R
E.  ( M + m ) R
C.
1
2
2
5
2
m
I disk = 12 MR 2
I rod (about center) =
2
R
ring
3
I solid sphere = 25 MR 2
M + m) R 2
1
2
1
2
hollow sphere
2
1
12
M
2
ML
Moments of inertia
are additive: I total = I wheel + I clay
The wheel is a disk so Iwheel = ½MR2 and the clay is a
point mass so Iclay = mR2: I = I
+ I = 1 MR 2 + mR 2
total
wheel
clay
2
9
Clicker question 3
Set frequency to BA
A solid sphere, a ring, and a cylinder, each with mass M and
radius R, are each rotating about an axis through the center of
the object with angular velocity ω. Which one has the largest
kinetic energy?
A.  solid sphere
B.  ring
K = 12 Iω 2
C.  cylinder
D.  All will have the same
E.  Impossible to tell I ring = MR 2 I disk = 12 MR 2 I solid sphere = 2 MR 2
5
I hollow sphere = 23 MR 2
I rod (about center) = 121 ML2
They have the same angular velocity ω, but different moments
of inertia: I
K solid sphere < K cylinder < K ring
solid sphere < I cylinder < I ring so
10
Parallel axis theorem
The moments of inertia for a ring, disk, solid sphere, and
hollow sphere were for an axis through the center of mass
What if the axis is not through the center of mass?
If the axis is parallel to the center of mass axis then can
2
I
=
I
+
Md
use the parallel axis theorem P
.
CM
I CM = moment of inertia for an axis through the center of mass
d = distance between the two parallel axes
It should be clear from the formula that the
smallest momentum of inertia is always the one
where the axis goes through the center of mass.
11
Clicker question 4
Set frequency to BA
Consider a rod of uniform density with an axis of rotation
through its center and an identical rod with the axis of rotation
through one end. Which has the larger moment of inertia?
A.  I C > I E
B.  I E > I C
C.  I C = I E
C
The moment of inertia for a particle
is MR2 so the moment of inertia
increases as M or R increases.
E
axis
axis
For E, the average R is
greater than for C.
Or can use parallel axis theorem: I P = I CM + Md 2
For parallel axes, the smallest moment of inertia
is when the axis is through the center of mass
I rod (about center) =
1
12
2
ML
I rod (about end) =
1
12
2
ML + M
! L $2
# &
"2%
= 13 ML2
12