Rotational Motion
(1) Kinematics
Everything’s analogous to linear kinematics
Define angular properties properly
and derive the equations of motion
by analogy
Motion on a Wheel
Computer Hard Drive
A computer hard drive typically rotates at
5400 rev/minute
Find the:
• Angular Velocity in rad/sec
• Linear Velocity on the rim (R=3.0cm)
• Linear Acceleration
It takes 3.6 sec to go from rest to 5400
rev/min, with constant angular acceleration.
• What is the angular acceleration?
Computer Hard Drive
Numbers worked out:
Examples
Consider two points
on a rotating wheel.
One on the inside (P)
and the other at the
end (b):
• Which has greater
angular velocity?
• Which has greater
linear velocity?
b
R1
R2
Rotation and Translation
Rolling without Slipping
• In reality, car tires both rotate and
translate
• They are a good example of
something which rolls (translates,
moves forward, rotates) without
slipping
• Is there friction? What kind?
Derivation
• The trick is to pick your
reference frame correctly!
• Think of the wheel as sitting
still and the ground moving
past it with speed V.
Velocity of ground (in bike
frame) = -ωR
=> Velocity of bike (in ground
frame) = ωR
Bicycle comes to Rest
A bicycle with initial linear velocity V0 decelerates
uniformly (without slipping) to rest over a distance
d. For a wheel of radius R:
a) What is the angular velocity at t0=0?
b) Total revolutions before it stops?
c) Total angular distance traversed
by wheel?
(d) The angular acceleration?
(e) The total time until it stops?
Torque vs. Force
• Torque: rot. Force
• Remember: a ∝ F
• α∝?
• α∝F
• α ∝ R⊥
• α ∝ τ = R⊥ F
Torque – More general
• τ = R⊥ F
• τ = R F⊥
• τ = R F sinθ
Torque – More general
• +: clockwise
• Two Torques, opposite
• τ = -R1 F1 + R2 F2 sin 60°
A better way to define Torque
r
r
r
τ = R×F
This gives us the magnitude
and the direction
Vector Cross Product
r r r
C = A× B
C = A B Sin Θ
Direction from Right - Hand - Rule
" Swing A into B"
Check :
r r
A× A
r r
r r
A × B vs. B × A
Example of Cross Product
The location of a
body is length r from
the origin and at an
angle θ from the xaxis. A force F acts
on the body purely
in the y direction.
What is the Torque
on the body?
x
z
y
θ
Rotational Dynamics
• What plays the role of mass in rotation?
•
•
•
•
•
•
•
F = ma = mRα
τ = R F = mR2α
Rotational inertia: mR2
Στi = (ΣmiRi2) α
I = ΣmiRi2
Στ = I α
(Στ)CM = ICM αCM
Calculating Moments of Inertia
M
dm = ρ l dR =
dR
l
I = ∫ R dm
2
M
I=
l
(
M
2
2
3
1
R
dR
=
R
∫− l 2
3
l
l
)
M
=
− l2
l
l
2
⎛ l3 l3 ⎞ 1
⎜⎜ + ⎟⎟ = Ml 2
⎝ 24 24 ⎠ 12
A few helpful theorems
• Parallel Axis Theorem
I = ICM + M h2
• Perpendicular Axis Theorem
Iz = Ix + Iy
Only valid for flat object!
Angular Momentum
Angular Momentum
L = Iω
Momentum
p = mv
Στ = Iα = dL/dt
ΣF = ma = dp/dt
Στ=0 ⇒ L=const.
ΣF=0 ⇒ p=const.
Total Angular Momentum is conserved if Στ=0.
Note: L = I ω, Angular Momentum is a vector
Rotating Kinetic Energy
• K = Σ(1/2mivi2) = Σ(1/2 miRi2 ω2)
= ½ Σ(miRi2) ω2 = ½ I ω2
• Rotational Kinetic Energy: ½ I ω2
• W=ØF dl= ØF⊥Rdθ
= Øτdθ
• W=1/2 I ω22 - 1/2 I ω12
Rotation and Translation
• Translation: K = ½ mv2
• Rotation: K = ½ Iω2
• Both (e.g. rolling):
– K = ½ mvCM2 + ½ Iω2
this is what
we did before
Atwood’s Machine Revisited
A pulley with a fixed
center (at point O),
radius R0 and moment
of inertia I, has a
massless rope wrapped
around it (no slipping).
The rope has two
masses, m1 and m2
attached to its ends.
Assume m2>m1
e
w
at ore:
h
f
si w t be
is d a
h
T oke
lo
Now:
A pulley with a fixed center
(at point O), radius R0 and
moment of inertia I, has a
massless rope wrapped
around it (no slipping). The
rope has two masses, m1
and m2 attached to its
ends. Assume m2>m1
Or :
Vector, Right Hand Rule
Why does the Bicycle Wheel
Turn to the Right?
Angular Momentum
r
r
L = Iω
Newton’s Law for rotational motion:
r
r
r
r
dω d ( Iω ) d ( L ) dL
∑τ = Iα = I dt = dt = dt = dt
r
r dL
r
∑τ = Iα = dt
r
r
Remember,
r
r
r
τ = R×F
This gives us the magnitude
and the direction of Torque
Angular Momentum
r r r
L = r×p
Angular Motion of a Particle
Determine the
angular
momentum, L, of
a particle,with
mass m and
speed v, moving
in uniform
circular motion
with radius r.
Conservation of Angular
Momentum
r
r dL
∑τ = dt
if τ = 0 → L = Const
r
r
L = Iω
Man on a Disk
A person with mass
m stands on the
edge of a disk with
radius R and
moment ½MR2.
Neither is moving.
The person then
starts moving on the
disk with speed V.
Find the angular
velocity of the disk.
A bullet strikes a cylinder
A bullet of speed V
and mass m strikes a
solid cylinder of mass
M and inertia ½MR2,
at radius R and sticks.
The cylinder is
anchored at point 0
and is initially at rest.
What is ω of the
system after the
collision?
Is energy Conserved?
Kepler’s 2nd Law
2nd Law: Each
planet moves
so that an
imaginary line
drawn from the
Sun to the
planet sweeps
out area in
equal periods
of time.
Static Equilibrium
This is what we are familiar with:
1D
Fnet = 0
2D
This is what we need to look at:
τnet = 0
Crane example:
??
Another example:
Good example that requires
consideration of both forces and torques
Center of Gravity – remember CM !
Hook’s law – same as before!
F = -kΔL