Rotational Motion (1) Kinematics Everything’s analogous to linear kinematics Define angular properties properly and derive the equations of motion by analogy
Rotational Motion
(1) Kinematics
Everything’s analogous to linear kinematics
Define angular properties properly
and derive the equations of motionby analogy
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?
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?
bR1R2
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 RestA 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?
Vector Cross Product
A B vs.B A
A A
:CheckB" intoA Swing"
Rule-Hand-Right fromDirection SinB AC
B A C
rrrr
rr
rrr
××
×
Θ=
×=
Example of Cross ProductThe location of a body is length r from the origin and at an angle θ from the x-
axis. A force F
acts on the body purely in the y
direction.
What is the Torque on the body?
z
x
yθ
Rotational Dynamics
•
What plays the role of mass in rotation?
•
F = ma = mRα•
τ
= R F = mR2α
•
Rotational inertia: mR2
•
Στi
= (Σmi
Ri2) α
•
I = Σmi
Ri2
•
Στ
= I α•
(Στ)CM
= ICM
αCM
Calculating Moments of Inertia
( ) 233
2
121
24243
31 2
2
2
2
Mllll
Ml
MdRRl
MI Rl
l
l
l=⎟⎟
⎠
⎞⎜⎜⎝
⎛+===
−−∫
∫= dmRI 2 dRl
MdRdm l == ρ
A few helpful theorems
•
Parallel Axis TheoremI = ICM
+ M h2
•
Perpendicular Axis TheoremIz
= Ix
+ IyOnly valid for flat object!
Angular Momentum
Angular Momentum
MomentumL = Iω
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(1/2mi
vi2) = Σ(1/2 mi
Ri2 ω2)
= ½
Σ(mi
Ri2) ω2
= ½
I ω2
•
Rotational Kinetic Energy: ½
I ω2
•
W= F dl= F⊥Rdθ= τdθ
•
W=1/2 I ω22
-
1/2 I ω1
2
Rotation and Translation
•
Translation: K = ½
mv2
•
Rotation: K = ½
Iω2
•
Both (e.g. rolling): –
K = ½
mvCM
2
+ ½
Iω2
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
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 :
Angular Momentum
dtLdI
dtLd
dtLd
dtId
dtdII
rrr
rrrrrr
==
=====
∑
∑
ατ
ωωατ
)()(
ωrr
IL =Newton’s Law for rotational motion:
Angular Motion of a ParticleDetermine the angular momentum, L, of a particle,with mass m
and
speed v, moving in uniform circular motion with radius r.
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 cylinderA 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
Law2nd
Law:
Each
planet moves so that an imaginary line drawn from the Sun to the planet sweeps out area in equal periods of time.