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Rotational Dynamics
Angular Momentum.
Collisions.
o Rotational Dynamics
 Basic Concepts
 Rotation
 Angular speed
 Torque
 Angular Acceleration
 Nature of Angular Momentum (& Energy), origins
 Angular Momentum Conservation
 Angular Momentum for a system of bodies
o Parallel Axis Theorem
Lecture 8
o Aim of the lecture
 Concepts in Rotational Dynamics
 Angular speed, torque, acceleration
 Dependence on mass
 Dependence on radius of mass,
 Moment of Inertia
 Dependence on rotation speed
 Newton’s Second Law
 Conservation of angular momentum
o Main learning outcomes
 familiarity with
 w, t, dw/dt, I, L, including vector forms
 Use of energy and angular momentum conservation
 Calculation of moment of inertia
Basic Concepts – Angular Position
y
m
r
q
o Rotational Dynamics
 Systems rotating about fixed axis
 Use r and q to describe position
 Natural quantities for rotation
o For Rigid Body Rotation
 r is fixed
 m moves in a circle
The position of the mass is
(r,q) or (x,y) they both give same information
x
Centre of rotation (axis pointing out of page)
Basic Concepts – Angular Rotation
y
v
m
r
q
o Rotation – ‘massless’ rod with mass
 Mass moving with speed v
 r is fixed
 q changes with time
o This is Rigid Body Rotation
 m moves in a circle
 angular speed is called w
(see next slide)
The position of the mass
(r,q) depends on time (r,q) = (r,wt)
x
Centre of rotation (axis pointing out of page)
Basic Concepts – Angular Speed
y
v
m
r
q
o The mass is rotation round
 say f revolutions per second
 f is the frequency of rotation
 f is measured in Hertz, Hz
 the time for one revolution is 1/f
 q depends on f and time
o There are 2p radians in a circle, so
 the number of radians per second is
 2p f = w the angular speed
 w is measured in radians per second
 w = (change in angle)/(time taken)
(r,q) depends on time (r,q) = (r,wt)
x
Centre of rotation (axis pointing out of page)
For a 98 TVR cebera
car the maximum
6250rpm = 6250/60 revs per second
Engine rpm is 6250
= 104 Hz = f
(same for all colours)
= 104 x 2p = 655 rads/sec = w
For dancer, the maximum
rpm is much lower, about
60 rpm
60rpm
= 1 Hz
=f
= 2p rads/sec= w
o Rotation is extremely common, it is measured in
 rpm,
 Hertz (frequency of rotation) or
 radians per second (angular speed)
Then
here
Its not really rotating, so its angular speed is w = 0
Look
here
Basic Concepts – Angular Speed
y
v
m
o The mass is moving
 at speed v
 at radius r
 with angular speed w
In time Dt, m moves vDt
Which changes the angle (in radians) by vDt/r
So w = angle/time = vDt/rDt = v/r
r
w = v/r
q
(r,q) depends on time (r,q) = (r,wt)
x
Centre of rotation (axis pointing out of page)
w = v/r
In fact whilst w is the angular speed, there
is also a vector form, called the
Angular velocity. As above.
Its direction is along the axis of rotation,
such that the object is rotating clockwise
looking along the vector w
Basic Concepts - Torque
o To make an object rotate about an axis
o Must apply a torque,
o A force perpendicular to the radius
A torque in rotation
is like a force in linear motion
dA
dB
A torque is a perpendicular force
times a radius
tA = FAdA
tB = FBdB
Only the perpendicular
component matters
so here the torque is
t = rFcos(q)
Centre of rotation
Torque is a ‘twisting force’
Distance matters:
A high torque is
needed for car wheel bolts
This
is achieved with
•The same torque can be achieved
with
• a long lever and small forcea long lever
• a short lever and a large force
A short lever would not
work, wheel would
fall off UNLESS the
force was much bigger
Basic Concepts – Angular Acceleration
Linear:
Rotation:
F = mdv/dt = ma
t = Idw/dt = Iw = I
oYou can accelerate, ‘spin up’ a rotating object
 by applying a torque, t
 the rate of angular acceleration = t/I
where I is the moment of inertia (see later)
 Moment of inertia is like mass in the linear case
o Torque is actually a vector
 Direction is perpendicular to:
Force being applied
 It is parallel to:
The axis of rotation
Torque is pointing INTO
the page [the direction a
screw would be driven]
F
t = I dw/dt = I
This is the vector form for the
relationship between torque and
angular acceleration
( In advanced work, I is a tensor, but
in this course we will just use a constant )
Angular Momentum
o Conserved Quantities
 If there is a symmetry in nature, then
 There will be a conserved quantity associated with it.
• (the maths to prove this is beyond the scope of the course)
o Examples:
 Physics today is the same as physics tomorrow,
 TIME symmetry
 The conserved quantity is Energy
 Physics on this side of the room is the same as on the other side
 Linear Translational symmetry
 The conserved quantity is called Momentum
 Physics facing west is the same as physics facing east
 Rotational symmetry
 The conserved quantity is called Angular Momentum
o What is angular momentum?
 For a mass m rotating at speed v, and radius r the angular momentum, L is
Centre of
rotation
r
L = mrv (= Iw)
v
Angular Momentum
•This is (in fact) a familiar quantity:
•A spinning wheel is hard to alter direction
•A gyroscope is based on conservation of angular momentum
•The orbits of planets are proscribed by conservation of a.m.
•It is claimed that gyroscopic effects help balance a bicycle
Angular Momentum
L = mrv
Centre of
rotation
r
v
o The angular momentum will depend on
 the mass; the speed;
 AND the radius of rotation
 The radius of rotation complicates things
 extended masses (not just particles) have more than one radius
 each part of such a mass will have a different value for mr
 and a different speed

•The train rotates around the
turntable axis.
•The cab is close to the axis,
r is small, v is small
contributes little to L.
•The chimney is far from axis
high r, higher v
will contribute more.
To simplify things we use the angular rotation speed, w
Where w = radians per second or 2p (rotations per
second)
w = 2p (v/2pr)
= v/r in radians per second [see earlier slides]
So we can write v = wr
and L = (mr2)w
The quantity in brackets (mr2) is called the moment of
inertia
It is given the symbol I.
v∫
[
body ]
I = mr2
I=
r2dm
for a single mass
for an extended
Angular Momentum
L = mrv = (mr2)w
Centre of
rotation
r
v
o The moment of inertia, (mr2) is
 given the symbol I
 can be calculated for any rigid (solid) body
 depends on where the body is rotating around
 The usual formula for angular momentum is
L = Iw
 In rotational dynamics there is a mapping from linear mechanics
 replace m by I
 replace v by w
 replace P by L
 then many of the laws of linear mechanics can be used
 For example:
 momentum
P = mv
so L = Iw
 kinetic energy, E = mv2/2 so E = Iw2/2
Moments of Inertia
o
Some examples of moment of inertia are:
 A disk rotating around its centre I = mr2/2
(if it rotates about the y axis it is I = mr2/4)
 A sphere rotating around any axis through the centre, I =
2mr2/5
 A uniform road of length L
rotating around its centre I = mL2/12
 A simple point mass around an axis
is the same as a hollow cylinder
I = mr2
Vector Form
L
o
Angular Momentum is a vector
 The magnitude of the vector is L = Iw
 The direction is along the rotation axis
looking in the direction of clockwise
rotation

L = Iw
 Note that this vector does NOT define a position in
space
 Clearly w is also a vector quantity with a similar
definition
o For several objects considered together ‘a system’
o The total angular momentum is the sum of the individual momenta
L = Sli
Where:
 li is the angular momentum of the ith object
 S means ‘sum of’
 L is the total angular momentum
Parallel Axis theorm
L
o
Angular Momentum is a vector
 The angular momentum vector does NOT define a position in
space
 The parallel axis theorem says that any rotating body has the
same angular momentum around any axis. [NOT changing the
rotation axis!]
But NOT this
one
This spinning shell has the same A.M. about all the blue axes shown
(without moving the shell, position does not matter)
Conservation
o The total angular momentum is conserved
o Entirely analogous to linear momentum
o Spins in opposite directions have opposite signs
As pirouetting skater
pulls in arms, w increases
As the circling air drops lower, r
decreases so w goes up – a tornado