Rotational
Kinematics
Position

In translational motion,
position is represented by a
point, such as x.
x=3
x
0
5
linear
p/2

In rotational motion,
position is
represented by an
angle, such as q, and
a radius, r.
r
p
q
3p/2
0
angular
Displacement
Dx = - 4


Linear displacement is
represented by the
vector Dx.
Angular displacement
is represented by Dq,
which is not a vector,
but behaves like one
for small values.
Counterclockwise is
considered to be
positive.
x
0
5
linear
p/2
Dq
p
0
3p/2
angular
Tangential and angular
displacement


A particle that rotates
through an angle Dq
also translates through
a distance s, which is
the length of the arc
defining its path.
This distance s is related to the
angular displacement Dq by the
equation s = rDq.
s
Dq
r
Speed and velocity

The instantaneous velocity
has magnitude vT = ds/dt
and is tangent to the circle.

The same particle rotates
with an angular velocity w =
dq/dt.
The direction of the angular
velocity is given by the right
hand rule.
Tangential and angular
speeds are related by the
equation v = r w.


s
vT
Dq
vT
r
w is outward
according to
RHR
Acceleration





Tangential acceleration is
given by aT = dvT/dt.
This acceleration is parallel
or anti-parallel to the
velocity.
Angular acceleration of this
particle is given by a =
dw/dt.
Angular acceleration is
parallel or anti-parallel to
the angular velocity.
Tangential and angular
accelerations are related by
the equation a = r a.
s
vT
Dq
vT
r
w is outward
according to
RHR
Don’t forget centripetal
acceleration.
a = at2+ac2
Problem: Assume the particle is
speeding up.
a)
b)
c)
d)
e)
f)
What is the direction of the
instantaneous velocity, v?
What is the direction of the
angular velocity, w?
What is the direction of the
tangential acceleration, aT?
What is the direction of the
angular acceleration a?
What is the direction of the
centripetal acceleration, ac?
What is the direction of the overall
acceleration, a, of the particle?
What changes if
the particle is
slowing down?
First Kinematic Equation

v = vo + at (linear form)
 Substitute
angular velocity for velocity.
 Substitute angular acceleration for
acceleration.

w = wo + at (angular form)
Second Kinematic Equation

x = xo + vot + ½ at2 (linear form)
 Substitute
angle for position.
 Substitute angular velocity for velocity.
 Substitute angular acceleration for
acceleration.

q = qo + wot + ½ at2 (angular form)
Third Kinematic Equation

v2 = vo2 + 2a(x - xo)
 Substitute
angle for position.
 Substitute angular velocity for velocity.
 Substitute angular acceleration for
acceleration.

w2 = wo2 + 2a(q - qo)
Practice problem
The Beatle’s White Album is spinning at 33 1/3 rpm when the power is
turned off. If it takes 1/2 minute for the album’s rotation to stop, what is
the angular acceleration of the phonograph album? (-0.12 rad/s2)
Hwk: Chpt 10 # 1-4, 9,10,13,16,17
Rotational
Energetics
Inertia and Rotational Inertia



In linear motion, inertia is equivalent to mass.
Rotating systems have “rotational inertia”.
I = mr2 (for a system of particles)
rotational inertia (kg m2)
 m: mass (kg)
 r: radius of rotation (m)
 I:

Solid objects are more complicated; we’ll get to
those later. See page 278 for a “cheat sheet”.
Sample Problem

A 2.0-kg mass and a 3.0-kg mass are mounted on opposite ends a 2.0-m
long rod of negligible mass. What is the rotational inertia about the center of
the rod and about each mass, assuming the axes of rotation are
perpendicular to the rod?
Kinetic Energy

Bodies moving in a straight line have
translational kinetic energy
 Ktrans

Bodies that are rotating have rotational
kinetic energy
 Krot

= ½ m v 2.
= ½ I w2
It is possible to have both forms at once.
 Ktot
= ½ m v2 + ½ I w2
Practice problem
A 3.0 m long lightweight rod has a 1.0 kg mass attached to one end,
and a 1.5 kg mass attached to the other. If the rod is spinning at 20 rpm
about its midpoint around an axis that is perpendicular to the rod, what
is the resulting rotational kinetic energy? Ignore the mass of the rod.
Rotational Inertia
Rotational Inertia Calculations
I = mr2 (for a system of particles)
 I =  dm r2 (for a solid object)
 I = Icm + m h2 (parallel axis theorem)

 I:
rotational inertia about center of mass
 m: mass of body
 h: distance between axis in question and
axis through center of mass
Practice problem
A solid ball of mass 300 grams and diameter 80 cm is thrown at 28 m/s. As it
travels through the air, it spins with an angular speed of 110 rad/second.
What is its
a) translational kinetic energy?
b) rotational kinetic energy?
c) total kinetic energy?
Practice Problem
Derive the rotational inertia of a long thin rod of length L and mass M
about a point 1/3 from one end
a)
using integration of I =  r2 dm
b)
using the parallel axis theorem and the rotational inertia of a rod
about the center.
Practice Problem
Derive the rotational inertia of a ring of mass M and radius R about the
center using the formula I =  r2 dm.
Torque and Angular
Acceleration I
Equilibrium

Equilibrium occurs when there is no net force
and no net torque on a system.
 Static
equilibrium occurs when nothing in the system
is moving or rotating in your reference frame.
 Dynamic equilibrium occurs when the system is
translating at constant velocity and/or rotating at
constant rotational velocity.

Conditions for equilibrium:


t = 0
F = 0
Torque
Torque is the rotational analog of
force that causes rotation to begin.
Consider a force F on the beam that is applied a distance r from
the hinge on a beam. (Define r as a vector having its tail on the
hinge and its head at the point of application of the force.)
A rotation occurs due to the
combination of r and F. In
this case, the direction is
clockwise.
What do you think is the
direction of the torque?
Hinge (rotates)
r
Direction of rotation
Direction of torque is
INTO THE SCREEN.
F
Calculating Torque


The magnitude of the torque is proportional to
that of the force and moment arm, and torque is
at right angles to plane established by the force
and moment arm vectors. What does that sound
like?
t=rF
t : torque
 r: moment arm (from point of rotation to point of
application of force)
 F: force

Practice Problem
What must F be to achieve equilibrium? Assume
there is no friction on the pulley axle.
F
3 cm
2 cm
10 kg
2 kg
Torque and Newton’s 2nd Law

Rewrite F = ma for rotating systems
 Substitute
torque for force.
 Substitute rotational inertia for mass.
 Substitute angular acceleration for
acceleration.

t = I a
 t:
torque
 I: rotational inertia
 a: angular acceleration
Practice Problem
A 1.0-kg wheel of 25-cm radius is at rest on a fixed axis. A force of 0.45 N is applied
tangent to the rim of the wheel for 5 seconds.
a) After this time, what is the angular velocity of the wheel?
b)Through what angle does the wheel rotate during this 5 second period?
Rotational Dynamics
Workshop
Sample problem

Derive an expression for the acceleration of a flat disk of
mass M and radius R that rolls without slipping down a
ramp of angle q.
Practice problem
Calculate initial angular acceleration of rod
of mass M and length L.
Calculate initial acceleration of end of rod.
Sample problem
Calculate acceleration.
Assume pulley has mass M, radius R, and
is a uniform disk.
m2
m1
Rotational Dynamics
Lab
Tuesday, December 2, 2008
Wednesday, December 3, 2008
Announcements

Pass forward these assignments:
R: 10.5; P: 28,29,30

R: 10.6; P: 32,33,34
Exam repair must be done this week by
Thursday. You must do repairs in the
morning (7:00 AM) or lunch. One shot
only, so plan on staying the whole hour.
Demonstration

A small pulley and a larger attached
disk spin together as a hanging
weight falls. DataStudio will collect
angular displacement and velocity
information for the system as the
weight falls. The relevant data is:






Diameter of small pulley: 3.0 cm
Mass of small pulley: negligible
Diameter of disk: 9.5 cm
Mass of disk: 120 g
Hanging mass: 10 g
See if we can illustrate Newton’s 2nd
Law in rotational form.
Demonstration calculations
Rotational Dynamics
Lab
Work and Power in
Rotating Systems
Practice Problem
What is the acceleration of this system, and the magnitude of tensions T1 and T2?
Assume the surface is frictionless, and pulley has the rotational inertia of a
uniform disk.
mpulley = 0.45 kg
rpulley = 0.25 m
T1
T2
m1 = 2.0 kg
m2 = 1.5 kg
30o
Work in rotating systems

W = F • Dr (translational systems)
 Substitute
torque for force
 Substitute angular displacement for displacement

Wrot = t • Dq
 Wrot
: work done in rotation
 t : torque
 Dq: angular displacement

Remember that different kinds of work change
different kinds of energy.
 Wnet
= DK
Wc = -DU
Wnc = DE
Power in rotating systems


P = dW/dt (in translating or rotating systems)
P = F • v (translating systems)
 Substitute
torque for force.
 Substitute angular velocity for velocity.

Prot = t • w (rotating systems)
 Prot
: power expended
 t : torque
 w: angular velocity
Conservation of Energy

Etot = U + K = Constant
 (rotating
or linear system)
 For gravitational systems, use the center of
mass of the object for calculating U
 Use rotational and/or translational kinetic
energy where necessary.
Practice Problem
A rotating flywheel provides power to a machine. The flywheel is originally rotating
at of 2,500 rpm. The flywheel is a solid cylinder of mass 1,250 kg and diameter of 0.75
m. If the machine requires an average power of 12 kW, for how long can the flywheel
provide power?
Practice Problem
A uniform rod of mass M and length L rotates around a pin through one end. It is
released from rest at the horizontal position. What is the angular speed when it
reaches the lowest point? What is the linear speed of the lowest point of the rod at this
position?
Rolling without
Slipping
Rolling without slipping review
Conservation of Energy review
Introduction to angular momentum of a particle
Thursday, December 5, 2008
Rolling without slipping

Total kinetic energy of a body is the sum of the
translational and rotational kinetic energies.
K

= ½ Mvcm2 + ½ I w2
When a body is rolling without slipping, another
equation holds true:
 vcm

=wr
Therefore, this equation can be combined with
the first one to create the two following
equations:
= ½ M vcm2 + ½ Icm v2/R2
 K = ½ m w2R2 + ½ Icm w2
K
Sample Problem
A solid sphere of mass M and radius R rolls from rest down a ramp of
length L and angle q. Use Conservation of Energy to find the linear
acceleration and the speed at the bottom of the ramp.
Sample Problem

A solid sphere of mass M and radius R rolls from rest down a ramp
of length L and angle q. Use Rotational Dynamics to find the linear
acceleration and the speed at the bottom of the ramp.
Angular Momentum
of Particles
Sample Problem
A solid sphere of mass M and radius R rolls from rest down a ramp of
length L and angle q. Use Conservation of Energy to find the linear
acceleration and the speed at the bottom of the ramp.
Sample Problem

A solid sphere of mass M and radius R rolls from rest down a ramp
of length L and angle q. Use Rotational Dynamics to find the linear
acceleration and the speed at the bottom of the ramp.
Practice Problem
A hollow sphere (mass M, radius R) rolls without slipping down a ramp of
length L and angle q. At the bottom of the ramp
a) what is its translational speed?
b) what is its angular speed?
Angular Momentum
Angular momentum is a quantity that tells
us how hard it is to change the rotational
motion of a particular spinning body.
 Objects with lots of angular momentum
are hard to stop spinning, or to turn.
 Objects with lots of angular momentum
have great orientational stability.

Angular Momentum of a particle
For a single particle with known
momentum, the angular momentum can
be calculated with this relationship:
L=rp

 L:
angular momentum for a single particle
 r: distance from particle to point of rotation
 p: linear momentum
Practice Problem
a)
b)
Determine the angular momentum for the revolution of
the earth about the sun.
the moon about the earth.
Practice Problem
Determine the angular momentum for the 2 kg particle shown
a) about the origin.
y (m)
b) about x = 2.0.
5.0
5.0
-5.0
v = 3.0 m/s
x (m)
Angular Momentum of
Solid Objects
and Conservation of
Angular Momentum
Angular Momentum - solid
object


For a solid object, angular momentum is
analogous to linear momentum of a solid object.
P = mv (linear momentum)
 Replace
momentum with angular momentum.
 Replace mass with rotational inertia.
 Replace velocity with angular velocity.

L = I w (angular momentum)
 L:
angular momentum
 I: rotational inertia
 w: angular velocity
Practice Problem
Set up the calculation of the angular momentum for the rotation of the
earth on its axis.
Law of Conservation of Angular Momentum



The Law of Conservation of Momentum states
that the momentum of a system will not change
unless an external force is applied. How would
you change this statement to create the Law of
Conservation of Angular Momentum?
Angular momentum of a system will not change
unless an external torque is applied to the
system.
LB = LA (momentum before = momentum after)
Practice Problem
A figure skater is spinning at angular velocity wo. He brings his arms and legs
closer to his body and reduces his rotational inertia to ½ its original value.
What happens to his angular velocity?
Practice Problem
A planet of mass m revolves around a star of mass M in a highly elliptical orbit. At
point A, the planet is 3 times farther away from the star than it is at point B. How does
the speed v of the planet at point A compare to the speed at point B?
Demonstrations
Bicycle wheel demonstrations
 Gyroscope demonstrations
 Top demonstration

Precession
Practice Problem
A 50.0 kg child runs toward a 150-kg merry-go-round of radius 1.5 m, and jumps
aboard such that the child’s velocity prior to landing is 3.0 m/s directed tangent to the
circumference of the merry-go-round. What will be the angular velocity of the merrygo-round if the child lands right on its edge?
Angular momentum and torque


In translational systems, remember that
Newton’s 2nd Law can be written in terms of
momentum.
F = dP/dt
 Substitute
force for torque.
 Substitute angular momentum for momentum.

t = dL/dt
 t:
torque
 L: angular momentum
 t: time
So how does torque affect angular
momentum?




If t = dL/dt, then torque changes L with respect
to time.
Torque increases angular momentum when the
two vectors are parallel.
Torque decreases angular momentum when the
two vectors are anti-parallel.
Torque changes the direction of the angular
momentum vector in all other situations. This
results in what is called the precession of
spinning tops.
If torque and angular momentum are
parallel…
Consider a disk rotating
as shown. In what
direction is the angular
momentum?
Consider a force applied
as shown. In what
direction is the torque?
r
F
The torque vector is parallel to
the angular momentum vector.
Since t = dL/dt, L will increase
with time as the rotation speeds.
L is out
t is out
If torque and angular momentum are
anti-parallel…
Consider a disk rotating
as shown. In what
direction is the angular
momentum?
Consider a force applied
as shown. In what
direction is the torque?
r
F
The torque vector is anti-parallel
to the angular momentum vector.
Since t = dL/dt, L will decrease
with time as the rotation slows.
L is in
t is out
If the torque and angular momentum
are not aligned…



For this spinning
top, angular
momentum and
torque interact in a
more complex way.
Torque changes
the direction of the
angular
momentum.
This causes the
characteristic
precession of a
spinning top.
r
L
t = r  Fg
Fg

t
t = dL/dt

DL
Rotation Review
Practice Problem
A pilot is flying a propeller plane and the propeller appears to be rotating clockwise
as the pilot looks at it. The pilot makes a left turn. Does the plane “nose up” or “nose
down” as the plane turns left?