Chapter 9
Rotational Motion
Rotational Motion




Many interesting physical
phenomena are not “linear”
Ball at the end of a string revolving
Planets around Sun
Moon around Earth
New Physical Quantities:
•Angular Displacement
•Angular velocity
•Angular acceleration
The Radian


The radian is a unit
of angular measure
The radian can be
defined as the arc
length s along a
circle divided by
the radius r

s
 
r
57.3°
More About Radians



Whole circle
1 rev  360  2 rad
Comparing degrees and radians
360
1 rad 
 57.3
2
Converting from degrees to radians

 [rad] 
 [deg rees]
180
Example
49. Express it in radians,
revolutions.
2 rad
  49 deg
 0.855rad
360 deg
rev
  49 deg
 0.136rev
360 deg
rev
  0.855rad 
 0.136rev
2 rad
Angular Displacement



Axis of rotation is
the center of the
disk
Need a fixed
reference line
During time t, the
line on rotating
object moves
through angle θ
Angular Displacement, cont.




The angular displacement is defined
as the angle the object rotates
through during some time interval
   F  I
The unit of angular displacement is
the radian
Each point on the object undergoes
the same angular displacement
Average Angular Velocity

The average
angular velocity, ω,
of a rotating rigid
object is the ratio
of the angular
displacement to
the time interval


t

 F  I
t
   t
Angular velocity, cont.


The instantaneous angular velocity
Units of angular velocity are
radians/sec
• rad/s


Velocity will be positive if θ is
increasing (counterclockwise)
Velocity will be negative if θ is
decreasing (clockwise)
Average Angular Acceleration

The average angular acceleration
of an object is defined as the ratio of
the change in the angular velocity to
the time it takes for the object to
undergo the change:

F I
t
  F   I  t
Angular Acceleration, cont




Units of angular acceleration are rad/s²
Positive angular accelerations are in the
counterclockwise direction and negative
accelerations are in the clockwise direction
When a rigid object rotates about a fixed
axis, every portion of the object has the
same angular velocity and the same
angular acceleration
Angular speed
Angular Acceleration, final



The sign of the acceleration does not
have to be the same as the sign of
the angular velocity
The instantaneous angular
acceleration
Only consider rotations with uniform
angular acceleration
Example
A bicycle wheel makes 50 revolutions
in 0.5 min. Compute the angular
speed in rev/s, rad/s, deg/s.
Example
The bicycle wheel traveling at 1.67
rev/s decelerates uniformly to a stop
in 3 seconds. Compute the angular
deceleration in rev/s², rad/s², deg/s²
Analogies Between Linear and
Rotational Motion
Linear Motion with constant
acc.
(x,v,a)
vF  vI  at
vaverage 
1
vI  vF 
2
1 2
x  vI t  at
2
2
2
vF  vI
a
2x
Rotational Motion with fixed axis
and constant 
,,
F  I  t
1
 average   I   F 
2
1
   I t  t 2
2
 F   I2

2
2
Example
A car coasts to a stop with a uniform
deceleration of 1.2 rad/s² for its
wheel. If its initial angular speed was
1000 rev/min, how many revolutions
does the wheel make before coming
to a stop?
Relationship Between Angular and
Linear Quantities


Displacements

s  r
Speeds
vT  r
s

vT   r  r
t
t

Accelerations
aT  r

Every point on the
rotating object has
the same angular
motion
Every point on the
rotating object
does not have the
same linear motion
Examples



Pulley
Rolling wheel
A belt runs on a pulley with diameter
5 cm which revolves at a speed of
1000 rev/min. What length of belt
passes over the pulley each second?
Example
A 60 cm diameter wheel is rotating @
speed of 3.0 rev/s. How fast is this
car going?
Example
If the masses of Atwood’s machine are
accelerating at 4.8 m/s² and radius
of 0.3 m, what is ?
Centripetal Acceleration


An object traveling in a circle, even
though it moves with a constant
speed, will have an acceleration
The centripetal acceleration is due to
the change in the direction of the
velocity
Centripetal Acceleration, cont.



Centripetal refers
to “center-seeking”
The direction of the
velocity changes
The acceleration is
directed toward the
center of the circle
of motion
Centripetal Acceleration, final

The magnitude of the centripetal
acceleration is given by
v2
ac 
r
• This direction is toward the center of the
circle
Centripetal Acceleration and
Angular Velocity


The angular velocity and the linear
velocity are related (v = rω)
The centripetal acceleration can also
be related to the angular velocity
ac  r

 must be in rad/s
2
Forces Causing Centripetal
Acceleration

Newton’s Second Law says that the
centripetal acceleration is accompanied by
a force
2
• F = ma

v
F m
r
• F stands for any force that keeps an object
following a circular path



Tension in a string
Gravity
Force of friction
Examples


Ball at the end of
revolving string
(m=500 grams,
r=120 cm, 0.8
rev/s). Tension?
Fast car rounding
a curve
More on circular Motion


Length of circumference = 2R
Period T (time for one complete
circle)
2r

v
2
2
v
(2r )
a

r
r 2
a
4 2 r

2
Newton’s Law of Universal
Gravitation

Every particle in the Universe
attracts every other particle with a
force that is directly proportional to
the product of the masses and
inversely proportional to the square
of the distance between them.
m1m2
F G 2
R
Universal Gravitation, 2



G is the constant of universal
gravitational
G = 6.673 x 10-11 N m² /kg²
This is an example of an inverse
square law
Motion of Satellites



Consider only circular orbit
r  RE  h
Radius of orbit r:
Gravitational force is the centripetal
force.
mmE
v2
mE
2
F  ma  G 2  m
G
v
r
r
r
GmE
v
r
Motion of Satellites

2r

v
Period 
2r

GmE
32

Kepler’s 3rd Law
Synchronous Orbits
  24hr  86400s, G  6.67 10 ,
11
mE  6 10  r  4.23 10 m  2.6 10 miles
24
7
4
Communications Satellite

A geosynchronous orbit
• Remains above the same place on the earth
• The period of the satellite will be 24 hr


r = h + RE
Still independent of the mass of the satellite
r  4.23 10 m  2.6 10 miles
RE  6370km  4000miles
7
h  r  RE  22,000miles
4
Satellites and Weightlessness





weighting an object in an elevator
Elevator at rest: mg
Elevator accelerates upward:
m(g+a)
Elevator accelerates downward:
m(g+a) with a<0
Satellite: a=-g!!
Chapter 10
Dynamics of
Rotation
Force vs. Torque


Forces cause accelerations
Torques cause angular accelerations
• rotation

Force and torque are related
Torque, cont

Torque, , is the tendency of a force to
rotate an object about some axis
  Fl


 is the torque
F is the force
• symbol is the Greek tau



l is the lever arm
SI unit is N.m
Lever Arm: Perpendicular distance from
the pivot point to the line of force
Direction of Torque

Torque is a vector quantity
• We will treat only 2-d torque so no
need for vector notion.
• If the turning tendency of the force is
counterclockwise, the torque will be
positive (+)
• If the turning tendency is clockwise,
the torque will be negative (-)
Multiple Torques

When two or more torques are acting
on an object, the torques are added
• with the signs

If the net torque is zero, the object’s
rate of rotation doesn’t change
General Definition of Torque


The applied force is not always
perpendicular to the position vector
The component of the force
perpendicular to the object will cause
it to rotate
  FL sin 
l  L sin 

F is the force
L is distance
between pivot and
point of action


 is the angle
Moment of Inertia


The angular acceleration is inversely
proportional to the analogy of the
mass in a rotating system
This mass analog is called the
moment of inertia, I, of the object
• SI units are kg m2
I  mr
2
Newton’s Second Law for a
Rotating Object
  I


The angular acceleration is directly
proportional to the net torque
The angular acceleration is inversely
proportional to the moment of inertia
of the object
More About Moment of Inertia


There is a major difference between
moment of inertia and mass: the
moment of inertia depends on the
quantity of matter and its
distribution in the rigid object.
The moment of inertia also depends
upon the location of the axis of
rotation
Moment of Inertia of a Uniform
Ring


Image the hoop is
divided into a
number of small
segments, m1 …
These segments
are equidistant
from the axis
I  mi ri2  MR2
Other Moments of Inertia
Example
Wheel of radius R=20 cm and
I=30kg·m^2. Force F=40N acts
along the edge of the wheel.
1. Angular acceleration?
2. Angular speed 4s after starting
from rest?
3. Number of revolutions for the 4s?
Rotational Kinetic Energy


An object rotating about some axis
with an angular speed, ω, has
rotational kinetic energy Ekr=½Iω2
Energy concepts can be useful for
simplifying the analysis of rotational
motion
Ek  Ekt  Ekr
Total Energy of a System

Conservation of Mechanical Energy
Ek  Ekt  Ekr
EkI  E pI  EkF  E pF
• Remember, this is for conservative
forces, no dissipative forces such as
friction can be present
• Potential energies of any other
conservative forces could be added
Rolling down incline


Energy conservation
Linear velocity and angular speed are
related v=R
1 2 1 2
mgh  mv  I
2
2
1 2 1 I 2 1
I 2
mgh  mv  ( 2 )v  (m  2 )v
2
2 R
2
R

Smaller I, bigger v, faster!!
Work-Energy in a Rotating
System


In the case where there are
dissipative forces such as friction,
use the generalized Work-Energy
Theorem instead of Conservation of
Energy
(Ekt+Ekr+Ep)i+W=(Ekt+Ekt+Ep)f
Angular Momentum


Similarly to the relationship between
force and momentum in a linear
system, we can show the relationship
between torque and angular
momentum
Angular momentum is defined as
•L = I ω
• and
L
 
t
Angular Momentum, cont


If the net torque is zero, the angular
momentum remains constant
Conservation of Angular Momentum
states: The angular momentum of a
system is conserved when the net
external torque acting on the
systems is zero.
• That is, when
  0, Li  Lf or Iii  If f
Conservation Rules, Summary

In an isolated system, the following
quantities are conserved:
• Mechanical energy
• Linear momentum
• Angular momentum
Conservation of Angular
Momentum, Example

With hands and
feet drawn closer
to the body, the
skater’s angular
speed increases
• L is conserved, I
decreases, 
increases
Example
A 500 grams uniform sphere of 7.0 cm
radius spins at 30 rev/s on an axis
through its center.
 Moment of inertia
 Rotational kinetic energy
 Angular momentum
Example
A turntable is a uniform disk of metal
of mass 1.5 kg and radius 13 cm.
What torque is required to drive the
turntable so that it accelerates at a
constant rate from 0 to 33.3 rpm in 2
seconds?