Rotational Dynamics
Dawson High School Physics
Acknowledgements
© Mark Lesmeister/Pearland ISD
 Selected graphics and questions are from Cutnell and Johnson, Physics 9e:
Instructor Companion Site, © 2015 John Wiley and Sons.
Newton’s 2nd Law for Rotation
Newton’s Second Law for Rotation
 Consider a mass m in a rigid
body that is free to rotate.
 The mass is acted on by a
force F.
F
 What does the radial
component do?
 What does the tangential
force do?
 The mass will undergo a
tangential acceleration at.
.
r
Newton’s Second Law for Rotation

Ft  mat
F


at  r
Ft  mr
r
Newton’s Second Law for Rotation


Ft r  mr 
2
  (mr )
F
2
r
Newton’s Second Law for Rotation
 For a single particle of
mass m
  (mr )
2
F
r
Newton’s Second Law for Rotation


   mr 
2
i
   m r 
τ
  m r 
2
i

i
i i
2
EXT
i i
Newton’s Second Law for Rotation
Newton’s Second Law for Rotation

 NET , EXT  I
 Net Torque = Rotational Inertia
X angular acceleration
I   mi ri
2
i
.
© John Wiley and Sons
Discovery Lab: Rotational Inertia
 At Stations 1,3 and 5:
 Hold the meterstick horizontally in your hand at the 50 cm
mark. Try spinning the meterstick in a horizontal circle around
that axis. Do the same with the second meterstick.
 Which meterstick is easier to spin? Why? Discuss you answers.
 At Stations 2,4 and 6:
 Hold the meterstick at one end. Try spinning it in a horizontal
circle. Hold it at the other end and repeat.
 Around which axis is it easier to spin the meterstick? Why?
Discuss your answers.
Rotational Inertia
2
 I   mi ri
i
for a system of particles.
© John Wiley and Sons
Example: Rotational Inertia for a
System of Particles
 Calculate the rotational inertia of the following system of
particles about the given axis.
a
M
M
a
M
M
Example: Rotational Inertia for a
System of Particles
 Calculate the rotational inertia of the following system of
particles about the given axis.
2a
M
M
M
M
© 2015 John Wiley and
Sons
Newton’s 2nd Law for Rotation
Example 1
 A bicycle tire of radius 0.30 m
rotates at 40.0 rad/s. What
torque is required to bring the
tire to rest in 2.0 s? The tire has
a mass of 1.5 kg.
 Hint: Treat the tire as a hoop (I
= MR2)
 If the torque is applied at the
rim of the tire (r = 0.25 m) by
the brakes, how much force
must the brake pads apply?
Practice with Newton’s 2nd Law for
Rotation
Newton’s Second Law for Rotation

 NET , EXT  I
 Net Torque = Rotational
Inertia X angular acceleration

I   mi ri 2
i
for a system of particles.
© John Wiley and Sons
9.4.1. Two solid disks, which are free to rotate independently about the same axis
that passes through their centers and perpendicular to their faces, are initially at
rest. The two disks have the same mass, but one of has a radius R and the other
has a radius 2R. A force of magnitude F is applied to the edge of the larger
radius disk and it begins rotating. What force must be applied to the edge of the
smaller disk so that the angular acceleration is the same as that for the larger
disk? Express your answer in terms of the force F applied to the larger disk.
a) 0.25F
b) 0.50F
c) F
d) 1.5F
e) 2F
9.4.2. The corner of a rectangular piece of wood is attached to a rod that
is free to rotate as shown. The length of the longer side of the
rectangle is 4.0 m, which is twice the length of the shorter side.
Two equal forces with magnitudes of 22 N are applied to two of the
corners. What is the magnitude of the net torque on the block and
direction of rotation, if any?
a) 44 Nm, clockwise
F
b) 44 Nm, counterclockwise
c) 88 Nm, clockwise
d) 88 Nm, counterclockwise
e) zero Nm, no rotation
F
9.4.3. Consider the following three objects, each of the same mass and
radius:
(1) a solid sphere
(2)
a solid disk
(3)
a hoop
All three are released from rest at the top of an inclined plane. The three
objects proceed down the incline undergoing rolling motion without
slipping. In which order do the objects reach the bottom of the incline?
a) 1, 2, 3
b) 2, 3, 1
c) 3, 1, 2
d) 3, 2, 1
e) All three reach the bottom at the same time.
9.4.4. A long board is free to rotate about the pivot shown in each of
the four configurations shown. Weights are hung from the board
as indicated. In which of the configurations, if any, is the net
torque about the pivot axis the largest?
a) 1
b) 2
c) 3
d) 4
e) The net torque is the same for
all four situations.
9.4.5. The drawing shows a yo-yo in contact with a tabletop. A string
is wrapped around the central axle. How will the yo-yo behave if
you pull on the string with the force shown?
a) The yo-yo will roll to the left.
b) The yo-yo will roll to the right.
c) The yo-yo will spin in place, but not roll.
d) The yo-yo will not roll, but it will move to the left.
e) The yo-yo will not roll, but it will move to the right.
Newton’s Second Law for Rotation
Example 2
 (Based on Ch. 9, Problem 42 from Cutnell
and Johnson. )A 15.0 m length of hose is
wound around a reel, initially at rest. The
rotational inertia of the reel is 0.44 kg m2,
and its radius is 0.160 m. When the reel is
turning, friction at the axle exerts a torque
of magnitude 3.40 Nm on the reel. If the
hose is pulled so that the tension in it
remains a constant 25.0 N, how long does it
take to completely unwind the hose from
the reel? Neglect the mass and thickness of
the hose on the reel, and assume the hose
unwinds without slipping.
Newton’s 2nd Law for Rotation Practice
 http://cnx.org/content/m42179/latest/?collection=col11
406/1.7
Newton’s 2nd Law for Rotation Lab
  I
Tr  I
a
Tr  I
r
F  ma
mg  T  ma
mg  ma  T
m( g  a )  T
a
m( g  a ) r  I
r
Newton’s 2nd Law for Rotation Lab
 Use the LoggerPro file
  I
a
m( g  a ) r  I
r
classfiles/Lesmeister/Newton’s
2nd Law for Rotation.
 Use 200 g as the smallest mass.
 Correct fixable sources of
uncertainty before you collect
data.
Warm-up
 A box slides down a ramp
of height h.
 If no energy is lost to
friction, how fast will the
box be moving when it
reaches the bottom of the
ramp? Express your
answer in terms of h and
fundamental constants.
Newton’s 2nd Law for Rotation:
Example 3
 A hoop and a disc, both of
mass M and radius R, are
rolled down a ramp.
 IHOOP=MR2
 IDISC=1/2 MR2
 Which object will
experience a greater
angular acceleration?
 Calculate the ratio of the
angular accelerations.
Rolling Objects: Non-slip condition
 If a wheel rolls without slipping, the center of mass of the wheel
will have the same velocity and acceleration as a point on the edge
of the wheel.
Rolling Objects
sCM  R
vCM  R
aCM  R
Rotational Work and Kinetic Energy
Rotational Work
 The formula for work can be adapted to rotational motion.
 WROT

 FT d
d  r
WROT
 WROT
 FT r
 
  FT r
Rotational Kinetic Energy

K ROT   12 mi vti2   12 mi (ri ) 2

K ROT 

K ROT  12 I 2
1
2
2 2
m
r
 ii
Rotational Kinetic Energy Example
 Calculate the rotational kinetic energy of the Earth, and
compare that to the kinetic energy of the Earth’s motion
around the Sun. Assume the Earth is a sphere of radius 6.4 x
106 m and mass 6.0 x 1024 kg. The radius of the Earth’s orbit
is 1.5 x 1011 m.
Energy of Rolling Objects
 Rolling objects have rotational kinetic energy as well translational
kinetic energy.
2
KTOT  12 I 2  12 mvCM
Conservation of Energy in Rotational
Motion
 In the absence of
external forces,
energy is conserved.
 Kinetic energy consists
of both rotational and
translational parts.
Conservation of Energy in Rotational
Motion
 A 5 kg. ball rolls down
a ramp that is 1.0 m
high. I=2/5 MR2
 If the ball rolls without
slipping, how fast will
it be moving (velocity
of the center of mass)
when it reaches the
bottom of the ramp?
9.4.5. The drawing shows a yo-yo in contact with a tabletop. A string
is wrapped around the central axle. How will the yo-yo behave if
you pull on the string with the force shown?
a) The yo-yo will roll to the left.
b) The yo-yo will roll to the right.
c) The yo-yo will spin in place, but not roll.
d) The yo-yo will not roll, but it will move to the left.
e) The yo-yo will not roll, but it will move to the right.
9.5.1. Four objects start from rest and roll without slipping down a ramp.
The objects are a solid sphere, a hollow cylinder, a solid cylinder,
and a hollow sphere. Each of the objects has the same radius and the
same mass, but they are made from different materials. Which
object will have the greatest speed at the bottom of the ramp?
a) Since they are all starting from rest, all of the objects will have the
same speed at the bottom as a result of the conservation of
mechanical energy.
b) solid cylinder
c) hollow cylinder
d) solid sphere
e) hollow sphere
9.5.2. A bowling ball is rolling without slipping at constant speed
toward the pins on a lane. What percentage of the ball’s total
kinetic energy is translational kinetic energy?
a) 50 %
b) 71 %
c) 46 %
d) 29 %
e) 33 %
9.5.3. A hollow cylinder is rotating about an axis that passes through the center of
both ends. The radius of the cylinder is r. At what angular speed  must the
this cylinder rotate to have the same total kinetic energy that it would have if
it were moving horizontally with a speed v without rotation?
v2
a)  
2r
b)  
v
2
r
v
c)  
r
v
d)  
2r
v2
e)   2
r
9.5.4. Two solid cylinders are rotating about an axis that passes
through the center of both ends of each cylinder. Cylinder A has
three times the mass and twice the radius of cylinder B, but they
have the same rotational kinetic energy. What is the ratio of the
angular velocities, A/B, for these two cylinders?
a) 0.25
b) 0.50
c) 1.0
d) 2.0
e) 4.0
9.5.5. Consider the drawing. A rope is wrapped around one-third of the
circumference of a solid disk of radius R = 2.2 m that is free to rotate about an
axis that passes through its center. The force applied to the rope has a
magnitude of 35 N; and the disk has a mass M of 7.5 kg. Assuming the force is
applied horizontally as shown and the disk is initially at rest, determine the
amount of rotational work done until the time when the end of the rope reaches
the top of the disk?
a) 140 N
b) 160 N
c) 180 N
d) 210 N
e) 250 N
Angular Momentum
Angular Momentum of a Single Particle
 The angular momentum of a particle about a point O is
L  rP sin 
P=mv

r
O
Angular Momentum for Circular Motion
P  mvT  mr
L  rP sin   r (mr ) sin   mr 
2
L  I
Angular Momentum
 For a single particle in circular motion around a point, or a
rigid object rotating on an axis of symmetry, the angular
momentum is:
L  Iω
I  rotational inertia
ω  angular velocity
Torque and Angular Momentum
 We can derive an “Impulse-momentum Theorem” for
rotation.

  I  I
t

I ( f  i )
t

t  L
( I f  Ii )
t
Torque and Angular Momentum
Practice
 F= 2.50 N
 r= 0.260 m
 t = 0.150 s.
 M=4.00 kg
 Find f if it starts from
rest, assuming friction is
negligible.
Angular Momentum Practice
 http://cnx.org/content/m42182/latest/?collection=col11
406/1.7
 Mass of the Earth = 5.979 X 1024 kg
 Radius of the Earth= 6.376 X 106 m
 Radius of the Earth’s orbit = 1.496 X 1011 m
 Mass of the Moon = 7.348 X 1022 kg
 Radius of the Moon= 1.734 X 106 m
 Radius of the Moon’s orbit = 3.84 X 108 m
Conservation of Angular
Momentum
What If the net torque on an object or
system is 0?

 The net external torque =
L
rate of change of angular


 EXT t
momentum.


L  0
Li  L f
 If the net external torque is
zero, the angular momentum
doesn’t change.
 Angular momentum is
conserved in an isolated
system.
Conservation of Angular Momentum
 If the net external torque acting on a system is zero, the
total angular momentum of the system is constant.
Conservation of Angular Momentum
Example 1
Given I 0 , 0 and I , find .
Conservation of Angular Momentum
Example 2
 (Based on Example 15, p.
264 in the textbook.) An
Earth satellite has a
distance of closest
approach (perigee) of 8.37
X 106 m, and a farthest
distance (apogee) of 25.1 X
106 m. The speed of the
satellite at perigee is 8450
m/s. Find the speed at
apogee.
Conservation of Angular Momentum
Example 3: Kepler’s Second Law
 Kepler’s Second Law states that a line joining a planet and the
sun sweeps out equal areas in equal times.
Conservation of Angular Momentum
Example 4: Stability of a Motorcycle
© OpenStax College
Conservation of Angular Momentum
Practice Problem 1
 Two disks rotate about a
frictionless shaft. Disk 1 has a
rotational speed of i and a
rotational inertia of I1. It drops
onto another disk of rotational
inertia I2 that is initially at rest.
The disks exert frictional forces
on each other and soon rotate
with the same speed f. Find
f.
Conservation of Angular Momentum:
Practice Problem 2
 A puck of mass 0.0500 kg and
moving at 30 m/s hits a stick that
is loosely nailed to a frictionless
table. The stick is 1.20 m long
and has a mass of 2.0 kg. The puck
adheres to the stick and they
rotate together around the nail?
What is the final angular velocity
of the puck and stick system?
Conservation of Angular Momentum
Practice Problem 3
 A particle of mass m moves with
speed v0 in a circle of radius r0 .
The particle is attached to a
string that passes through a hole
in the table. The string is pulled
downward so the mass moves in
a circle of radius r. Find the final
velocity.
 Ans. v= (r0/r) v0
Translational and Angular Formulas
Translational
Angular
FNET  ma

p  mv
τ NET  Iα
L  Iω
KETRANS  mv
1
2
2
KE ROT  I
1
2
2
Conservation of Angular Momentum
Practice
 http://cnx.org/content/m42182/latest/?collection=col11
406/1.7
9.6.1. A star is rotating about an axis that passes through its center. When
the star “dies,” the balance between the inward pressure due to the
force of gravity and the outward pressure from nuclear processes is
no longer present and the star collapses inward; and its radius
decreases with time. Which one of the following choices best
describes what happens as the star collapses?
a) The angular velocity of the star remains constant.
b) The angular momentum of the star remains constant.
c) The angular velocity of the star decreases.
d) The angular momentum of the star decreases.
e) Both angular momentum and angular velocity increase.
9.6.2. A solid sphere of radius R rotates about an axis that is
tangent to the sphere with an angular speed . Under the
action of internal forces, the radius of the sphere increases to
2R. What is the final angular speed of the sphere?
a) /4
b) /2
c) 
d) 2
e) 4
9.6.3. While excavating the tomb of Tutankhamen (d. 1325 BC),
archeologists found a sling made of linen. The sling could hold a stone
in a pouch, which could then be whirled in a horizontal circle. The stone
could then be thrown for hunting or used in battle. Imagine the sling
held a 0.050-kg stone; and it was whirled at a radius of 1.2 m with an
angular speed of 2.0 rev/s. What was the angular momentum of the
stone under these circumstances?
a) 0.14 kg  m2/s
b) 0.90 kg  m2/s
c) 1.2 kg  m2/s
d) 2.4 kg  m2/s
e) 3.6 kg  m2/s
9.6.4. Joe has volunteered to help out in his physics class by sitting on a stool that easily
rotates. As Joe holds the dumbbells out as shown, the professor temporarily applies a
sufficient torque that causes him to rotate slowly. Then, Joe brings the dumbbells close
to his body and he rotates faster. Why does his speed increase?
a) By bringing the dumbbells inward, Joe exerts a torque on the stool.
b) By bringing the dumbbells inward, Joe decreases the moment of inertia.
c) By bringing the dumbbells inward, Joe increases the angular momentum.
d) By bringing the dumbbells inward, Joe increases the moment of inertia.
e) By bringing the dumbbells inward, Joe decreases the angular momentum.
9.6.5. Joe has volunteered to help out in his physics class by sitting on
a stool that easily rotates. Joe holds the dumbbells out as shown as
the stool rotates. Then, Joe drops both dumbbells. How does the
rotational speed of stool change, if at all?
a) The rotational speed increases.
b) The rotational speed decreases,
but Joe continues to rotate.
c) The rotational speed remains
the same.
d) The rotational speed quickly
decreases to zero rad/s.
9.6.6. Joe has volunteered to help out in his physics class by sitting on a stool that easily
rotates. Joe holds the dumbbells out as shown as the stool rotates. Then, Joe drops both
dumbbells. Then, the angular momentum of Joe and the stool changes, but the angular
velocity does not change. Which of the following choice offers the best explanation?
a) The force exerted by the dumbbells acts in
opposite direction to the torque.
b) Angular momentum is conserved, when no
external forces are acting.
c) Even though the angular momentum decreases,
the moment of inertia also decreases.
d) The decrease in the angular momentum is balanced
by an increase in the moment of inertia.
e) The angular velocity must increase when the
dumbbells are dropped.
9.6.7. Sarah has volunteered to help out in her physics class by sitting on a stool that
easily rotates. The drawing below shows the view from above her head. She
holds the dumbbells out as shown as the stool rotates. Then, she drops both
dumbbells. Which one of the four trajectories illustrated best represents the
motion of the dumbbells after they are dropped?
9.6.8. Two ice skaters are holding hands and spinning around their combined center
of mass, represented by the small black dot in Frame 1, with an angular
momentum L. When the skaters are at the position shown in Frame 2, they
release hands and move
in opposite directions as
shown in Frame 3.
What is the angular
momentum of the
skaters in Frame 3?
a)
zero kg  m2/s
b) a value that is greater than zero kg  m2/s, but less than L
c)
a value less than L and decreasing as they move further apart
d) a value that is greater than L
e)
L