Chapter 11 Rolling, Torque and Angular Momentum

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Chapter 11 Rolling Motion, Torque and Angular
Momentum

• Rolling motion (axis of rotation is moving)
• Kinetic Energy of rolling motion
• Rolling motion on an incline
• Torque
• Angular momentum
• Angular momentum is conserved

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Rolling motion of a particle on a
wheel (Superposition of rolling and linear motion)
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11-2 Rolling motion
Smooth rolling There is no slipping
Linear speed of center of mass
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11-2 Rolling motion
The angular velocity of any point on the wheel is
the same.
The linear speed of any point on the object
changes as shown in the diagram!! For one instant
(bottom), point P has no linear speed. For one
instant (top), point P has a linear speed of
2vCOM
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11-3 Kinetic Energy of Rolling
Superposition principle Rolling motion
Pure translation Pure rotation
Kinetic energy of rolling motion
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Sample Problem 11-1
Approximate each wheel on the car Thrust SSC as a
disk of uniform thickness and mass M 170 kg,
and assume smooth rolling. When the cars speed
was 1233 km/h, what was the kinetic energy of
each wheel?
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11-4 Forces of Rolling
Friction and Rolling
If the wheel rolls without sliding (smooth
rolling) and is accelerating, then from
, (smooth rolling) where is the
linear acceleration of the center of mass and a
is the angular acceleration.
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Rolling Down a Ramp
The positive direction here is chosen to be down
the plane.
a is counterclockwise and positive.
where acom points down plane ve
Therefore from (2) ?
and substituting this in (1) ?
Note that a positive acom points down plane.
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Demo
A ring and and disk of equal mass and diameter
are rolling down a frictionless incline. Both
start at the same position which one will be
faster at the end of the incline?
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Sample Problem 11-2

• A uniform ball, of mass M 6.00 kg and radius R,
  rolls smoothly from rest down a ramp at angle q
  30.0
• (a)  The ball descends a vertical height h 1.20
  m to reach the bottom of the ramp. What is its
  speed at the bottom?

A positive vcom points down plane.
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(b) What are the magnitude and direction of the
friction force on the ball as it rolls down the
ramp?
A positive acom points down plane.
A positive fs means that the direction we
selected for fs (up) is correct! fsR is a
clockwise torque (ve)
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11-5 Yo-Yo

• The yo-yo can be considered as a rolling down a
  ramp
• Instead of rolling down a ramp at angle q with
  the horizontal, the yo-yo rolls down a string at
  angle q 90 with the horizontal.
• Instead of rolling on its outer surface at radius
  R, the yo-yo rolls on an axle of radius Ro.
• Instead of being slowed by frictional force fs,
  the yo-yo is slowed by the net force T on it from
  the string.

So we would again get the same expression for the
acceleration as for rolling with q 90.
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11-6 Torque and the vector product
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Sample Problem 11.3
What is the torque, with respect to the origin O,
due to each force?
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Sample Problem 11.3
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11-7 Angular momentum of a particle

• The SI unit of angular momentum l is .
• Angular momentum is a vector, the direction is
  determined by the right hand rule.
• The magnitude of angular momentum is
• where f is the angle between and when
  these two vectors are arranged tail to tail.

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Sample Problem 11-4
The RHR indicates that l1 is positive.
RHR right hand rule
The RHR indicates that l2 is negative.
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11-8 Newtons Second Law in Angular Form
(single particle)
(single particle)

• Note that the torque and angular momentum
  must be defined with respect to the same origin.

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11-9 Angular momentum of a system of Particles
(L total angular momentum)

• is the net torque on the ith particle.
  is the sum of all the torque(internal
  and external) on the system. However the internal
  torques sums to zero. Let represent the net
  external torque on the system.

( system of particles )
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11-10 Angular momentum of a rigid object
rotating about a fixed axis
Well consider an object that is rotating about
the z-axis. The angular momentum of the object
is given by
Note that in this case L and w are along the z
axis. Also note the analog formula for linear
momentum p mv
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11-11 Conservation of angular momentum
The total angular momentum of a system is
constant in both magnitude and direction if the
resultant external torque acting on the system is
zero.
If the system undergoes an internal
rearrangement, then
If the object is rotating about a fixed axis (say
z-axis), then
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Demo
A students stands still on a rotating platform
and holds two texts on outstretched arms. He
brings the arms closer. What happens? Discuss
A students stands still on a rotatable platform
and holds a spinning wheel. The bicycle wheel is
spinning in the clockwise direction when viewed
from above. He flips the wheel over. What
happens?
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TABLE 11-1     More Corresponding Variables and
Relations for Translational and Rotational Motiona

Translational  
Rotational  

Force  
Torque  
Linear momentum  
Angular momentum  
Linear momentumb  
Angular momentumb  
Linear momentumb  
Angular momentumc  
Newton's second lawb  
Newton's second lawb  
Conservation lawd  
Conservation lawd  

a  See also Table 10-3.b  For systems of
particles, including rigid bodies.c  For a rigid
body about a fixed axis, with L being the
component along that axis.d  For a closed,
isolated system.
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P22
Force F 2i-3k acts on a particle with position
vector r 0.5j-2.0k relative to the origin. In
unit vector notation, what is the resulting
torque on the pebble about (a) the origin and (b)
the point (2.0, 0, -3.0)?
P72
A uniform solid ball rolls smoothly along a floor
and up a ramp inclined at 15.0. It is
momentarily stops when it has rolled 1.50 m along
the ramp. What was its initial speed?
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P85
In fig. 11.-62, a constant horizontal force Fapp
of magnitude 12 N is applied to a uniform solid
cylinder by a fishing line wrapped around the
cylinder. The mass of the cylinder is 10 kg, its
radius is 0.10 m, and the cylinder rolls smoothly
on the horizontal surface. (a) What is the
magnitude of the acceleration of the com of the
cylinder? (b) What is the magnitude of the
angular acceleration of the cylinder about the
com? (c) In unit vector notation, what is the
frictional force acting on the cylinder?
P90
A uniform rod rotates in a horizontal plane about
a vertical axis through one end. The rod is 6.00
m long, weighs 10.0 N, and rotates at 240
rev/min. Calculate (a) its rotational inertia
about the axis of rotation and (b) the magnitude
of its angular momentum about the axis.
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P48
A cockroach of mass 0.17 kg runs counterclockwise
around the rim of a lazy Susan (a circular disk
mounted on a veritcal axle) that has radius 15
cm, rotational inertia 5.0 x 10-3 kgm2, and
frictionless bearings. The cockroachs speed
(relative to the ground) is 2.0 m/s, and the lazy
Susan turns clockwise with angular velocity wo
2.8 rad/s. The cockroach finds a bread crumb on
the rim and, of course, stops. (a) What is the
angular speed of the lazy Susan after the
cockroach stops?