Rotational Motion and

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Rotational Motion and Angular Momentum
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• Angular Quantities
• Vector Nature of Angular Quantities
• Constant Angular Acceleration
• Torque
• Vector Cross Product
• Rotational Dynamics Torque and Rotational
  Inertia
• Solving Problems in Rotational Dynamics

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• Determining Moments of Inertia
• Rotational Kinetic Energy
• Rotational Plus Translational Motion Rolling
• Why Does a Rolling Sphere Slow Down?

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• Angular MomentumObjects Rotating About a Fixed
  Axis
• Angular Momentum of a Particle
• Angular Momentum and Torque for a System of
  Particles General Motion
• Angular Momentum and Torque for a Rigid Object

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• Conservation of Angular Momentum
• The Spinning Top and Gyroscope
• Rotating Frames of Reference Inertial Forces
• The Coriolis Effect

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Angular Quantities
In purely rotational motion, all points on the
object move in circles around the axis of
rotation (O). The radius of the circle is R.
All points on a straight line drawn through the
axis move through the same angle in the same
time. The angle ? in radians is defined where
l is the arc length.
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Angular Quantities
Birds of preyin radians. A particular birds eye
can just distinguish objects that subtend an
angle no smaller than about 3 x 10-4 rad. (a) How
many degrees is this? (b) How small an object can
the bird just distinguish when flying at a height
of 100 m?
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Solution
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Angular Quantities
Angular displacement The average angular
velocity is defined as the total angular
displacement divided by time The instantaneous
angular velocity
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Angular Quantities
The angular acceleration is the rate at which the
angular velocity changes with time The
instantaneous acceleration
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Angular Quantities
Every point on a rotating body has an angular
velocity ? and a linear velocity v. They are
related
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Angular Quantities
Is the lion faster than the horse? On a rotating
carousel or merry-go-round, one child sits on a
horse near the outer edge and another child sits
on a lion halfway out from the center. (a) Which
child has the greater linear velocity? (b) Which
child has the greater angular velocity?
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Angular Quantities
Objects farther from the axis of rotation will
move faster.
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Angular Quantities
If the angular velocity of a rotating object
changes, it has a tangential acceleration
Even if the angular velocity is constant, each
point on the object has a centripetal
acceleration
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Angular Quantities
Here is the correspondence between linear and
rotational quantities
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Angular Quantities
Angular and linear velocities and
accelerations. A carousel is initially at rest.
At t 0 it is given a constant angular
acceleration a 0.060 rad/s2, which increases
its angular velocity for 8.0 s. At t 8.0 s,
determine the magnitude of the following
quantities (a) the angular velocity of the
carousel (b) the linear velocity of a child
located 2.5 m from the center (c) the tangential
(linear) acceleration of that child (d) the
centripetal acceleration of the child and (e)
the total linear acceleration of the child.
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Solution
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Angular Quantities
The frequency is the number of complete
revolutions per second Frequencies are
measured in hertz The period is the time one
revolution takes
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Angular Quantities
Hard drive. The platter of the hard drive of a
computer rotates at 7200 rpm (rpm revolutions
per minute rev/min). (a) What is the angular
velocity (rad/s) of the platter? (b) If the
reading head of the drive is located 3.00 cm from
the rotation axis, what is the linear speed of
the point on the platter just below it? (c) If a
single bit requires 0.50 µm of length along the
direction of motion, how many bits per second can
the writing head write when it is 3.00 cm from
the axis?
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Angular Quantities
Given ? as function of time. A disk of radius R
3.0 m rotates at an angular velocity ? (1.6
1.2t) rad/s, where t is in seconds. At the
instant t 2.0 s, determine (a) the angular
acceleration, and (b) the speed v and the
components of the acceleration a of a point on
the edge of the disk.
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Vector Nature of Angular Quantities
The angular velocity vector points along the axis
of rotation, with the direction given by the
right-hand rule. If the direction of the rotation
axis does not change, the angular acceleration
vector points along it as well.
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Constant Angular Acceleration
The equations of motion for constant angular
acceleration are the same as those for linear
motion, with the substitution of the angular
quantities for the linear ones.
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Constant Angular Acceleration
Centrifuge acceleration. A centrifuge rotor is
accelerated from rest to 20,000 rpm in 30 s. (a)
What is its average angular acceleration? (b)
Through how many revolutions has the centrifuge
rotor turned during its acceleration period,
assuming constant angular acceleration?
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Solution
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Torque
To make an object start rotating, a force is
needed the position and direction of the force
matter as well. The perpendicular distance from
the axis of rotation to the line along which the
force acts is called the lever arm.
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Torque
A longer lever arm is very helpful in rotating
objects.
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Torque
Here, the lever arm for FA is the distance from
the knob to the hinge the lever arm for FD is
zero and the lever arm for FC is as shown.
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Torque
The torque is defined as
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Torque
Torque on a compound wheel.
Two thin disk-shaped wheels, of radii RA 30 cm
and RB 50 cm, are attached to each other on an
axle that passes through the center of each, as
shown. Calculate the net torque on this compound
wheel due to the two forces shown, each of
magnitude 50 N.
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Solution
The torque due to FA tends to accelerate the
wheel counterclockwise, whereas the torque due to
FB tends to accelerate the wheel clockwise.
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Vector Cross Product
The vector cross product is defined as
The direction of the cross product is defined by
a right-hand rule
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Vector Cross Product
The cross product can also be written in
determinant form
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Vector Cross Product
Some properties of the cross product
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Torque as a Vector
Torque can be defined as the vector product of
the force and the vector from the point of action
of the force to the axis of rotation
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Torque as a Vector
For a particle, the torque can be defined around
a point O
Here, is the position vector from the particle
relative to O.
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Torque as a Vector
Torque vector. Suppose the vector is in the xz
plane, and is given by (1.2 m) 1.2 m)
Calculate the torque vector if (150 N) .
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Torque and Rotational Inertia
Knowing that , we see that
This is for a single point mass what about an
extended object? As the angular acceleration is
the same for the whole object, we can write
R
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Torque and Rotational Inertia
The quantity is called the rotational inertia
of an object. The distribution of mass matters
herethese two objects have the same mass, but
the one on the left has a greater rotational
inertia, as so much of its mass is far from the
axis of rotation.
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Torque and Rotational Inertia
The rotational inertia of an object depends not
only on its mass distribution but also the
location of the axis of rotationcompare (f) and
(g), for example.
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Solving Problems in Rotational Dynamics

1. Draw a diagram.
2. Decide what the system comprises.
3. Draw a free-body diagram for each object under
   consideration, including all the forces acting on
   it and where they act.
4. Find the axis of rotation calculate the torques
   around it.

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Solving Problems in Rotational Dynamics
5. Apply Newtons second law for rotation. If the
rotational inertia is not provided, you need to
find it before proceeding with this step. 6.
Apply Newtons second law for translation and
other laws and principles as needed. 7. Solve. 8.
Check your answer for units and correct order of
magnitude.
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Determining Moments of Inertia
If a physical object is available, the moment of
inertia can be measured experimentally. Otherwise,
if the object can be considered to be a
continuous distribution of mass, the moment of
inertia may be calculated
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Determining Moments of Inertia
Cylinder, solid or hollow. (a) Show that the
moment of inertia of a uniform hollow cylinder of
inner radius R1, outer radius R2, and mass M, is
I ½ M(R12 R22), if the rotation axis is
through the center along the axis of symmetry.
(b) Obtain the moment of inertia for a solid
cylinder.
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Solution
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Determining Moments of Inertia
The parallel-axis theorem gives the moment of
inertia about any axis parallel to an axis that
goes through the center of mass of an object
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Determining Moments of Inertia
Parallel axis. Determine the moment of inertia of
a solid cylinder of radius R0 and mass M about an
axis tangent to its edge and parallel to its
symmetry axis.
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Solution
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Determining Moments of Inertia
The perpendicular-axis theorem is valid only for
flat objects.
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Rotational Kinetic Energy
The kinetic energy of a rotating object is given
by By substituting the rotational quantities,
we find that the rotational kinetic energy can be
written A object that both translational and
rotational motion also has both translational and
rotational kinetic energy
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Rotational Kinetic Energy
When using conservation of energy, both
rotational and translational kinetic energy must
be taken into account.
All these objects have the same potential energy
at the top, but the time it takes them to get
down the incline depends on how much rotational
inertia they have.
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Rotational Kinetic Energy
The torque does work as it moves the wheel
through an angle ?
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Rotational Plus Translational Motion Rolling
In (a), a wheel is rolling without slipping. The
point P, touching the ground, is instantaneously
at rest, and the center moves with velocity
. In (b) the same wheel is seen from a reference
frame where C is at rest. Now point P is moving
with velocity . The linear speed of the wheel
is related to its angular speed
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Why Does a Rolling Sphere Slow Down?
A rolling sphere will slow down and stop rather
than roll forever. What force would cause
this? If we say friction, there are problems

• The frictional force has to act at the point of
  contact this means the angular speed of the
  sphere would increase.
• Gravity and the normal force both act through
  the center of mass, and cannot create a torque.

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Why Does a Rolling Sphere Slow Down?
The solution No real sphere is perfectly rigid.
The bottom will deform, and the normal force will
create a torque that slows the sphere.
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Objects Rotating About a Fixed Axis
The rotational analog of linear momentum is
angular momentum, L
Then the rotational analog of Newtons second law
is
This form of Newtons second law is valid even if
I is not constant.
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Objects Rotating About a Fixed Axis
In the absence of an external torque, angular
momentum is conserved
More formally,the total angular momentum of a
rotating object remains constant if the net
external torque acting on it is zero.
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Objects Rotating About a Fixed Axis
This means
Therefore, if an objects moment of inertia
changes, its angular speed changes as well.
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Angular Momentum of a Particle
A particles angular momentum. What is the
angular momentum of a particle of mass m moving
with speed v in a circle of radius r in a
counterclockwise direction?
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Angular Momentum of a Particle
The angular momentum of a particle about a
specified axis is given by
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Angular Momentum of a Particle
If we take the derivative of , we find
Since
we have
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Objects Rotating About a Fixed Axis
Object rotating on a string of changing length.
A small mass m attached to the end of a string
revolves in a circle on a frictionless tabletop.
The other end of the string passes through a hole
in the table. Initially, the mass revolves with a
speed v1 2.4 m/s in a circle of radius R1
0.80 m. The string is then pulled slowly through
the hole so that the radius is reduced to R2
0.48 m. What is the speed, v2, of the mass now?
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Solution
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Objects Rotating About a Fixed Axis
Clutch. A simple clutch consists of two
cylindrical plates that can be pressed together
to connect two sections of an axle, as needed, in
a piece of machinery. The two plates have masses
MA 6.0 kg and MB 9.0 kg, with equal radii R0
0.60 m. They are initially separated. Plate MA
is accelerated from rest to an angular velocity
?1 7.2 rad/s in time ?t 2.0 s. Calculate (a)
the angular momentum of MA, and (b) the torque
required to have accelerated MA from rest to ?1.
(c) Next, plate MB, initially at rest but free to
rotate without friction, is placed in firm
contact with freely rotating plate MA, and the
two plates both rotate at a constant angular
velocity ?2, which is considerably less than ?1.
Why does this happen, and what is ?2?
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Objects Rotating About a Fixed Axis
Neutron star. Astronomers detect stars that are
rotating extremely rapidly, known as neutron
stars. A neutron star is believed to form from
the inner core of a larger star that collapsed,
under its own gravitation, to a star of very
small radius and very high density. Before
collapse, suppose the core of such a star is the
size of our Sun (r 7 x 105 km) with mass 2.0
times as great as the Sun, and is rotating at a
frequency of 1.0 revolution every 100 days. If it
were to undergo gravitational collapse to a
neutron star of radius 10 km, what would its
rotation frequency be? Assume the star is a
uniform sphere at all times, and loses no mass.
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Solution
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Objects Rotating About a Fixed Axis
Angular momentum is a vector for a symmetrical
object rotating about a symmetry axis it is in
the same direction as the angular velocity vector.
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Objects Rotating About a Fixed Axis
Running on a circular platform. Suppose a 60-kg
person stands at the edge of a 6.0-m-diameter
circular platform, which is mounted on
frictionless bearings and has a moment of inertia
of 1800 kgm2. The platform is at rest initially,
but when the person begins running at a speed of
4.2 m/s (with respect to the Earth) around its
edge, the platform begins to rotate in the
opposite direction. Calculate the angular
velocity of the platform.
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Objects Rotating About a Fixed Axis
Spinning bicycle wheel. Your physics teacher is
holding a spinning bicycle wheel while he stands
on a stationary frictionless turntable. What will
happen if the teacher suddenly flips the bicycle
wheel over so that it is spinning in the opposite
direction?
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Angular Momentum and Torque for a System of
Particles
The angular momentum of a system of particles can
change only if there is an external
torquetorques due to internal forces cancel.
This equation is valid in any inertial reference
frame. It is also valid for the center of mass,
even if it is accelerating
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Angular Momentum and Torque for a Rigid Object
For a rigid object, we can show that its angular
momentum when rotating around a particular axis
is given by
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Angular Momentum and Torque for a Rigid Object
Atwoods machine. An Atwood machine consists of
two masses, mA and mB, which are connected by an
inelastic cord of negligible mass that passes
over a pulley. If the pulley has radius R0 and
moment of inertia I about its axle, determine the
acceleration of the masses mA and mB, and compare
to the situation where the moment of inertia of
the pulley is ignored.
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Angular Momentum and Torque for a Rigid Object
Bicycle wheel. Suppose you are holding a bicycle
wheel by a handle connected to its axle. The
wheel is spinning rapidly so its angular momentum
points horizontally as shown. Now you suddenly
try to tilt the axle upward (so the CM moves
vertically). You expect the wheel to go up (and
it would if it werent rotating), but it
unexpectedly swerves to the right! Explain.
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Angular Momentum and Torque for a Rigid Object
A system that is rotationally imbalanced will not
have its angular momentum and angular velocity
vectors in the same direction. A torque is
required to keep an unbalanced system rotating.
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Angular Momentum and Torque for a Rigid Object
Torque on unbalanced system. Determine the
magnitude of the net torque tnet needed to keep
the illustrated system turning.
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Solution
?
?
I?
L
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Conservation of Angular Momentum
If the net torque on a system is constant,
The total angular momentum of a system remains
constant if the net external torque acting on the
system is zero.
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Conservation of Angular Momentum
Bullet strikes cylinder edge. A bullet of mass m
moving with velocity v strikes and becomes
embedded at the edge of a cylinder of mass M and
radius R0. The cylinder, initially at rest,
begins to rotate about its symmetry axis, which
remains fixed in position. Assuming no frictional
torque, what is the angular velocity of the
cylinder after this collision? Is kinetic energy
conserved?
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The Spinning Top and Gyroscope
A spinning top will precess around its point of
contact with a surface, due to the torque created
by gravity when its axis of rotation is not
vertical.
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The Spinning Top and Gyroscope
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The Spinning Top and Gyroscope
The angular velocity of the precession is given
by
This is also the angular velocity of precession
of a toy gyroscope, as shown.
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Rotating Frames of Reference Inertial Forces
There is an apparent outward force on objects in
rotating reference frames this is a fictitious
force, or a pseudoforce. The centrifugal force
is of this type there is no outward force when
viewed from an inertial reference frame.
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The Coriolis Effect
The Coriolis effect is responsible for the
rotation of air around low-pressure areas
counterclockwise in the Northern Hemisphere and
clockwise in the Southern. The Coriolis
acceleration is
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Summary

• Angles are measured in radians a whole circle
  is 2p radians.
• Angular velocity is the rate of change of
  angular position.
• Angular acceleration is the rate of change of
  angular velocity.
• The angular velocity and acceleration can be
  related to the linear velocity and acceleration.
• The frequency is the number of full revolutions
  per second the period is the inverse of the
  frequency.

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Summary

• The equations for rotational motion with
  constant angular acceleration have the same form
  as those for linear motion with constant
  acceleration.
• Torque is the product of force and lever arm.
• The rotational inertia depends not only on the
  mass of an object but also on the way its mass is
  distributed around the axis of rotation.
• The angular acceleration is proportional to the
  torque and inversely proportional to the
  rotational inertia.

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Summary

• An object that is rotating has rotational
  kinetic energy. If it is translating as well, the
  translational kinetic energy must be added to the
  rotational to find the total kinetic energy.

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Summary

• Angular momentum of a rigid object

• Newtons second law

• Angular momentum is conserved.
• Torque

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Summary

• Angular momentum of a particle

• Net torque

• If the net torque is zero, the vector angular
  momentum is conserved.