Torque and Angular Momentum

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Chapter 8

• Torque and Angular Momentum

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Torque and Angular Momentum

• Rotational Kinetic Energy
• Rotational Inertia
• Torque
• Work Done by a Torque
• Equilibrium (revisited)
• Rotational Form of Newtons 2nd Law
• Rolling Objects
• Angular Momentum

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Rotational KE and Inertia
For a rotating solid body
For a rotating body vi ?ri where ri is the
distance from the rotation axis to the mass mi.
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Moment of Inertia
The quantity
is called rotational inertia or moment of inertia.
Use the above expression when the number of
masses that make up a body is small. Use the
moments of inertia in the table in the textbook
for extended bodies.
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Moments of Inertia
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Moment of Inertia
Example The masses are m1 and m2 and they are
separated by a distance r. Assume the rod
connecting the masses is massless. Q (a) Find
the moment of inertia of the system below.
r1 and r2 are the distances between mass 1 and
the rotation axis and mass 2 and the rotation
axis (the dashed, vertical line) respectively.
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Moment of Inertia-Axis Dependent
Take m1 2.00 kg, m2 1.00 kg, r1 0.33 m , and
r2 0.67 m.
(b) What is the moment of inertia if the axis is
moved so that is passes through m1?
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Moment of Inertia
What is the rotational inertia of a solid iron
disk of mass 49.0 kg with a thickness of 5.00 cm
and a radius of 20.0 cm, about an axis through
its center and perpendicular to it?
From the table
The moment of inertia involves the mass, the
square of a characteristic length, perpendicular
to the axis of rotation, and a dimensionless
factor ß.
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Torque
A torque is caused by the application of a force,
on an object, at a point other than its center of
mass or its pivot point.
Q Where on a door do you normally push to open
it? A Away from the hinge.
A rotating (spinning) body will continue to
rotate unless it is acted upon by a torque.
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Torque - Force Components
Torque method 1
Top view of door
r the distance from the rotation axis (hinge)
to the point where the force F is applied.
F? is the component of the force F that is
perpendicular to the door (here it is Fsin?).
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Torque
The units of torque are Newton-meters (Nm) (not
joules!).
By convention

• When the applied force causes the object to
  rotate counterclockwise (CCW) then ? is positive.
• When the applied force causes the object to
  rotate clockwise (CW) then ? is negative.

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Torque
Torque method 2
r? is called the lever arm and F is the magnitude
of the applied force.
Lever arm is the perpendicular distance to the
line of action of the force.
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Torque - Lever Arm
Top view of door
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Torque Problem
The pull cord of a lawnmower engine is wound
around a drum of radius 6.00 cm, while the cord
is pulled with a force of 75.0 N to start the
engine. What magnitude torque does the cord
apply to the drum?
F75 N
R6.00 cm
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Torque Problem
Calculate the torque due to the three forces
shown about the left end of the bar (the red X).
The length of the bar is 4m and F2 acts in the
middle of the bar.
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Torque Problem
The lever arms are
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Torque Problem
The torques are
The net torque is 65.8 Nm and is the sum of the
above results.
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Work done by the Torque
The work done by a torque ? is
where ?? is the angle (in radians) that the
object turns through.
Following the analogy between linear and
rotational motion Linear Work is Force x
displacement. In the rotational picture force
becomes torque and displacement becomes the angle
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Work done by the Torque
A flywheel of mass 182 kg has a radius of 0.62 m
(assume the flywheel is a hoop).
(a) What is the torque required to bring the
flywheel from rest to a speed of 120 rpm in an
interval of 30 sec?
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Work done by the Torque
(b) How much work is done in this 30 sec period?
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Equilibrium
The conditions for equilibrium are
Linear motion
Rotational motion
For motion in a plane we now have three equations
to satisfy.
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Using Torque
A sign is supported by a uniform horizontal boom
of length 3.00 m and weight 80.0 N. A cable,
inclined at a 35? angle with the boom, is
attached at a distance of 2.38 m from the hinge
at the wall. The weight of the sign is 120.0 N.
What is the tension in the cable and what are the
horizontal and vertical forces exerted on the
boom by the hinge?
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Using Torque
This is important! You need two components for
F, not just the expected perpendicualr normal
force.
FBD for the bar
Apply the conditions for equilibrium to the bar
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Using Torque
Equation (3) can be solved for T
Equation (1) can be solved for Fx
Equation (2) can be solved for Fy
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Equilibrium in the Human Body
Find the force exerted by the biceps muscle in
holding a one liter milk carton with the forearm
parallel to the floor. Assume that the hand is
35.0 cm from the elbow and that the upper arm is
30.0 cm long. The elbow is bent at a right
angle and one tendon of the biceps is attached at
a position 5.00 cm from the elbow and the other
is attached 30.0 cm from the elbow. The weight
of the forearm and empty hand is 18.0 N and the
center of gravity is at a distance of 16.5 cm
from the elbow.
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MCAT type problem
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Newtons 2nd Law in Rotational Form
Compare to
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Rolling Object
A bicycle wheel (a hoop) of radius 0.3 m and mass
2 kg is rotating at 4.00 rev/sec. After 50 sec
the wheel comes to a stop because of friction.
What is the magnitude of the average torque due
to frictional forces?
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Rolling Objects
An object that is rolling combines translational
motion (its center of mass moves) and rotational
motion (points in the body rotate around the
center of mass).
For a rolling object
If the object rolls without slipping then vcm
R?.
Note the similarity in the form of the two
kinetic energies.
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Rolling Example
Two objects (a solid disk and a solid sphere) are
rolling down a ramp. Both objects start from
rest and from the same height. Which object
reaches the bottom of the ramp first?
This we know - The object with the largest linear
velocity (v) at the bottom of the ramp will win
the race.
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Rolling Example
Apply conservation of mechanical energy
Solving for v
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Rolling Example
Example continued
Note that the mass and radius are the same.
The moments of inertia are
For the disk
Since Vspheregt Vdisk the sphere wins the race.
For the sphere
Compare these to a box sliding down the ramp.
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The Disk or the Ring?
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How do objects in the previous example roll?
FBD
Both the normal force and the weight act through
the center of mass so ?? 0. This means that
the object cannot rotate when only these two
forces are applied.
x
The round object wont rotate, but most students
have difficulty imagining a sphere that doesnt
rotate when moving down hill.
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Add Friction
FBD
Also need acm ?R and
The above system of equations can be solved for v
at the bottom of the ramp. The result is the
same as when using energy methods. (See text
example 8.13.)
It is the addition of static friction that makes
an object roll.
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Angular Momentum
Units of p are kg m/s
Units of L are kg m2/s
When no net external forces act, the momentum of
a system remains constant (pi pf)
When no net external torques act, the angular
momentum of a system remains constant (Li Lf).
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Angular Momentum Example
A turntable of mass 5.00 kg has a radius of 0.100
m and spins with a frequency of 0.500 rev/sec.
Assume a uniform disk. What is the angular
momentum?
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Angular Momentum Example
A skater is initially spinning at a rate of 10.0
rad/sec with I2.50 kg m2 when her arms are
extended. What is her angular velocity after
she pulls her arms in and reduces I to 1.60 kg m2?
The skater is on ice, so we can ignore external
torques.
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The Vector Nature of Angular Momentum
Angular momentum is a vector. Its direction is
defined with a right-hand rule.
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The Right-Hand Rule
Curl the fingers of your right hand so that they
curl in the direction a point on the object
moves, and your thumb will point in the direction
of the angular momentum.
Angular Momentum is also an example of a vector
cross product
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The Vector Cross Product
The magnitude of C C ABsin(F) The direction of
C is perpendicular to the plane of A and B.
Physically it means the product of A and the
portion of B that is perpendicular to A.
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The Cross Product by Components
Since A and B are in the x-y plane A x B is along
the z-axis.
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Memorizing the Cross Product
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The Gyroscope Demo
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Angular Momentum Demo
Consider a person holding a spinning wheel. When
viewed from the front, the wheel spins CCW.
Holding the wheel horizontal, they step on to a
platform that is free to rotate about a vertical
axis.
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Angular Momentum Demo
Initially there is no angular momentum about the
vertical axis. When the wheel is moved so that
it has angular momentum about this axis, the
platform must spin in the opposite direction so
that the net angular momentum stays zero.
This is a result of conserving angular momentum.
Is angular momentum conserved about the direction
of the wheels initial, horizontal axis?
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It is not. The floor exerts a torque on the
system (platform person), thus angular momentum
is not conserved here.
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Summary

• Rotational Kinetic Energy
• Moment of Inertia
• Torque (two methods)
• Conditions for Equilibrium
• Newtons 2nd Law in Rotational Form
• Angular Momentum
• Conservation of Angular Momentum

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