Download Angular Momentum (AIS)

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Cutnell & Johnson, Wiley Publishing, Physics
5th Ed.
Torque (  )
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Copywrited by Holt, Rinehart, & Winston
• Torque is the product of a force and the
perpendicular distance from the axis of
rotation to the line of action of the force (aka
the lever arm).
  F l
• when F is not perpendicular to a line from the
axis of rotation
  
  r  F or
where F can be written as Fsinθ
  Fl sin 
Torque (  )
• Units: Newton meter (Nm)
• Positive: Counter Clockwise (ccw) Rotation
• Max Torque when Force is 90° to the line of
action.
Cutnell & Johnson, Wiley Publishing, Physics 5th Ed.
Copywrited by Holt, Rinehart, & Winston
Cutnell & Johnson, Wiley Publishing, Physics 5th Ed.
Equilibrium
• A rigid body is considered in equilibrium is it has
zero translational acceleration and zero
angular acceleration.
OR
2 conditions required for equilibrium
F  0
  0
If object has forces acting
in both the horizontal and
vertical directions then:
F y  0
Fx  0
Copyright ©2007 Pearson Prentice Hall, Inc.
Torque Summation
• In summing torques since the net torque is the
same anywhere on the body, it does not matter
which point of rotation we choose for our
summation of torques
• Best to choose a point that is easiest to solve
(The location of one of our unknown forces)
Copyright ©2007 Pearson Prentice Hall, Inc.
Copyright ©2007 Pearson Prentice Hall, Inc.
Diver
A diver weighing 530N is at the
end of a diving board with a
length of 3.90m. The board
has a negligible weight and is
bolted at one end with a
fulcrum supporting it at 1.40 m
from the end. Find the force
of the bolt and the fulcrum on
the board.
Cutnell & Johnson, Wiley Publishing, Physics 5th Ed.
Moment of Inertia
 i  
i
2
mi ri 
i
• Alpha is the same for all particles of the
object and when removed, we get the
MOMENT OF INERTIA. This is a property
of the object just like Inertia (mass).
I 
2
m
r
 ii
or  net  I
i
Cutnell & Johnson, Wiley Publishing, Physics 5th Ed.
• Different shapes have different
values of I.
Moment of Inertia
Experiments show that I is directly proportional
• to the mass.
• The distribution of mass in the body.
To illustrate this consider two wheels having
equal mass but different mass distribution.
A
B
B
If both of these wheels accelerate from rest to the same angular
velocity ω in the same time t.
• The angular acceleration, α, must be the same for both
wheels. Also, the total angle turned through must be the
same.
• But, when moving with angular velocity ω, the particles of
wheel B are moving faster than the particles of wheel A.
• Therefore, B possesses more kinetic energy than wheel A.
• More work is done accelerating wheel B than wheel A.
• A greater torque was needed to accelerate B than A. and so
the moment of inertia of wheel B is greater than the moment
of inertia of wheel A.
Cutnell & Johnson, Wiley Publishing, Physics 5th Ed.
A
Moment of Inertia
• The moment of inertia of a body is directly
proportional to its mass and increases as
the mass is moved further from the axis of
rotation.
• The fact that I depends on mass distribution
means that the same body can have different
moments of inertia depending on which axis
of rotation we consider.
Cutnell & Johnson, Wiley Publishing, Physics 5th Ed.
Cutnell & Johnson, Wiley Publishing, Physics 5th Ed.
Moment of Inertia Proof
The moment of inertia of any body can be
found by adding together the moments
of inertia of all its component particles.
I body  mi ri
2
Using this idea gives the following results:
Copywrited by Holt, Rinehart, & Winston
• Point at which a body’s weight can be
considered to act when calculating the
torque due to weight.
• Xcg=location of the center of gravity
  Wtot xcg  W1 x1  W2 x2  ...
W1 x1  W2 x2  ...
xcg 
W1  W2  ...
A synonymous concept is the center of mass and
sometimes these terms are interchanged.
Cutnell & Johnson, Wiley Publishing, Physics 5th Ed.
Cutnell & Johnson, Wiley Publishing, Physics 5th Ed.
Center of Gravity
Center of Gravity
• If an extended object is to be balanced,
it must be supported through its center
of gravity.
Copyright ©2007 Pearson Prentice Hall, Inc.
Center of Gravity
Copyright ©2007 Pearson Prentice Hall, Inc.
• The center of gravity can be physically found
by suspending an object by a point and
tracing the vertical axis it hangs along. Hang
the object from a different point and again
tracing its vertical axis. Where these two
lines cross would be the center of gravity.
Cutnell & Johnson, Wiley Publishing, Physics 5th Ed.
Copywrited by Holt, Rinehart, & Winston
CG of an Arm
• Calculate the center of gravity of the
following arm.
• W1=17 N
• W2=11 N
• W3=4.2 N
Cutnell & Johnson, Wiley Publishing, Physics 5th Ed.
Comparing Linear and
Rotational Dynamics
We can derive new equations
by simply substituting
rotational quantities for
linear quantities.
For example:
Power is the time-rate of doing
work.
dW
d
P
 
 
dt
dt
• p = mv gives us
L = Iω
Linear
quantity
Angular (rotational)
quantity
d

vo
1
vf
2
a

F
T (Torque)
m
I (Moment of Inertia)
p
What is the last quantity
derived called?
Angular momentum
L
Angular Momementum
(L)
• equivalent to linear momentum
L = Iω
• A "principle of conservation of angular
momentum" also exists.
• With arms and leg outstretched (A), the
moment of inertia, Iinitial, is relatively large.
• Suppose the skater has a low initial
angular velocity ωinitial.
• The skater then gradually decreases her/his moment of
inertia by bringing arms and leg nearer to the axis of
rotation (B and C).
• Her/his angular velocity is observed to increase.
• This is easily explained if we consider that the person’s
angular momentum does not change.
Iinitial × initial = Ifinal × final
• If Iinitial > Ifinal then ωfinal > ωinitial
Cutnell & Johnson, Wiley Publishing, Physics 5th Ed.
Conservation of Angular
Momentum
Conservation of Angular
Momentum
Angular Momentum
L = pr
• linear momentum multiplied by its
distance from the point considered
• Notice that we can therefore calculate
the angular momentum of a body about
any point even if the body is not moving
in a circular path.
A High Diver
• A diver can change
his/her rate of rotation
in a similar way. The
diver starts out with low
angular velocity with
body straight and arms
outstretched. The
distribution of mass is
then changed in order
to vary the angular
velocity.
Cat
• Cats use the principle of
conservation of angular
momentum in order to
rotate themselves "in
mid-air". A cat will
always land on its feet,
if given enough (but not
too much!) height.
Comparing Linear and Rotational Dynamics
Copywrited by Holt, Rinehart, & Winston
Copywrited by Holt, Rinehart, & Winston
Cutnell & Johnson, Wiley Publishing, Physics 5th Ed.
Rotational Kinetic
Energy
• An Object can experience
Translational (linear) motion and/or
Rotational motion. Both motions
have related kinetic energies.
• Translational kinetic energy is ½ mv2
• The Rotational Kinetic energy is
based upon Translation KE:
KE 
1
2
mv
2

1
m (r  ) 
2
2
1
2
I
2
Riddle of the Sphinx
Beside you are two balls. One is hollow,
the other is solid. Tell me which is which.
• The balls are of equal weight.
– The hollow ball is weighted inside with
metal, so they are both exactly the same
weight.
• A plank of wood may be used.
How will you avoid being the Sphinx’s
lunch?
Pulley with Mass
A 10 kg pulley of radius 0.5 m is
connected to a 5.0 kg mass through
a string as shown. If the mass is let
go from rest
(a) What is its linear acceleration?
(b) What is the angular acceleration
of the pulley?