Download Angular Momentum

1J-29 Overhanging Blocks
How far out from the table can a stack of bricks be balanced ?
Length = L, CG = L/2
Length = 3L/2, CG = 3L/4
Length = 7L/4, CG = 25L/24
How do the blocks
stay balanced when
the top block extends
beyond the bottom
block ?
Δx
Blocks can over hang but the Center of Gravity
of a block must be inside the block below
For 6 blocks max extension Δx :
Δx = L/2 + L/4 + L/6 + L/8 + L/10 + L/12 = 1.22(L)
IN REALITY EACH BLOCK HAS TO BE MOVED SLIGHTLY BACK TO AVOID TIPPING, SO
THE TOTAL EXTENSION WILL BE A LITTLE LESS.
2/25/2011
Rotational Inertia and Newton’s
Second Law
• In linear motion, net force and mass determine the
acceleration of an object.
• For rotational motion, torque determines the rotational
acceleration.
• The rotational counterpart to mass is rotational inertia or
moment of inertia.
– Just as mass represents the resistance to a change in linear
motion, rotational inertia is the resistance of an object to change
in its rotational motion.
– Rotational inertia is related to the mass of the object.
– It also depends on how the mass is distributed about the axis of
rotation.
Simplest example:
a mass at the end of a light rod
• A force is applied to the mass
in a direction perpendicular to
the rod.
• The rod and mass will begin to
rotate about the fixed axis at
the other end of the rod.
• The farther the mass is from
the axis, the faster it moves for
a given rotational velocity.
Simplest example:
a mass at the end of a
light rod
• To produce the same rotational
acceleration, a mass at the end
of the rod must receive a larger
linear acceleration than one
nearer the axis.
• F = ma
– It is harder to get the system
rotating when the mass is at the
end of the rod than when it is
nearer to the axis.
– I case the distance are equal, it’s
harder to move a heavier mass.
Rotational Inertia and Newton’s
Second Law
• Newton’s second law for linear motion:
Fnet = ma
• Newton’s second law for rotational motion:
∙R=m∙
∆
∆
∙
= ∙

∙R=m∙
∙
net = I
– The rotational acceleration produced is equal to the torque
divided by the rotational inertia.
Rotational Inertia and Newton’s
Second Law
• For an object with its mass concentrated at a point:
– Rotational inertia = mass x square of distance from axis
– I = mr2
• The total rotational inertia of an object like a merry-goround can be found by adding the contributions of all the
different parts of the object.
Two 0.2-kg masses are located at either end of a 1m long, very light and rigid rod as shown. What is
the rotational inertia of this system about an axis
through the center of the rod?
a)
b)
c)
d)
0.02 kg·m2
0.05 kg·m2
0.10 kg·m2
0.40 kg·m2
I = mr2
= (0.2 kg)(0.5m)2 x 2
= 0.10 kg·m2
Rotational
inertias for
more complex
shapes:
Angular Momentum
• Linear momentum is mass (inertia) times linear velocity:
p = mv
• Angular momentum is rotational inertia times rotational
velocity:
L = I
– Angular momentum may also be called rotational momentum.
– A bowling ball spinning slowly might have the same angular
momentum as a baseball spinning much more rapidly, because of
the larger rotational inertia I of the bowling ball.
Conservation of Angular Momentum
net = I = ∙
∆
∆
=
∆
∆
=
∆
∆
i.e. the direction of the angular
momentum change is the same as that
of the net toque.
When net =
∆
0,
∆
= 0, i.e. L = const.
Conservation of Angular Momentum
Inertia m : Fnet  ma
If Fnet  0,
p  mv
p  constant
 net  I
L  I
 0, L  constant
Inertia I :
If  net
Kinetic Energy
=
1
=
2
=
Conservation of Angular Momentum
1Q- 23 Conservation of angular momentum
Changing the moment of inertia of a skater
How does conservation
of angular momentum
manifest itself ?
I = 2mR2
=
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Physics
214
Physics 214
Fall
2009
Fall 2009
14
14
Kepler’s Second Law
• Kepler’s second law says that
the radius line from the sun
to the planet sweeps out
equal areas in equal times.
• The planet moves faster in its
elliptical orbit when it is
nearer to the sun than when
it is farther from the sun.
Kepler’s Second Law
• This is due to conservation of
angular momentum.
• The gravitational force acting
on the planet produces no
torque about an axis through
the sun because the lever
arm is zero: the force’s line of
action passes through the
sun.
Kepler’s Second Law
• When the planet moves
nearer to the sun, its
rotational inertia about the
sun decreases.
• To conserve angular
momentum, the rotational
velocity of the planet about
the sun must increase.
Angular momentum is a vector
• The direction of the rotational-velocity vector is given by
the right-hand rule.
• The direction of the angular-momentum vector is the same
as the rotational velocity.
Inertia I, rotational velocity 
Angular momentum : L  I
1Q-32 Stability Under Rotation
Example of Gyroscopic Stability: Swinging a spinning Record
Why does the Record not
“flop around” once it is
set spinning ?
L
L
The dominant physical law is conservation of angular momentum. With
no torque the vector L, perpendicular to the plane of rotation always
points in the same direction.
SINCE THERE IS NO TORQUE ABOUT THE CENTER OF ROTATION OF THE RECORD,
THE ANGULAR MOMENTUM VECTOR CANNOT CHANGE. THIS IS GYROSCOPIC
STABILITY. THIS IS A VERY SIMPLE GYROSCOPE AND SOPHISTICATED GYROSCOPES
ARE USED TO STEER AIRCRAFT AND ORIENT THE HUBBLE TELESCOPE
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2/25/2011
Physics
214
Physics 214
Fall
2009
Fall 2009
19
19
1Q-21 Conservation of angular momentum
Conservation of angular momentum using a spinning wheel
What happens
when the wheel
is inverted ?
The dominant physical law is conservation of angular momentum. To
change the angular momentum of the wheel requires an external torque.
So although we can change the direction of the angular momentum of
the wheel the force we use is internal to the wheel/stool system so the
the stool rotates to keep the net angular momentum the same
To turn the wheel requires significant force and work is needed. The energy of the
final system is greater than the initial energy by the amount of work that is done.
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20
Angular momentum and bicycles
• The wheels have angular
momentum when the
bicycle is moving.
• For straight line motion, the
direction of the angularmomentum vector is the
same for both wheels and is
horizontal.
• To tip the bike over, the
direction of the vector must
change, requiring a torque.
Angular momentum and bicycles
• If the bike is not perfectly upright, a gravitational torque
acts about the line of contact of the tires with the road.
• As the bike begins to fall, it acquires a rotational velocity
and angular
momentum about
this axis.
• If the bike tilts to the
left, the change in
angular momentum
points straight back.
Angular momentum and bicycles
• the change in angular momentum caused by the
gravitational torque adds to the angular momentum
already present from the rotating tires.
• This causes a
change in the
direction of the
total-angular
momentum vector
which can be
accommodated by
turning the wheel.
1Q-30 Bicycle Wheel Gyroscope
Gyroscopic action and precession
L
What happens
to the wheel,
does it fall
down?
F = mg
F
mg
The counterclockwise torque adds to L and
produces a precession, providing L is large
and the torque is small
The torque causes the vector L to precess and changes the direction of
the angular momentum vector which is perpendicular to the plane of
rotation. This is a very large top.
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24
A student sits on a stool holding a bicycle wheel with a rotational
velocity of 5 rad./s about a vertical axis. The rotational inertia of
the wheel is 2 kg·m2 about its center and the rotational inertia of
the student and wheel and platform about the rotational axis of the
platform is 6 kg·m2. What is the initial angular momentum of the
system?
a)
b)
c)
d)
10 kg·m2/s upward
25 kg·m2/s downward
25 kg·m2/s upward
50 kg·m2/s downward
L = I = (2 kg·m2)(5 rad./s)
= 10 kg·m2/s
upward from plane of wheel
Quiz: If the student flips the axis of the wheel, reversing the
direction of its angular-momentum vector, what is the rotational
velocity of the student and stool about their axis after the wheel is
flipped?
a)
b)
c)
d)
1.67 rad/s
3.33 rad/s
60 rad/s
120 rad/s
 = L / I = (20 kg·m2/s) / (6 kg·m2)
= 3.33 rad/s
in direction of original angular velocity