angular momentum of a rigid object rotating about a fixed axis but for any rigid object the...
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Angular Momentum of a rigid object rotating about a fixed axis
IL
dtId
dtdL
But for any rigid object the rotational inertia is a constant
dtd
IdtdL
IdtdL
dtdLNewton’s
Second LawAnalogous to
dtpd
Fnet
What if the system is isolated and closed?
Isolated – no external torques Closed – no change in the mass
dtdL
dtdL0
constant L
Law of Conservation of Angular Momentum
In any closed, isolated system, the angular momentum is constant
Conservation of Angular Momentum Examples
1. The spinning volunteer.
fi LL
ffii II
iI i fI f
Conservation of Angular Momentum Examples
together. couple thenThey first. the as direction
same the in at spinning set is , radius and mass with
disk, second The . at spinning set is , radius and mass
withdisk, first The unit. one as rotate and couplethey that so together brought
be can and axle same the on bearings frictionlow on mounted are disks Two 3.
min.90050004
min45050002
rev m. kg.
rev m. kg.
a. What is their angular speed after coupling?
fi LL
fii Lll 21
fii IIII 212211
fii RmmRRmmR
22
22
12 2
21
21
221
21
fii 32 21
Conservation of Angular Momentum Examples
together. couple thenThey first. the as direction
same the in at spinning set is , radius and mass with
disk, second The . at spinning set is , radius and mass
withdisk, first The unit. one as rotate and couplethey that so together brought
be can and axle same the on bearings frictionlow on mounted are disks Two 3.
min.90050004
min45050002
rev m. kg.
rev m. kg.
a. What is their angular speed after coupling?
fii 32 21
32 21 ii
f
3
min.9002min450 revrevf
min750revf
Conservation of Angular Momentum Examples
together. couple thenThey first. the as direction
same the in at spinning set is , radius and mass with
disk, second The . at spinning set is , radius and mass
withdisk, first The unit. one as rotate and couplethey that so together brought
be can and axle same the on bearings frictionlow on mounted are disks Two 3.
min.90050004
min45050002
rev m. kg.
rev m. kg.
coupling? after speed angular their is whatrotation, sdisk' first the of
direction opposite the in at spinning set is disk second the instead If b. rev. min900
3
min.9002min450 revrevf
min450revf
Conservation of Angular Momentum Examples
2. Two children, each with mass M, sit on opposite ends of a narrow board with length L and mass M. The board is pivoted at its center and is free to rotate in a horizontal circle without friction. (Treat the board as a thin rod.)
M M
l
childboardtotal III 22
2
22
121
l
MMlItotal
22
21
121
MlMlItotal
2
127
MlItotal
2l
a. What is the rotational inertia of the board plus the children about a vertical axis through the center of the board?
Conservation of Angular Momentum Examples
2. Two children, each with mass M, sit on opposite ends of a narrow board with length L and mass M. The board is pivoted at its center and is free to rotate in a horizontal circle without friction. (Treat the board as a thin rod.)
b. What is the magnitude and direction of the angular momentum of the system if it is rotating with angular speed ωo in a clockwise direction as seen from above?
l
IL
oMlL 2127 Downward
M M
Conservation of Angular Momentum Examples
While the system is rotating, the children pull themselves toward the center of the board until they are half as far from the center as before.
c. What is the ratio of the new rotational inertia to the initial rotational inertia?
M M
l
4l
2
127
MlIi 2
2
42
121
l
MMlI f
2
245
MlI f
2
2
127245
Ml
Ml
I
I
i
f
145
i
f
I
I
Conservation of Angular Momentum Examples
While the system is rotating, the children pull themselves toward the center of the board until they are half as far from the center as before.
d. What is the resulting angular speed in terms of ωo?
M M
l
4l
fi LL
ffoi II
of
if I
I
of 514
Conservation of Angular Momentum Examples
While the system is rotating, the children pull themselves toward the center of the board until they are half as far from the center as before.
e. What is the change in kinetic energy of the system as a result of the children changing their position? (From where does the added kinetic energy come?)
M M
l
4l
kikfk EEE
22
21
21
iiffk IIE
222
2
127
21
514
245
21
ook MlMlE
22
4021
ok MlE The added energy comes from the work done by the
children when pulling themselves forward.
*Note: L = constant
Ek = increases