question 1 problem sheet
Post on 03-Apr-2015
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Dr. Suhas P Veetil
Q1) A cylinder with moment of inertia I1 rotates about a vertical, frictionless axle with angular
velocity ωi. A second cylinder; this one having a moment of inertia of I2and initially not rotating,
drops onto the first cylinder. Because of friction between the surfaces, the two eventually reach
the same angular speed ωf.
(a) Calculate ωf.
(b) Show that the kinetic energy of the system decreases in this interaction by calculating
the ratio of the final to initial rotational energy. Express your answer in terms of ωi.
ANSWER:
Given that the moment of inertia of each wheel is I1 andI2
initial angular velocity of first wheel is ωi
initial angular velocity of second wheel is 0
final combined angular velocity of two wheels is ωf
Since there is no external force on the system then the angular momentum is conserved.
( )
( )
1 1 1
1
1 1
i f
if
I I I
I
I I
ω ω
ωω
= +
=+
( )
( ) ( ) ( )( )
( )
( ) ( )
2
1
2
2 11 2 1 2
1 1
1
1 2
1Initial Kinetic energy,
2
1 1Final Kinetic energy,
2 2
iinitial
iffinal
final
initial
KE I
IKE I I I I
I I
KE I
KE I I
ω
ωω
=
= + = +
+
=+
Total initial
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