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