Two disks of identical mass but different radii (r and 2r) are spinning on frictionless bearings at the same angular speed ?0, but in opposite directions. The two disks are brought slowly together. The resulting frictional force between the surfaces eventually brings them to a common angular velocity. a) what is there final angular speed?b) what is the change in the rotational kinetic energy?

Explanation:

Moment of inertia of a disc = 1/2 M R²

Since mass is same for both and radius are r and 2r, their moment of inertia can be in the ratio of 1: 4 . Let them be I and 4I . Angular speed are ω₀ and   - ω₀ .

We shall apply law of conservation of angular momentum .

initial total angular momentum

I x ω₀ - 4I x ω₀ = - 3Iω₀

Let final common angular momentum be ω

total final angular momentum = ( I + 4I ) ω

Applying law of conservation of angular momentum

( I + 4I ) ω =  - 3Iω₀

ω = - 3 / 5 ω₀ .

b )

Initial total rotational K E

= 1/2 I ω₀² + 1/2 4I ω₀²

= 1/2 x5I ω₀²

Final total rotational K E

= 1/2 ( I + 4I ) ( - 3 / 5 ω₀ )²

= 1/2 x 9 / 5 I ω₀²

= 9 / 10I ω₀²

change in rotational kinetic energy = 9 / 10I ω₀² - 1/2 x5I ω₀²

(9/10 - 5/2) xI ω₀²

=( .9 - 2.5 )I ω₀²

= - 1.6 I ω₀² Ans

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