Question

in the uniform circular motion of an object , if radius decreases by36%, the angular speed is?

Answer 1

Applying Law of conservation of Angular momentum ,
initial Angular momentum = Final angular momentum
Iw = I'w'
where I is moment of inertia (initial ) , I' is the final moment of inertia , w is initial angular velocity and w' is final angular velocity .
and ,
I = mr²
where m is mass of object and r is initial radius .
I'= mr'²
where r' is the final radius .
and there is 36% decrease in initial radius(r) then ;
r' = r - (r×0.36 )
=> r' = 0.64r
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Now combining all the stuff we get ;
mr²w = mr'²w'
=> r²w = r'²w'
using r'= 0.64r . we get ;
r²w = (0.64r)²w'
=> w' = w/(0.64)²
=> w' = 2.44 w
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% increase in angular velocity =
{(2.44w - w )/w } ×100
= 144%
HENCE;
IF WE DECREASE RADIUS BY 36% THEN THE ANGULAR VELOCITY GETS INCREASED BY 144%
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Hope it helps!