When the earth shrinks without reducing its mass What change will be there in the duration of a day?

A man stands on a rotating platform, with his arms stretched horizontally holding a 5 kg weight in each hand. The angular speed of the platform is 30 revolutions per minute. The man then brings his arms close to his body with the distance of each weight from the axis changing from 90cm to 20cm. The moment of inertia of the man together with the platform may be taken to be constant and equal to 7.6 kg m2. 

(a) What is his new angular speed? (Neglect friction.) 

(b) Is kinetic energy conserved in the process? If not, from where does the change come about?

(a) Moment of inertia of the man-platform system = 7.6 kg m2

Moment of inertia when the man stretches his hands to a distance of 90 cm,

2 × m r2 = 2 × 5 × (0.9)2 

            = 8.1 kg m2 

Initial moment of inertia of the system, Ii2 

Angular speed, ωi = 300 rev/min 

Angular momentum, Li = Iiωi  =  15.7 × 30                         ...(i) 

Moment of inertia when the man folds his hands to a distance of 20 cm

2 × mr2 = 2 × 5 (0.2)22 

Final moment of inertia, If = 7.6 + 0.4 = 8 kg m2  

Final angular speed = ωf 

Final angular momentum, Lf = Ifωf = 0.79 ωf                   .... (ii) 

From the conservation of angular momentum, we have

                       Iiωi  =  Ifωf 

ωf =

Kinetic energy is not conserved in the given process. But, the kinetic energy increases as there is a decrease in the moment of inertia. 

The additional kinetic energy comes from the work done by the man to fold his hands toward himself.

Text Solution

If both assertion and reason are true and reason is the correct explanation of assertion.If both assertion and reason are reason are true but reason is not the correct explanation of assertion.If assertion is true but reason is falseIf both assertion and reason are false.

Answer : A

Solution : (a) When earth shrinks its angular momentum remains constant i.e., <br> `L = I omega = (2)/(5) mR^2 xx (2 pi)/(T) = constant` <br> `T prop I prop R^2` it means if size of earth changes then its moment of inertia changes <br> If radius is half so time period will become `T//4`/ hence `24//4 = 6 Hr`.