System of Particles and Rotational Motion Test - 13

I = 2/5 MR²  = 8000 gm cm² , using parallel axis theorem the moment of  Inertia about tangent to sphere is  =  2/5 MR²+ MR²  = 7/5 MR².  =2.8 × 10⁴ gm cm² 

As  the Moment of inertia of earth considered as sphere is I = 2/5 MR² , thus according to law of conservation of angular momentum  as the radius contracts to half , thus  new moment of inertia of earth will be I/4 , thus the angular velocity will increase  4 times  and making the length of the day to 6 hrs. 

According to law of conservation of angular momentum if no external torque is applied on a body in rotation than  its angular momentum remains conserved.

Using theorem of parallel axis, the axis is shifted by L/2 distance from centre of Mass where the Moment of Inertia is ML²/12, thus the moment of inertia of a thin rod about an axis passing through one end and perpendicular to its length is  ML²/12 + ML²/4 =1/3 ML²

Moment of inetia of axis passing through its center and perpendicular to its plane:

 I = ½ MR²

Using perpendicular axis theorem   Ix +Iy = Iz  ,so  2I_d = ½ MR²

Moment of inetia of  along its diameter  I_d = ¼MR²

^{And the radius of gyration}    MK²  = ¼MR²   

^{K = R/2}