Given, initial volume of earth,
Vi=V Final volume of earth,
Vf=V∕8 and
Ri,Rf are initial and final radii of earth.
From Kepler's third law of planetary motion,
T2∝R3. . . (i)
where,
T is time-period.
Since, volume,
V=πR3 ∴V∝R3 ⇒== ⇒8=()3⇒=2 By using law of conservation of angular momentum,
Li=Lf where,
Li,Lf are initial and final angular momentum, respectively.
Iiωi=Ifωf where,
Ii,If and
ωi,ωf are initial and final moment of inertia and angular speed, respectively.
⇒Ii=If ⇒= Tf=Ti() Moment of inertia of earth
I(earth) =MR2 ∴Tf=Ti()=Ti()2 =24()2 =24×=6h