What will be the duration of the day if earth suddenly contracts to half of its present radius?

If the radius of earth contracts to half of its value today, (the mass remaining unchanged), then the duration of the day will be 6 hrs.

Explanation:

`"L"_1 = "I"_1  omega_1 = 2/5 "MR"^2  omega_1`      .....(i)

Now its radius becomes half without change in its mass

`"L"_2 = "I"_2  omega_2 = 2/5 "M" ("R"/2)^2 omega_2`     ....(ii)

Let angular momentum be conserved,

L1 =L2

`"Hence,"  2/5 "MR"^2 omega_1`

` = 1/4 xx 2/5 "MR"^2 omega_2`

`"i.e."  omega_2 = 4 omega_1`

`"T"_2/"T"_1 = omega_1/omega_2 = 1/4`

`"T"_2 = "T"_1/4 = 24/4`

` = 6  "hours"`