If the earth suddenly contract what will be the effect on the duration of the day give your answer with proper reasoning 1
Answers
We can consider two scenarios: one with a constant mass and the other with change in mass proportional to the change in volume.
In both cases, the same principle applies: the conservation of angular momentum. Angular momentum is
L=I×ωL=I×ω
where II is the inertia moment and ωω is the angular velocity. The angular momentum LL is conserved, meaning it must stay the same value, so if the inertia II decreases by some factor, the speed ωω has to increase by the same factor.
For a sphere (let’s assume the Earth is a perfect sphere): I=25mR2I=25mR2 (mm is the mass and RR is the radius)
So let’s find out how II changes in both scenarios.
Scenario 1: constant mass
If the radius is halved, that means the inertia moment is reduced by a factor 4 (because of the R2R2). So the angular speed ωω must be increased by a factor 4, which means a day would be 6 hours instead of 24.
Scenario 2: changing mass
if the mass changes proportionnally to the volume, it means the mass is reduced by a factor 8 (because the volume of a sphere is proportional to R3R3). So the inertia moment is reduced by a factor 4 because of the radius, and by a factor 8 because of the mass. So the total reduction is by a factor 32! which means the angular speed is increased by a factor 32, so a full day is only 45 minutes long !