Suddenly the radius of the world became half of the initial, what is the change in the length of the day?

Question

Suddenly the radius of the world became half of the initial, what is the change in the length of the day?

Answer 1

If you're assuming the mass remains the same, then for momentum to be conserved, w1*I1 = w2*I2 or w2 = w1*I1/I2.

Since I varies as the radius squared, w2 = w1*(1/½)² = 4*w1.

Ld2 = (1/4)Ld1 = 1/4(24 hr) = 6 hr

Delta Ld = 6 - 24 = -18 hr.

Answer 2

The radius does not determine the length of the say.  How fast the earth rotates does.  So if rotation speed stayed the same, the length of the day would stay the same.

Answer 3

Dorians answer is excellent and I am just summarizing.

The speed of rotation "varies inversely" with the square of the radius.

So 1/2 radius = {1/(1/2)}^2  = 2^2 = 4 times greater speed of rotation.

double the radius 2 x radius = {1/2}^2 = 1/4 the speed of rotation.

it can be really helpful to understand the idea of an inverse relationship.

There's a good video lecture on this idea from MIT's opencourseware at the link below. See the link to lecture 1, the section on "dimensional analysis" It's not too hard!...

Answer 4

nothing. the earth will still be going the same rotational speed.

i think.

Answer 5

This is a (not so easy) question from a conservation of angular momentum.

For an object with a fixed mass that is rotating about a fixed symmetry axis, the magnitude of angular momentum L is expressed as the product of the moment of inertia I of the object and its angular frequency ω:

L = Iω.

If we treat Earth (the world) as a solid sphere of radius r and mass m, then its moment of inertia would be I = 2m x sqr(r) / 5, while its angular frequency can be calculated from the equation ω = 2π / T, where T is the period of the Earth = 24 h.

If the Earth would shrink to half its diameter, then its moment of inertia would also change because of the new radius. If we denote a new radius with r', then r' = r / 2 and a new moment of inertia I' = 2m x sqr(r') / 5 = 2m x sqr(r/2) / 5 = 2m x sqr(r) / 20 = m x sqr(r) / 10, which means that the moment of inertia would be 4-times smaller.

Angular momentum is an important quantity in this question because it is conserved. This means that a new angular momentum would be the same as before the Earth has shrunk. If the moment of inertia is 4-times smaller this can only mean that, for the product L = Iω to remain the same, the angular frequency of the Earth would have to increase by factor 4. The new angular frequency would then be ω' = 4ω = 2π / T' or T' = 2π / ω' = 4ω = 4 x 2π / T. If we solve for T', this means that T would be T / 4 or 24 h / 4 = 6 h. The length of the day would decrease 4-times, or rather, there would be 3 hours of day and 3 hours of night.

By the way, it is fairly impossible for a planet like Earth to be faced with an event like this is, but it actually happens with massive stars that collapse and become neutron stars and similar objects in a universe. Because their change of radius is much more bigger than it's just been discussed, their period of rotation becomes much smaller; these stars rotate very rapidly and make one rotation in just one second or even a few miliseconds.

Answer 6

it would be longer though not sure how much.

If I can dumb it down the ball is smaller, but it's still a ball.

Maybe gets more sun as the ratios have changed.

Answer 7

12 hours, i think?

Answer 8

Not enough information. Since you are assuming something that violates numerous conservation laws of physics, you must specify which few remaining ones you wish to be upheld in order to answer your question. Energy? Angular momentum? Baryon number?