How Would a Reduced Earth Radius Affect Its Rotation and Orbit?

[xxxyyyy]

Regarding Earth's radius...

I am new to this forum and looking for the answer to the following question...

What will be the effect of reduction in Earth's radius by a factor of 2 on the following:
a) Earth's rotation speed about its own axis
b) Earth's rotation speed about the SUN
c) impact on orbit around the run?

regards.

 

[Integral]

What do you think?

 

[DaMastaofFisix]

So long as you know the principles that are being addresed here and their fundamental formulas, you can thinks these thorugh pretty quickly. It should be no sweat once you think about it hard enough.

 

[xxxyyyy]

This is what I think... probably someone can correct.

The rotational kinetic energy = I*w^2/2 where I = moment of inertia and w = angular velocity. I = M*R*R where M = mass (no change) and R = radius of earth.

Thus, Energy = M*R*R*w*w/2... If R1 and R2 are the new radii and w1 and w2 are the angular velocities, we should have...

R1*R1*w1*w1 = R2*R2*w2*w2, where R2 = R1/2. thus
4*w1^2 = w2^2,
Or w2 = w1 * 2... thus the angular speed of rotation will increase.. thus the days will become shorter in the sense the Earth will able to complete one revolution about the self axis in a shorter time.

How about the nature of orbit around the sun? Any clues here?

 

[DaMastaofFisix]

Umm, not too shabby. Kind of a strange mathematicl approach, given that the monemt of interia is PROPRTIONAL to MR^2, but not necessarily equal to it. In the case of a uniform sphere (as you should assume the Earth to be), the MOI is (2/5)MR^2. But when it was all said and done, your CONCLUSION was correct, not neccesarily your math. First question, the principle that needed to be addressed was conservation of ANGULAR MOMENTUM, with an emphasis on the chane in moment of intertia due to the decrease in both the mass and the radius of the earth.

As for the second question, still angular momentum, but now you can address it with keppler's laws of planetary motion and the law of periods. Also might want to dig into unifrom circular motion.

As for the third question...same stuff applies...Keppler's laws. GO for it!

 

[xxxyyyy]

Since only the Earth's radius has changed and not the mass, only the density has become higher. I feel that there should not be any impact on the rotation around the sun.

but as shown above, Earth will probably spin faster on its axis and hence it will be completing one rotation in less than 24 hrs as compared to now...

any comments?